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Supply quality management with wholesale price and revenue-sharing contracts under horizontal competition
European Journal of Operational Research 206 (2010) 329–340
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Production, Manufacturing and Logistics
Supply quality management with wholesale price and revenue-sharing contracts under horizontal competition Fouad El Ouardighi a,*, Bowon Kim b a b
ESSEC Business School, Avenue Bernard Hirsch, B.P. 105, 95021 Cergy Pontoise, France KAIST Graduate School of Management, 207-43 Cheongryangri Dongdaemoon-Ku, Seoul 130-722, South Korea
a r t i c l e
i n f o
Article history: Received 10 January 2009 Accepted 26 February 2010 Available online 3 March 2010 Keywords: Design quality Supply quality management Horizontal competition Wholesale price contract Revenue sharing contract
a b s t r a c t In a number of industries (e.g., the airplane industry, aerospace industry, auto industry, or computer industry), certain suppliers essentially have a monopoly on the production technology for key components, and inevitably manufacturers in these industries have common suppliers. A key part of manufacturers’ work with suppliers concerns improving the quality of their respective products, which gives rise to a collaborative activity usually termed as ‘‘supply quality management”. When the manufacturers are competitors, they do not wish to see a common supplier dividing his involvement in quality improvement unequally between themselves and their rivals. However, as the suppliers collaborate with several manufacturers, it is highly questionable whether their efforts will be strictly equivalent for each manufacturer. In this paper, a non-cooperative dynamic game is formulated in which a single supplier collaborates with two manufacturers on design quality improvements for their respective products. The manufacturers compete for market demand both on price and design quality. The paper analyzes how each party should allocate resources for quality improvement over time. In order to take into account the potential coordinating power of the compensation scheme adopted in this type of decentralized setting, we compare the possible outcomes under a wholesale price contract and a revenue-sharing contract. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction In a number of industries (e.g., the airplane industry, aerospace industry, auto industry, or computer industry), certain suppliers essentially have a monopoly on the production technology of key components, and inevitably manufacturers in these industries have common suppliers. In the plane making business, for instance, the two major rival players Boeing and Airbus share up to twelve strategic suppliers (Michaels and Lunsford, 2004) so that their supply chains are both competing and interdependent. A key part of manufacturers’ work with suppliers concerns improving the quality of their respective products, which gives rise to a collaborative activity usually termed as ‘‘supply quality management”. When the manufacturers are competitors, they do not wish to see a common supplier dividing his involvement in quality improvement unequally between themselves and their rivals. However, as the suppliers cooperate with several manufacturers, it is highly questionable whether their efforts will be strictly equivalent for each manufacturer. Given the empirical evidence on the decisive importance of quality management for supply chain success (Morash, 2001; Smets, 2004), it is interesting to study how vertical coordination affects the players’ optimal allocation of efforts for quality improvement in the context of horizontal competition at the manufacturing level. It has been shown that coordinated supply chains perform better in quality improvement than uncoordinated supply chains. For instance, Kim and El Ouardighi (2007) consider the problem of optimal effort allocation between design quality improvement for an existing product and development of a new product, in a one manufacturer-one supplier supply chain. Using qualitative and numerical methods, they find that the uncoordinated chain pays more attention to new product development, while the coordinated chain focuses more on the quality of the existing product. Given that the degree of coordination in a supply chain can affect the chain members’ quality improvement effort, it is difficult to consider the issue of supply quality management in competing and interdependent supply chains without taking into account the impact of coordination, if any, in a decentralized setting.
* Corresponding author. Tel.: + 33 1 34 43 33 20; fax: + 33 1 34 43 30 01. E-mail address: [email protected] (F. El Ouardighi). 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.02.035
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Supply chain contracts are generally considered as a useful tool to bring supply chain actors in a decentralized setting to operate in coordination. Different models of supply chain contracts have been developed in the literature, and one model considered ‘very attractive’ is the revenue-sharing contract (RSC) (Cachon and Lariviere, 2005). Under a RSC, each manufacturer pays the supplier a fixed transfer price for each unit purchased, plus a fixed percentage of her own sales revenue. The RSC is viewed as a valuable alternative to the wholesale price contract (WPC) (Cachon and Lariviere, 2001), in which the supplier merely charges the retailer a fixed per-unit price. Although the WPC is commonly observed in practice because it is simple to implement and cheap to administer, it can engender a double marginalization effect. This effect arises when a supplier sells a product to a retailer who is facing a downward sloping demand curve (e.g. Bresnahan and Reiss, 1985): the retailer is not concerned with the supplier’s profit when fixing the retail price, and as a result the retail price is higher than in a centralized chain. The advantage of the RSC over a WPC is that it mitigates this double marginalization effect (e.g., Chen et al., 2001) and can significantly increase the supply chain profits, for example by more than 10% in the video rental industry (Mortimer, 2008). To date, comparisons between RSCs and WPCs in supply quality management have been limited to one important but specific dimension of quality, namely conformance quality, which refers to the extent to which a product conforms to a given design quality standard (Garvin, 1988). In the setup of a one supplier/one retailer supply chain in which improved conformance quality can enhance current customer loyalty, El Ouardighi et al. (2008) show that chain members should invest more in quality under a RSC than under a WPC over time. This paper extends knowledge of the situation in which a supplier is serving two price-competing manufacturers. To do so, we use another essential and complementary dimension of quality, that is, design quality, which refers to the set of product attributes or features that enhance the match with the customer’s needs (Garvin, 1988). In a setup consisting of interdependent, competing chains we investigate how the compensation scheme adopted (WPC or RSC) can affect design quality competition at the manufacturing level via the way the players allocate their quality improvement efforts. In doing so, we analyze how any mitigation of the double marginalization effect through a RSC affects supply quality management in the context of horizontal competition at the manufacturing level. The paper is organized as follows. Section 2 presents a dynamic game model where a single supplier collaborates with two competing manufacturers in improving the quality of their respective products. Section 3 analyzes the three players’ differential game in a decentralized setting and compares the outcomes obtained under WPC and RSC. To gain additional insights into the optimal dynamic behavior of the players, a numerical study is conducted in Section 4. Section 5 draws the key managerial implications from the analytical and numerical analyses. 2. A stylized non-cooperative differential game model We consider a situation with three players: two duopolistic manufacturers and a monopolist supplier. This configuration belongs to the category of interdependent, competing supply chains (El Ouardighi et al., 2009), that is, supply chains which share common members and compete for market demand. Both manufacturers purchase a similar part from their supplier, which is used in their respective finished products. The manufacturers’ products are in competition for the final demand market. Each manufacturer invests in quality improvement for her product, and the monopolist supplier is supposed to collaborate in each manufacturer’s quality improvement activity. Quality here means design quality, i.e., the set of product attributes or features that enhance the match with the customer’s needs (Garvin, 1988). In this sense, quality improvement activity aims to increase the desirability of the product, notably through the implementation and use of dedicated tools such as quality function deployment (QFD), Taguchi arrays, Ishikawa diagrams, etc. Although the adoption of such tools is generally costly in terms of instantaneous effort, it does not necessarily incur an extra cost per unit produced. Let Q i ðtÞ > 0 represent the design quality level at time t for manufacturer i’s product, for which quality improvement is given by:
Q_ 1 ðtÞ ¼ u1 ðtÞ þ v 1 ðtÞ Q 1 ð0Þ ¼ Q 10 > 0; Q_ 2 ðtÞ ¼ u2 ðtÞ þ v 2 ðtÞ Q 2 ð0Þ ¼ Q 20 > 0;
ð1Þ ð2Þ
where ui ðtÞ P 0 and v i ðtÞ P 0 denote the respective effort input by manufacturer i and the monopolist supplier to manufacturer i’s product quality, i ¼ 1; 2. Note that a similar representation of the evolution of design quality is used by Mukhopadhyay and Kouvelis (1997) in the context of a dynamic game model of duopolistic competition. We assume that manufacturer i’s final demand, Di ðtÞ; i ¼ 1; 2, is determined both by price and quality competition on the duopoly market, as follows:
Di ðtÞ ¼ a bðpi ðtÞ p3i ðtÞÞ þ dðQ i ðtÞ Q 3i ðtÞÞ;
ð3Þ
where a 0, and b and d are symmetric positive constants for the sake of simplicity. In Eq. (3), manufacturer i’s demand is: - A decreasing function of the differential between her own price, pi ðtÞ P 0, and the rival firm’s price, p3i ðtÞ P 0, and - An increasing function of the differential between her own quality, Q i ðtÞ, and the rival firm’s quality, Q 3i ðtÞ; i ¼ 1; 2. In the case where both manufacturers have constant design quality ðQ_ i ¼ 0; 8 iÞ, Eq. (3) would reduce to a standard Bertrand demand function where manufacturer i’s demand shows a linear decrease in own price and a linear increase in the rival manufacturer’s price. Manufacturer i’s demand equation thus provides an extension to the situation in which both manufacturers also compete on design quality. We assume that each manufacturer enters into either a WPC or a RSC with the supplier. A two-parameter scheme is therefore used, where: The first parameter is the transfer price (supposed constant), ci > 0, paid by manufacturer i to the supplier for each unit purchased, and The second parameter is the supplier’s share (supposed constant) in manufacturer i’s sales revenue, xi ; xi 2 ½0; 1½; i ¼ 1; 2. The parameter values are such that ci > 0 and xi ¼ 0 for a WPC, while ci > 0 and xi 20; 1½ for a RSC. The assumption of fixed contract parameters lies with the fact that, in the real world, when companies sign a contract, it is usual for them to specify the terms of contract, as
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they want to design their contract in a predictable manner as much as possible. Predetermining the transfer price and the revenue-sharing factor (if any) as fixed parameters is part of such effort. On the other hand, note that the adoption of a RSC presupposes that two non-trivial issues are solved, i.e., how the supplier can monitor manufacturer i’s sales, and how the value of the supplier’s share in each manufacturer’s revenue is determined. In the absence of any empirical evidence that design quality improvement positively affects the supplier’s production cost, transfer price and product quality are presumed to be independent for each manufacturer (e.g., Kim and El Ouardighi, 2007). Also, in order to keep the model analytically tractable, the manufacturers’ operating costs are supposed to be both equal to zero. Finally, the assumption that the duopolist manufacturers will potentially have different transfer prices derives from the possible substitutability between the transfer price and the supplier’s share in each manufacturer’s revenue, if any. i 20; 1, manufacturer i’s instantaneous profit can thus be expressed as i 1 xi ; x Denoting the manufacturer i’s revenue share by x follows:
i pi ðtÞ ci Di ðtÞ ½x
bi ui ðtÞ2 ; 2
ð4Þ
where bi ui ðtÞ2 =2 denotes manufacturer i’s quadratic cost function of own improvement effort, bi being a positive constant, i ¼ 1; 2 (e.g., El Ouardighi and Pasin, 2006). Note that the results would remain qualitatively the same if a general cost function fi ðui ðtÞÞ were adopted, with @fi =@ui > 0; @ 2 fi =@u2i > 0; f i ð0Þ ¼ 0. We consider a finite planning horizon and assume that the time value of money is negligible. Manufacturer i’s objective criterion can thus be expressed as follows:
Z
T
(
) bi ui ðtÞ2 i pi ðtÞ ci ½a bðpi ðtÞ p3i ðtÞÞ þ dðQ i ðtÞ Q 3i ðtÞÞ dt þ hi ðQ i ðTÞ Q 3i ðTÞÞ; ½x 2
0
ð5Þ
where hi ðQ i ðTÞ Q 3i ðTÞÞ is the salvage value for the differential between own quality and the rival manufacturer’s quality, i ¼ 1; 2. Manufacturer i’s salvage value is interpreted as a goodwill reward (resp., goodwill penalty) for a larger (resp., lower) own terminal quality level than the rival’s. On the other hand, the monopolist supplier’s objective criterion is:
Z
T
( 2 X
0
i¼1
) 2 2 X X di v i ðtÞ2 dt þ ½xi pi ðtÞ þ ci ½a bðpi ðtÞ p3i ðtÞÞ þ dðQ i ðtÞ Q 3i ðtÞÞ gi Q i ðTÞ; 2 i¼1 i¼1
ð6Þ
where di v i ðtÞ2 =2 is the supplier‘s quadratic cost function of own improvement effort, di being a positive constant, and gi Q i ðTÞ > 0 is the salvage value (i.e., goodwill reward) for manufacturer i’s quality, i ¼ 1; 2. For simplicity, the rest of the paper assumes symmetrical value for the supplier’s improvement effort cost coefficients (i.e., d1 ¼ d2 ¼ dÞ. 3. Supply chain contract and equilibrium strategies We focus on the decentralized setting of the game in which each player aims to maximize her own objective function. The game defined by (5) and (6) under constraints (1) and (2) involves more than one state variable, which precludes the search for closed-loop equilibrium, although it is more attractive from a strategic viewpoint. We confine our interest to the search for an open-loop Nash equilibrium, which implies that all the players precommit to their control actions throughout the game (Dockner et al., 2000). As it fulfils the time-consistency requirement, analysis of open-loop Nash equilibrium can serve as a benchmark for the assessment of more complex strategies. First, we characterize manufacturer i’s optimal control problem of determining his strategies pi ðtÞ (sales price) and ui ðtÞ (improvement effort). Skipping the time index for convenience, manufacturer i’s Hamiltonian is:
i pi ci ½a bðpi p3i Þ þ dðQ i Q 3i Þ H i ¼ ½x
bi u2i þ li1 ðui þ v i Þ þ li2 ðu3i þ v 3i Þ; 2
ð7Þ
where lij lij ðtÞ; i ¼ 1; 2; j ¼ 1; 2, are manufacturer i’s costate variables. Necessary conditions for optimality are:
i ½a bðpi p3i Þ þ dðQ i Q 3i Þ bðx i pi ci Þ ¼ 0; Hipi ¼ x
ð8Þ
Hiui
ð9Þ
i 1
¼ bi ui þ l ¼ 0:
According to (8) and (9), manufacturer i’s Hamiltonian is strictly concave with respect to manufacturer i’s control variables. From (8), it can be shown that @p@pi ¼ 1=2 > 0, that is, manufacturer i should increase her price in proportion to an increase in the rival’s price. In other words, 3i there should be a positive (horizontal) interdependency between the manufacturers’ sales prices. Lemma 1. Manufacturer i’s optimal non-cooperative price, i ¼ 1; 2, satisfies:
pi ¼
a 1 2ci c3i d þ þ ðQ Q 3i Þ : þ i x 3i b i b 3 x
ð10Þ
Proof. See the Appendix. h This rule positively relates manufacturer i’s price to the differential between own quality and the rival’s quality, i.e., a manufacturer with a larger (resp., lower) quality level than her rival should set a higher (resp., lower) sale price. Manufacturer i’s pricing policy is implicitly
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dynamical, i.e., the rate of change of manufacturer i’s sales price should vary proportionally with the relative instantaneous efforts to manufacturer i’s product quality, as:
p_ i ¼
d ðui þ v i u3i v 3i Þ: 3b
Finally, for similar quality levels, manufacturer i’s sales price is positively influenced not only by her own contract parameter values, but also to a certain extent by the rival’s. Lemma 2. Assuming symmetric manufacturers, it holds that:
pij0
cijx
cij0
i ¼0
ð11Þ ; i ¼ 1; 2.
Proof. See the Appendix. h According to (11), the optimal sales price should be lower under a RSC than under a WPC whenever the supplier’s share in manufacturer i’s sales revenue is not larger than the rate of decrease of the transfer price from a WPC to a RSC. This result is in line with the existing literature (e.g., Kogan and Tapiero, 2007), since it shows that the RSC effectively mitigates the double marginalization effect. Plugging (10) into (3) gives manufacturer i’s optimal demand:
Di ¼ a
1 ci c3i dðQ i Q 3i Þ ; b i x 3i 3 x
ð12Þ
i ¼ 1; 2, that is expected to be positive. If the manufacturers’ contract parameters are similar, a manufacturer with a larger (resp., lower) quality level than her rival should benefit from larger (resp., lower) sales. On the other hand, for similar quality levels, manufacturer i’s demand is negatively (resp., positively) influenced by her own (resp., the rival’s) contract parameter values. Note that the rate of change of manufacturer i’s demand should also evolve as a linear function of the relative instantaneous efforts to manufacturer i’s product quality, that is:
d _ ¼ ðui þ v i u3i v 3i Þ i ¼ 1; 2: D_ i ðtÞ ¼ bpðtÞ 3 Lemma 3. The dynamics of the non-cooperative manufacturer i’s improvement policy, i ¼ 1; 2, is:
u_ i ¼
i i 2dx 1 c c 2dx a b i 3i dðQ i Q 3i Þ ¼ Di : 3 3bbi xi x3i 3bbi
ð13Þ
Proof. See the Appendix. h Whatever the nature of the contract between manufacturer i and the supplier (either a WPC or a RSC), the non-cooperative manufacturer i’s improvement effort should be initially high and decreasing over time until ui ðTÞ ¼ hi =bi is reached, i ¼ 1; 2. If one manufacturer has a larger value on both contract parameters than the rival, or a larger value than the rival on one parameter and a similar value on the other parameter, her improvement effort should start from a lower initial value than the rival. In other words, the larger the compensation offered by a manufacturer to the supplier, the lower the manufacturer’s improvement effort. Comparing the specific case where one manufacturer has a WPC and the other a RSC (e.g., c1 > c2 and 0 ¼ x1 < x2 ), all other things being equal (i.e., b1 ¼ b2 \ Q 10 ¼ Q 20 \ h1 ¼ h2 ), it can be shown that the improvement effort decline rate for the manufacturer with a WPC should be larger than that for the manufacturer with a RSC (i.e., u_ 1 < u_ 2 ), that is, a manufacturer using a WPC should make a greater quality improvement effort than a manufacturer using a RSC. We now turn to the determination of optimal improvement effort strategies v 1 ðtÞ and v 2 ðtÞ for the supplier. Substituting (10) for optimal sales prices into (6), the supplier’s Hamiltonian is:
Hs ¼
2 2 2 X v 2i X 1 X ð3 xi Þci c3i ci c3i xi 3a þ b þ wi ðui þ v i Þ; þ þ dðQ i Q 3i Þ 3a b þ dðQ i Q 3i Þ d i 3i i x 3i 9b i¼1 xi x x x 2 i¼1 i¼1
ð14Þ
where wi wi ðtÞ; i ¼ 1; 2, are the supplier’s costate variables. The supplier’s necessary conditions for optimality are:
Hsv i ¼ dv i þ wi ¼ 0 i ¼ 1; 2:
ð15Þ
The supplier’s Hamiltonian is strictly concave with respect to the supplier’s control variables. Lemma 4. The dynamics of the non-cooperative supplier’s contribution policy to manufacturer i’s quality improvement, i ¼ 1; 2, is:
v_ i ¼
( ! ) X d 2ðxi x3i Þ ci c3i ½3a þ dðQ i Q 3i Þ þ 3 2 : xi 9d b xi x3i i
ð16Þ
Proof. See the Appendix. h At the end of the planning horizon, the supplier’s contribution effort to manufacturer i’s quality improvement should equal i ¼ 1; 2. The supplier’s contribution policy depends on the value of the contract parameters with the manufacturers (Table 1). If the supplier has similar contract parameter values with both manufacturers, i.e., both c1 ¼ c2 and x1 ¼ x2 , her contribution to both manufacturers’ quality improvement activity should be constant over the entire planning horizon, i.e., v i ðtÞ ¼ gi =d; 8t 2 ½0; T; i ¼ 1; 2. This
v i ðTÞ ¼ gi =d;
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c1 < c2 c1 ¼ c2 c1 > c2
x1 < x2
x1 ¼ x2
x1 > x2
v_ 1 P 0; v_ 2 6 0j06xi 63=4 v_ 1 P 0; v_ 2 6 0j06xi 63=4 v_ 1jx1 ¼0 > 0; v_ 2jc2 >0 < 0
v_ 1 P 0; v_ 2 6 0j06xi 63=4 v_ i ¼ 0 v_ 1 6 0; v_ 2 P 0j06xi 63=4
v_ 1jc >0 < 0; v_ 2jx ¼0 > 0 v_ 1 6 0; v_ 2 P 0j06x 63=4 v_ 1 6 0; v_ 2 P 0j06x 63=4 1
2
i i
result prevails under any symmetric configuration (either WPC or RSC). A symmetric compensation scheme should lead the monopolist supplier to provide each of the competing manufacturers with a constant contribution which does not depend on the contract parameters, but rather on her own marginal salvage value for each manufacturer’s terminal quality. Given this symmetry effect, whenever the manufacturers are in a position to offer similar contractual conditions to the monopolist supplier, their only lever to ensure that the supplier will make a positive contribution to their respective quality improvement activities is to provide her with a maximum incentive to achieve their respective terminal design quality level. On the other hand, if the supplier has either a larger value on both contract parameters with one manufacturer than with the other, or a larger value on one contract parameter with one manufacturer and a similar value on the other contract parameter with both manufacturers, her contribution effort to this manufacturer’s quality improvement activity should be initially high and decreasing over time, while her contribution effort to the other manufacturer’s quality improvement activity should be initially low and increasing over time. Finally, if the supplier has a larger (resp., lower) transfer price and a lower (resp., larger) revenue share with one manufacturer than with the other, her contribution effort path to both manufacturers’ quality improvement activity is ambiguous. However, in the specific case where one manufacturer has a WPC and the other manufacturer has a RSC (e.g., c1 > c2 and 0 ¼ x1 < x2 ), it can be shown, for similar manufacturers (i.e., Q 10 ¼ Q 20 \ h1 ¼ h2 ) that the rates of change of the supplier’s contribution to the competing manufacturers have opposite signs, i.e., v_ 1jWPC > 0; v_ 2jRSC < 0Þ. As a consequence, a manufacturer using a RSC should receive a larger contribution from the monopolist supplier than a rival manufacturer having a WPC. To summarize our results from the qualitative analysis, the monopolist supplier should contribute less to the quality improvement activity of the manufacturer offering the less rewarding contractual conditions, and vice versa. Meanwhile, the less rewarding manufacturer should make a greater quality improvement effort to compensate for the supplier’s smaller effort, in order to maintain its competitive position with respect to the rival manufacturer. Clearly, depending on the contractual conditions, there should be a direct substitutability between the monopolist supplier’s respective contributions to the competing manufacturers, as well as between each manufacturer’s quality improvement effort and the monopolist supplier’s contribution effort. In this sense, a manufacturer’s improvement effort path is indirectly determined by the rival manufacturer’s contractual conditions for the monopolist supplier, since she has to compensate for the supplier’s shortfall. This result is notably valid when one manufacturer has a WPC and the other has a RSC. Proposition 1 below characterizes the explicit behavior of the three players over time for any feasible contract parameter value. Proposition 1. The non-cooperative quality improvement policies of the duopolist manufacturers and the monopolist supplier are:
u1 ðtÞ ¼ ðk1 C h1 ÞðT tÞ þ k1 fA½sinðwTÞ sinðwtÞ B½cosðwTÞ cosðwtÞg=w þ h1 =b1 ;
ð17Þ
u2 ðtÞ ¼ ðh2 þ k2 CÞðT tÞ k2 fA½sinðwTÞ sinðwtÞ B½cosðwTÞ cosðwtÞg=w þ h2 =b2 ;
ð18Þ
v 1 ðtÞ ¼ ðm þ nCÞðT tÞ þ nfA½sinðwTÞ sinðwtÞ B½cosðwTÞ cosðwtÞg=w þ g1 =d; v 2 ðtÞ ¼ ðnC mÞðT tÞ þ nfA½sinðwTÞ sinðwtÞ B½cosðwTÞ cosðwtÞg=w þ g2 =d;
ð19Þ
xi d h1 1 with A ¼ Q 10 Q 20 C; B ¼ w cosðwTÞ bh22 þ g1 d g2 þ A tanðwTÞ; C ¼ h1 hw222m ; hi ¼ 23b b1 i P ffiffiffiffiffiffiffiffiffiffi p P ð32 i xi Þ c1 2ðx1 x2 Þd2 c2 ; w ¼ k ; n ¼ ; i ¼ 1; 2. i 1 2 i 3 x x 9db
h
ð20Þ a1 b
3
ci
i x
i xc3i ; ki ¼ 3i
i d2 2x 9bbi
d ; m ¼ 3d
h
2ðx1 x2 Þa þ b
Proof. See the Appendix. According to Proposition 1, under any symmetric configuration (either WPC or RSC) and for perfectly identical manufacturers (i.e., b1 ¼ b2 \ Q 10 ¼ Q 20 \ h1 ¼ h2 \ g1 ¼ g2 ), manufacturer i’s improvement effort, that is:
ui ðtÞ ¼
i daðT tÞ 1 2x þ hi ; bi 3b
ð21Þ
i ¼ 1; 2; should be initially high and monotonically decreasing over time until its terminal value ui ðTÞ ¼ hi =bi , while the supplier’s contribution should remain time-invariant at v i ðtÞ ¼ gi =d. Accordingly, the total efforts dedicated to manufacturer i’s quality improvement activity should be strictly lower under symmetric RSC than under symmetric WPC. Whatever the compensation scheme, the manufacturer i’s improvement effort is positively influenced by the marginal sensitivity of demand for quality, but negatively impacted by the marginal sensitivity of demand with respect to price. Also, the longer the planning horizon, the greater the manufacturer i’s improvement effort. Under similar conditions but asymmetric manufacturers’ marginal salvage values (i.e.,b1 ¼ b2 \ Q 10 ¼ Q 20 \ h1 – h2 \ g1 ¼ g2 Þ, manufacturer i’s improvement effort, that is:
ui ðtÞ ¼
i adðT tÞ 1 4x cosðwtÞ þ hi þ h3i þ ðhi h3i Þ ; 2bi 3b cosðwTÞ
ð22Þ
i ¼ 1; 2; should be initially high and periodically instead of monotonically decreasing over time. If the planning horizon is not too large, i.e., qffiffiffiffiffi i such that T < 34dp bb i , the manufacturer with the larger marginal salvage value should make a larger improvement effort than the rival over x h i 3i Þ cosðwtÞ Note that the requirement of not too large planning horizon is in line with our time, with a differential of magnitude ðhi h 2b cosðwTÞ . i
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assumption of negligible time value of money. For the case of larger planning horizon, the effect of discounting should be taken into account, which is not the case in our model. In this respect, the case of larger planning horizon is disregarded hereafter. The partial derivative of manufacturer i’s improvement effort with respect to her own revenue share is given by:
uix i ¼
1 4adðT tÞ ðhi h3i Þ sin ½wðT tÞ : þ 2bi 3b cos2 ðwTÞ
The manufacturer with the larger marginal salvage value should make a larger improvement effort under symmetric WPC over time than under symmetric RSC. Under such conditions, the rival manufacturer, i.e., with the lower marginal salvage value, may nevertheless have a larger improvement effort under symmetric RSC than under symmetric WPC. The previous results generalize to any symmetric compensation scheme (either WPC or RSC) with both asymmetric manufacturers’ marginal salvage values and supplier’s marginal salvage values (i.e., b1 ¼ b2 \ Q 10 ¼ Q 20 \ h1 – h2 \ g1 –g2 ), whenever the asymmetry in the manufacturers’ marginal salvage values is consistently reflected in that of the supplier’s marginal salvage values, i.e., gi >jhi >h3i g3i . This is a plausible assumption since the manufacturer with a larger terminal goodwill reward is more likely to provide the supplier with a larger incentive on the terminal design quality level than the rival manufacturer. Manufacturer i’s improvement effort, i ¼ 1; 2, then writes:
ui ðtÞ ¼
i adðT tÞ 1 2x 1 hi h3i gi g3i cosðwtÞ : 1 þ þ hi bi 3b 2 cosðwTÞ bi d
ð23Þ
Here also, given that the supplier’s contribution effort to the manufacturers’ quality improvement activity should be constant over time under any symmetric configuration, the manufacturer with the larger marginal salvage value and the larger supplier’s marginal salvage value should hold the leading position for quality improvement effort, as described above. Consider now a symmetric configuration with perfectly identical manufacturers except for their initial quality levels (i.e., b1 ¼ b2 \ Q 10 – Q 20 \ h1 ¼ h2 \ g1 ¼ g2 ). In these conditions, manufacturer i’s improvement effort, that is:
( ) i d 2aðT tÞ ðQ 10 Q 20 Þ sin ½wðT tÞ hi x pffiffiffiffiffiffiffiffiffiffiffiffi ; ui ðtÞ ¼ þ þ 3 bi b bi i bbi cosðwTÞ x
ð24Þ
i ¼ 1; 2; should be initially high and periodically decreasing over time. For not too large planning horizon, the improvement effort of the man pffiffiffiffi x dðQ 10 Q 20 Þ sin ½wðTtÞ ; 8t < T. ffiffiffiffiffi ufacturer with the greater initial design quality level should be larger than the rival’s, with a gap of i p 3 bbi cosðwTÞ On the whole, three effects have been identified, that is: - The main effect, as described in Eq. (21), which combines the influences of the potential demand, the sensitivity of demand with respect to price and quality, and the supplier’s share in the manufacturer i’s revenue, - The effect of the differential between the manufacturers’ marginal salvage values and the supplier’s marginal salvage values, as shown in Eq. (23), and - The effect of the differential between the manufacturers’ initial design quality levels, as given in Eq. (24). These influences add to the asymmetry effect due to the compensation schemes, if any. Note that the length of the planning horizon has a significant influence on the outcome of the quality duopolistic competition game, which requires taking into account the discounting effect for large enough planning horizon. Finally, from (19) and (20), it can be shown that the supplier’s contribution policy to each manufacturer’s quality improvement activity under asymmetric compensation schemes may have a non-monotonic path, notably if the manufacturers’ marginal salvage values and the supplier’s marginal salvage values are different. Proposition 2 below determines the optimal path for the manufacturers’ design quality. Proposition 2. The non-cooperative design quality path for each manufacturer’s product is:
h1 g1 1 n þ k1 n þ k1 þ ½h1 ðn þ k1 ÞC m½2T t þ ½A cosðwtÞ þ B sinðwtÞ A; ð25Þ ½A sinðwTÞ B cosðwTÞ t þ b1 d 2 w w2 h2 g2 1 n k2 n k2 ½A sinðwTÞ B cosðwTÞ t þ ½A cosðwtÞ þ B sinðwtÞ A: ð26Þ Q 2 ðtÞ ¼ Q 20 þ þ ½h2 ðn k2 ÞC þ m½2T t þ b2 d 2 w w2 Q 1 ðtÞ ¼ Q 10 þ
Proof. See the Appendix. Under symmetric WPC and perfectly identical manufacturers (i.e., b1 ¼ b2 \ Q 10 ¼ Q 20 \ h1 ¼ h2 \ g1 ¼ g2 ), manufacturer i’s design quality writes:
Q i ðtÞ ¼ Q i0 þ
i da hi g i x þ þ ð2T tÞ t; bi d 3bi b
ð27Þ
i ¼ 1; 2; that is, manufacturer i’s design quality should be increasing at a decreasing rate over time. Further, design quality under symmetric WPC should be larger than under symmetric RSC at any time period, and whatever the length of the planning horizon. Under similar conditions but asymmetric manufacturers’ marginal salvage values and supplier’s marginal salvage values (i.e.,b1 ¼ b2 \ Q 10 ¼ Q 20 \ h1 – h2 \ g1 – g2 ), manufacturer i’s design quality is given by:
"
Q i ðtÞ ¼ Q i0 þ
# pffiffiffiffiffiffiffi 2 i da 1X hi gi x 3 bbi sinðwtÞ hi h3i gi g3i þ : þ þ ð2T tÞ t þ pffiffiffiffiffiffi i cosðwTÞ 2 i¼1 bi d 3bi b bi d 4d x
ð28Þ
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Under not too large planning horizon, the quality level of the manufacturer with the larger marginal salvage value and/or the larger supplier’s marginal salvage value should be larger than that of the rival, 8t 6 T. Under symmetric manufacturers except for initial quality levels (i.e., b1 ¼ b2 \ Q 10 – Q 20 \ h1 ¼ h2 \ g1 ¼ g2 ), manufacturer i’s design quality is given by:
Q i ðtÞ ¼ Q i0 þ
i dað2T tÞ hi gi x 1 þ þ t þ ðQ i0 Q 3i0 Þ½wt sinðwTÞ þ cosðwtÞ 1: 3bbi 2 bi d
ð29Þ
Here also, for not too large planning horizon, the quality level of the manufacturer with the greater initial design quality level should be larger than the rival’s at any time period. For each manufacturer, the design quality path may then follow either monotonous or oscillatory increasing trend until:
Q i ðTÞ ¼ Q i0 þ
hi gi 1 n þ oj ki n þ oj k1 þ hi ðn þ oj ki ÞC oj m T þ ½A cosðwTÞ þ B sinðwTÞ A ½AsinðwTÞ B cosðwTÞ T þ bi d 2 w w2
ð30Þ
is reached at the end of the planning horizon, i ¼ 1; 2; j ¼ 1; 2; o1 ¼ 1 and o2 ¼ 1. Lemma 5. Under perfectly symmetric configurations, it holds that:
Q i ðTÞjxi ¼0 > Q i ðTÞj0
i ¼ 1; 2:
ð31Þ
Proof. See the Appendix. According to (31), the terminal design quality under a symmetric WPC configuration is greater than under a symmetric RSC configuration. This result is due to the fact that both manufacturers make less quality improvement effort under a RSC than under a WPC, while the monopolist supplier’s contribution effort to both manufacturers’ improvement activities remains similar under any symmetric configuration. On the whole, by mitigating the double marginalization effect, a compensation scheme based on RSC can indeed lead to lower sales prices than under a WPC, but this price reduction should be associated with a lower terminal design quality level. This result contradicts those obtained for a different dimension of quality, namely conformance quality (El Ouardighi et al., 2008), where it is shown that a RSC gives rise to a higher pricing policy than a WPC, but is associated with a higher conformance quality policy. This suggests that the way players are compensated may have conflicting impacts on different quality dimensions of a product. 4. Numerical illustrations Numerical examples are provided to illustrate the players’ non-cooperative policies in quality improvement under WPC and RSC. The preference between the two contract arrangements is also evaluated for both the supplier and the manufacturers. The parameter values of the game are given in Table 2. For a sensitivity analysis, a broad range of values is used for all parameters, as well as for the initial values of state variables. In order to test the robustness of our results with respect to the length of the planning horizon, the duration of the game isffi set at a sufficiently large extent, that is, 20 time units, t 2 ½0; 20, which for the whole range of parameters pffiffiffiffiffiffi pffiffiffiffiffiffi i. values is such that T > 3p bbi =4d x The evaluation of the impact of the compensation structure on players’ competitive behavior leads us to consider three potential configurations, that is: - Both manufacturers agree on a WPC with the supplier, i.e., c1 > 0; c2 > 0 and x1 ¼ x2 ¼ 0 (configuration 1, or C1), - Both manufacturers agree on a RSC with the supplier, i.e., c1 > 0; c2 > 0; x1 > 0 and x2 > 0 (C2), - Manufacturer 1 agrees on a RSC and manufacturer 2 agrees on a WPC with the supplier, i.e., c1 > 0; c2 > 0; x1 > 0 and x2 ¼ 0 (C3). The following four figures depict the typical dynamics of control and state variables under each configuration, within the ranges of values defined above. Fig. 1 shows that both manufacturers have a monotonically decreasing quality improvement effort policy over time. Both manufacturers make a greater improvement effort over time under symmetric WPCs than under symmetric RSCs. However, the manufacturer with an asymmetric WPC makes more effort than under a symmetric WPC, while the manufacturer with an asymmetric RSC makes less effort than under the symmetric RSC. The asymmetric configuration cannot be seen as a simple combination of the two symmetric configurations. In Fig. 2, due to a symmetry effect, the path of the supplier’s contribution effort to both manufacturers’ quality improvement activity is unaffected by the compensation structure, and constant paths are observed in both symmetric configurations. In the asymmetric configuration, the supplier’s contribution paths take opposite directions: the supplier’s contribution effort for the manufacturer with an asymmetric WPC shows a monotonic increase from a low initial level, and the supplier’s contribution effort for the manufacturer with an asymmetric RSC shows a monotonic decrease from a high initial level. The asymmetric configuration thus gives rise to unequal treatment of the competing manufacturers by the monopolist supplier, which generates substitutability between each manufacturer’s quality improvement effort and the monopolist supplier’s contribution effort. This effect is ultimately reflected in a negative mutual dependency between the duopolist manufacturers’ respective quality improvement efforts, as described in the previous figure. Table 2 Parameter values. Parameter
ci
xi
a
b
d
bi
d
hi
gi
Q i0
Low value High value
50 80
0 0.2
100 1000
0.5 2
0.5 2
10 20
10 20
100 200
50 100
1 10
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70 u1|C1
60
ui(t)
u2|C1
50
u1|C2
40
u1|C3
u2|C2 u2|C3
30 20 10 0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
t Fig. 1. Manufacturers’ quality improvement effort over time.
16 14 12 10
vi(t)
8
v1|C1
6
v2|C1 v1|C2
4
v2|C2
2
v1|C3 v2|C3
0 0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
t Fig. 2. Supplier’s contribution efforts over time.
According to Fig. 3, each manufacturer should apply a constant pricing policy over time under both symmetric configurations. Furthermore, the pricing policy should be higher under a symmetric WPC than under a symmetric RSC. Under the asymmetric configuration, the manufacturers’ pricing policies should start from close initial values and then evolve in opposite directions. Under an asymmetric WPC, the sales price should increase to reach a significantly higher level than under a symmetric WPC, while under asymmetric RSC, it should decrease to a lower level than under a symmetric RSC. According to Fig. 4, the design quality should follow an increasing path in all the configurations. With symmetric WPCs, quality should be larger than under symmetric RSCs. All other things being equal, this result is valid whatever the marginal sensitivity of sales to price or quality. Duopolist manufacturers should therefore adopt a price-reduction strategic orientation under a symmetric RSC, and a qualityimprovement strategic orientation under a symmetric WPC. In the asymmetric configuration, the manufacturer with a WPC also performs better than her rival with a RSC. This suggests that the enhanced contribution effort of the monopolist supplier to the manufacturer with a RSC is outweighed by the improvement effort of the manufacturer with a WPC. Hence, the manufacturer with a WPC sets a higher sale price than the manufacturer with a RSC not because of a lack of coordination with the monopolist supplier, but rather because she holds a competitive advantage on quality.
250 200 150 p1|C1
pi(t)
p2|C1
100
p1|C2 p2|C2
50
p1|C3 p2|C3
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
t Fig. 3. Sales price policies over time.
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900
Qi(t)
800
Q1|C1
700
Q2|C1
600
Q1|C2 Q2|C2
500
Q1|C3
400
Q2|C3
300 200 100 0 0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
t Fig. 4. Design quality paths over time. Table 3 Players’ total payoff under WPC and RSC. Manufacturer 2
Manufacturer 1
WPC ðc1 ¼ 80; x1 ¼ 0Þ RSC ðc1 ¼ 50; x1 ¼ 0:2Þ
WPC ðc2 ¼ 80; x2 ¼ 0Þ
RSC ðc2 ¼ 50; x2 ¼ 0:2Þ
J1 ¼ 291969; J 2 ¼ 291969; JS ¼ 405294 J1 ¼ 255078; J2 ¼ 272328; J S ¼ 445464
J 1 ¼ 272328; J2 ¼ 255078; J S ¼ 445464 J 1 ¼ 242813; J2 ¼ 242813; J S ¼ 490905
In Table 3, the players’ total payoff is determined for each of the three configurations under specific values of the contract parameters. It is noteworthy that in all configurations, the greater the sensitivity of demand with respect to quality, the lower the manufacturers’ profits, and the larger the supplier’s profit. Furthermore, the greater the sensitivity of demand with respect to price, the lower the profits for both manufacturer and supplier. On the other hand, the symmetric WPC configuration is the least profitable to the supplier, while the symmetric RSC configuration is the most beneficial. In other words, the symmetric WPC configuration represents an economic equilibrium for the manufacturers, while the symmetric RSC configuration represents an economic equilibrium for the supplier. Assuming that both manufacturers have a WPC, there is a disincentive for both of them to adopt a RSC, so that both the asymmetric configuration and the symmetric RSC configuration appear unfeasible. Table 3 implies that there might be a potential conflict between the manufacturers and their supplier. If collaboration between the manufacturers is permitted, each manufacturer has a great incentive to set up a WPC with the supplier (cell WPC-WPC in Table 3) so as to maximize her profit. On the other hand, the supplier will try to sign a RSC with each manufacturer separately so as to maximize her own profit (cell RSC-RSC in Table 3). Even if explicit collaboration between the manufacturers were not allowed, a rational decision-maker, i.e., manufacturer, would be able to see the potential gain in aligning her strategy with her counterpart’s. Similarly, a rational supplier would be aware of the possibility of such implicit collaboration between the manufacturers (or at least, of each manufacturer’s implicit assumption that collaboration would be possible). As such, our analysis enables the players, both the manufacturers and the supplier, to assess the limitations as well as the potential of their strategic choices, and to use compromise to find a better solution giving all players profits larger than in extreme situations (i.e., either WPC-WPC or RSC-RSC). 5. Conclusions This paper has adopted the perspective of interdependent, competing supply chains, where a monopolist supplier dynamically collaborates with duopolist manufacturers in improving the design quality of their respective products. It was assumed that the manufacturers compete for the final demand both on sale price and design quality. The transactions between each manufacturer and the supplier are determined under a two-parameter approach. The objective of the paper was twofold: - First, to determine the three players’ optimal effort allocation for quality improvement in a decentralized context; - Second, to evaluate how the compensation structure adopted influences the competitive dynamics of the three players’ game. Using analytical means, it has been shown that a supplier should take advantage of her monopolistic position by applying a discriminatory contribution policy to the respective quality improvement activities of the competing manufacturers. It is thus not in the best interests of rival manufacturers to offer similar contractual conditions to the monopolist supplier, since it gives rise to a symmetry effect which leads the supplier to determine her contribution effort to each manufacturer’s quality improvement activities as a function not of the contract parameters, but rather of her salvage value for the manufacturer’s terminal design quality level (i.e., goodwill reward). If identical contractual conditions are unavoidable, the manufacturers should negotiate them at their lowest value, and provide the supplier with a maximum incentive to contribute to their respective terminal design quality level. Supply quality competition then emerges as a combination of supply quality management and horizontal market competition, where two forms of interdependencies take place: A negative horizontal direct interdependency between the contribution efforts of the monopolist supplier to the manufacturers’ respective improvement activities, i.e., the better the contractual conditions with one manufacturer relatively to the other manufacturer, the heavier (resp., the lower) the supplier’s contribution effort to the improvement activity of this manufacturer (resp., the other manufacturer).
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A negative vertical direct interdependency between each manufacturer’s improvement effort and the monopolist supplier’s improvement effort, i.e., the heavier (resp., the lower) the supplier’s contribution effort to one manufacturer’s improvement activity, the lower (resp., the heavier) the improvement effort of this manufacturer. These two interdependencies give rise to a negative horizontal indirect mutual dependency between the duopolist manufacturers’ respective improvement efforts, i.e., the manufacturer which offers the poorer contractual conditions should make a greater improvement effort to outweigh the unfavorable supplier’s effort contribution policy, in order to maintain its competitive position relatively to the rival manufacturer. A manufacturer’s improvement effort path should not be determined only by her own contractual conditions with the monopolist supplier, but also by the rival manufacturer’s contractual conditions. Given the contrast in pricing policy between the WPC and the RSC, the presence of a double marginalization effect could have been wrongly diagnosed. Justification for a higher price under a WPC is found in the existence of a competitive advantage on design quality: the WPC entails a quality premium, reflected in a higher sales price. This result contradicts the static analysis of Economides (1999) according to which quality is negatively affected by the presence of a double marginalization effect in a bilateral monopoly setting. Actually, we show that, under dynamic horizontal competition, the double marginalization effect appears as a potentially misleading concept; since it may lead to a price differential between competing manufacturers being mistakenly interpreted as a consequence of a lack of vertical coordination, rather than a competitive advantage on quality. The WPC is more beneficial to the duopolist manufacturers, while the RSC is more profitable for the monopolist supplier. There is a negative incentive for the manufacturers to adopt a RSC, and it is thus not a coordinating contract in the context of a monopolist supplier serving duopolist manufacturers. Although the three players’ joint profits can be significantly increased, turning the supply chain into a revenue chain (Cachon and Lariviere, 2001) may result in lower design quality levels, and be detrimental to the manufacturers’ profits. Our results clearly show that RSC is not necessarily the perfect solution for supply quality management under horizontal competition. Another important implication here is that ‘‘one should not take it for granted that a successful mechanism in a one shot game works in the same manner in a dynamic game” (Zaccour, 2008). Although a different definition of quality was used, the results obtained are similar to those in El Ouardighi et al. (2008) in the sense that this study also finds that the positive interaction between the supplier’s transfer price and the manufacturers’ prices, i.e., the double marginalization effect, is not clearly visible under a WPC. Interestingly, the comparison with El Ouardighi et al. (2008) also suggests that WPC performs better on design quality and worse on conformance quality than RSC and vice versa. The formulation of a general demand function which would include both design quality and conformance quality could therefore open up a promising avenue for future research, for a more accurate comparison of the respective strengths and limitations of WPC and RSC in terms of both supply quality management and profit distribution among the chains’ members. Appendix Proof of Lemma 1. Using (8) to determine pi and substituting the corresponding expression for the rival manufacturer, p3i ; i ¼ 1; 2, gives (10). h Proof of Lemma 2. Obvious from (10). h Proof of Lemma 3. Substituting (10) for optimal sales price into (7) yields:
Hi ¼
2 bi u2i ci c3i 3a b þ li1 ðui þ v i Þ þ li2 ðu3i þ v 3i Þ; þ dðQ i Q 3i Þ i x 3i 9b x 2 i x
i ¼ 1; 2; which reveals the linear quadratic structure of the game. The costate variables are derived as follows:
id 2x ci c3i 3a b þ dðQ i Q 3i Þ i x 3i 9b x id 2x ci c3i ¼ 3a b þ dðQ i Q 3i Þ i x 3i 9b x
l_ i1 ¼ HiQ i ¼
li1 ðTÞ ¼ hi ;
l_ i2 ¼ HiQ 3i
li2 ðTÞ ¼ hi ;
i ¼ 1; 2. Differentiating (9) with respect to time and substituting for l_ i1 and simplifying gives (13).
h
Proof of Lemma 4. The supplier’s costate variables are derived from (14) as follows:
( ! ) X c1 c2 _w1 ¼ Hs ¼ d ðx1 x2 Þ ½6a þ 2dðQ 1 Q 2 Þ þ 3 2 w1 ðTÞ ¼ g1 ; xi Q1 9 b x1 x2 i ( ! ) X c2 c1 _w2 ¼ Hs ¼ d ðx2 x1 Þ ½6a þ 2dðQ 2 Q 1 Þ þ 3 2 w2 ðTÞ ¼ g2 : xi Q2 9 b x2 x1 i Differentiating (15) with respect to time and substituting for w_ i ; i ¼ 1; 2, and simplifying gives (16).
h
Proof of Proposition 1. Subtracting (2) from (1), differentiating the resulting expression with respect to time, and substituting (13) and (16) for u_ i and v_ i ; i ¼ 1; 2, gives:
€ 2 ðtÞ þ ðk1 þ k2 ÞðQ ðtÞ Q ðtÞÞ ¼ h1 h2 2m: € 1 ðtÞ Q Q 1 2
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Given ki > 0; i ¼ 1; 2, this equation has the solution:
Q 1 ðtÞ Q 2 ðtÞ ¼ A cosðwtÞ þ B sinðwtÞ þ where w2 ¼
P
i ki ,
h1 h2 2m ; w2
for which only the positive root, i.e., w > 0, is considered. Using the initial conditions Q i0 > 0; i ¼ 1; 2, we get:
A ¼ Q 10 Q 20
h1 h2 2m : w2
Using the following derivations:
Q_ 1 ðtÞ Q_ 2 ðtÞ ¼ Aw sinðwtÞ þ Bw cosðwtÞ; Q_ 1 ðTÞ Q_ 2 ðTÞ ¼ u1 ðTÞ þ v 1 ðTÞ u2 ðTÞ v 2 ðTÞ ¼ h1 =b1 þ g1 =d h2 =b2 g2 =d; p modp, we obtain: and assuming T – 2w
B¼
h1 =b1 þ g1 =d h2 =b2 g2 =d h1 h2 2m tanðwTÞ: þ Q 10 Q 20 w cosðwTÞ w2
Substituting the expression of Q 1 ðtÞ Q 2 ðtÞ into (13) and (16), i ¼ 1; 2, gives:
u_ 1 ðtÞ ¼ h1 k1 ½C þ A cosðwtÞ þ B sinðwtÞ; u_ 2 ðtÞ ¼ h2 þ k2 ½C þ A cosðwtÞ þ B sinðwtÞ;
v_ 1 ðtÞ ¼ m n½C þ A cosðwtÞ þ B sinðwtÞ; v_ 2 ðtÞ ¼ m n½C þ A cosðwtÞ þ B sinðwtÞ; where:
C¼
h1 h2 2m : w2
These four equations are integrated using their boundary conditions to obtain (17)–(20). Proof of Proposition 2. To solve (1) and (2), we first compute U i ðtÞ ¼ ui0 þ yields:
Rt 0
h
ui ðsÞds and V i ðtÞ ¼ v i0 þ
Rt 0
v i ðsÞds, respectively, i ¼ 1; 2, which
1 k1 h1 k1 þ ½A sinðwTÞ B cosðwTÞ t þ K 1 ; ðh1 k1 CÞðt TÞ2 þ 2 ½A cosðwtÞ þ B sinðwtÞ þ 2 w b1 w 1 k2 h2 k2 2 ½A sinðwTÞ B cosðwTÞ t þ K 2 ; U 2 ðtÞ ¼ ðh2 þ k2 CÞðt TÞ 2 ½A cosðwtÞ þ B sinðwtÞ þ 2 w b2 w ng o 1 n n 2 V 1 ðtÞ ¼ ðm þ nCÞðt TÞ þ 2 ½A cosðwtÞ þ B sinðwtÞ þ 1 þ ½A sinðwTÞ B cosðwTÞ t þ K 3 ; 2 w d w ng o 1 n n 2 2 þ ½A sinðwTÞ B cosðwTÞ t þ K 4 ; V 2 ðtÞ ¼ ðm nCÞðt TÞ þ 2 ½A cosðwtÞ þ B sinðwtÞ þ 2 w d w
U 1 ðtÞ ¼
where K 1 ; . . . ; K 4 are constants of integration. Substituting the corresponding expressions in (1) and (2), respectively, and solving gives:
1 n þ k1 h1 g1 n þ k1 ½h1 ðn þ k1 ÞC mðt TÞ2 þ ½ A cosðwtÞ þ B sinðwtÞ þ þ þ ½ A sinðwTÞ B cosðwTÞ t þ L1 ; 2 w2 b1 d w 1 n k2 h2 g2 n k2 ½A cosðwtÞ þ B sinðwtÞ þ þ þ ½A sinðwTÞ B cosðwTÞ t þ L2 ; Q 2 ðtÞ ¼ ½h2 ðn k2 ÞC þ mðt TÞ2 þ 2 w2 b2 d w
Q 1 ðtÞ ¼
where L1 and L2 are constants of integration. Using the initial values Q i0 ; i ¼ 1; 2, to determine L1 and L2 leads to (25) and (26).
h
Proof of Lemma 5. Comparing the two equations in (27) and invoking the symmetry property (i.e., b1 ¼ b2 \ c1 ¼ c2 \ x1 ¼ x2 \ Q 10 ¼ Q 20 \ h1 ¼ h2 \ g1 ¼ g2 ) yields (28). h References Bresnahan, T.F., Reiss, P.C., 1985. Dealer and manufacturer margins. Rand Journal of Economics 16, 253–268. Cachon, G.P., Lariviere, M.A., 2001. Turning the supply chain into a revenue chain. Harvard Business Review, 20–21. March. Cachon, G.P., Lariviere, M.A., 2005. Supply chain coordination with revenue sharing contracts: Strengths and limitations. Management Science 51, 30–44. Chen, F., Federgruen, A., Zheng, Y., 2001. Coordination mechanisms for decentralized distribution systems with one supplier and multiple retailers. Management Science 47, 693–708. Dockner, E., Jørgensen, S., Van Long, N., Sorger, G., 2000. Differential Games in Economics and Management Science. Cambridge University Press, Cambridge. Economides, N., 1999. Quality choice and vertical integration. International Journal of Industrial Organization 17, 903–914. El Ouardighi, F., Pasin, F., 2006. Quality improvement and goodwill accumulation in a dynamic duopoly. European Journal of Operational Research 175, 1021–1032. El Ouardighi, F., Jørgensen, S., Pasin, F., 2008. A dynamic game model of operations and marketing management in a supply chain. International Game Theory Review 10, 373– 397. El Ouardighi, F., Jørgensen, S., Pasin, F., 2009. A dynamic game of decentralized interdependent, competing supply chains. W.P. CERESSEC, ESSEC Business School. Garvin, D., 1988. Managing Quality. Free Press, New York. Kim, B., El Ouardighi, F., 2007. Supplier–manufacturer collaboration on new product development. In: Jørgensen, S., Vincent, T., Quincampoix, M. (Eds.), Advances in Dynamic Games and Applications to Ecology and Economics. Birkhauser, Boston. Kogan, K., Tapiero, C.S., 2007. Supply Chain Games: Operations Management and Risk Valuation. Springer, New York.
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
Supply quality management with wholesale price and revenue-sharing contracts under horizontal competition Fouad El Ouardighi a,*, Bowon Kim b a b
ESSEC Business School, Avenue Bernard Hirsch, B.P. 105, 95021 Cergy Pontoise, France KAIST Graduate School of Management, 207-43 Cheongryangri Dongdaemoon-Ku, Seoul 130-722, South Korea
a r t i c l e
i n f o
Article history: Received 10 January 2009 Accepted 26 February 2010 Available online 3 March 2010 Keywords: Design quality Supply quality management Horizontal competition Wholesale price contract Revenue sharing contract
a b s t r a c t In a number of industries (e.g., the airplane industry, aerospace industry, auto industry, or computer industry), certain suppliers essentially have a monopoly on the production technology for key components, and inevitably manufacturers in these industries have common suppliers. A key part of manufacturers’ work with suppliers concerns improving the quality of their respective products, which gives rise to a collaborative activity usually termed as ‘‘supply quality management”. When the manufacturers are competitors, they do not wish to see a common supplier dividing his involvement in quality improvement unequally between themselves and their rivals. However, as the suppliers collaborate with several manufacturers, it is highly questionable whether their efforts will be strictly equivalent for each manufacturer. In this paper, a non-cooperative dynamic game is formulated in which a single supplier collaborates with two manufacturers on design quality improvements for their respective products. The manufacturers compete for market demand both on price and design quality. The paper analyzes how each party should allocate resources for quality improvement over time. In order to take into account the potential coordinating power of the compensation scheme adopted in this type of decentralized setting, we compare the possible outcomes under a wholesale price contract and a revenue-sharing contract. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction In a number of industries (e.g., the airplane industry, aerospace industry, auto industry, or computer industry), certain suppliers essentially have a monopoly on the production technology of key components, and inevitably manufacturers in these industries have common suppliers. In the plane making business, for instance, the two major rival players Boeing and Airbus share up to twelve strategic suppliers (Michaels and Lunsford, 2004) so that their supply chains are both competing and interdependent. A key part of manufacturers’ work with suppliers concerns improving the quality of their respective products, which gives rise to a collaborative activity usually termed as ‘‘supply quality management”. When the manufacturers are competitors, they do not wish to see a common supplier dividing his involvement in quality improvement unequally between themselves and their rivals. However, as the suppliers cooperate with several manufacturers, it is highly questionable whether their efforts will be strictly equivalent for each manufacturer. Given the empirical evidence on the decisive importance of quality management for supply chain success (Morash, 2001; Smets, 2004), it is interesting to study how vertical coordination affects the players’ optimal allocation of efforts for quality improvement in the context of horizontal competition at the manufacturing level. It has been shown that coordinated supply chains perform better in quality improvement than uncoordinated supply chains. For instance, Kim and El Ouardighi (2007) consider the problem of optimal effort allocation between design quality improvement for an existing product and development of a new product, in a one manufacturer-one supplier supply chain. Using qualitative and numerical methods, they find that the uncoordinated chain pays more attention to new product development, while the coordinated chain focuses more on the quality of the existing product. Given that the degree of coordination in a supply chain can affect the chain members’ quality improvement effort, it is difficult to consider the issue of supply quality management in competing and interdependent supply chains without taking into account the impact of coordination, if any, in a decentralized setting.
* Corresponding author. Tel.: + 33 1 34 43 33 20; fax: + 33 1 34 43 30 01. E-mail address: [email protected] (F. El Ouardighi). 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.02.035
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Supply chain contracts are generally considered as a useful tool to bring supply chain actors in a decentralized setting to operate in coordination. Different models of supply chain contracts have been developed in the literature, and one model considered ‘very attractive’ is the revenue-sharing contract (RSC) (Cachon and Lariviere, 2005). Under a RSC, each manufacturer pays the supplier a fixed transfer price for each unit purchased, plus a fixed percentage of her own sales revenue. The RSC is viewed as a valuable alternative to the wholesale price contract (WPC) (Cachon and Lariviere, 2001), in which the supplier merely charges the retailer a fixed per-unit price. Although the WPC is commonly observed in practice because it is simple to implement and cheap to administer, it can engender a double marginalization effect. This effect arises when a supplier sells a product to a retailer who is facing a downward sloping demand curve (e.g. Bresnahan and Reiss, 1985): the retailer is not concerned with the supplier’s profit when fixing the retail price, and as a result the retail price is higher than in a centralized chain. The advantage of the RSC over a WPC is that it mitigates this double marginalization effect (e.g., Chen et al., 2001) and can significantly increase the supply chain profits, for example by more than 10% in the video rental industry (Mortimer, 2008). To date, comparisons between RSCs and WPCs in supply quality management have been limited to one important but specific dimension of quality, namely conformance quality, which refers to the extent to which a product conforms to a given design quality standard (Garvin, 1988). In the setup of a one supplier/one retailer supply chain in which improved conformance quality can enhance current customer loyalty, El Ouardighi et al. (2008) show that chain members should invest more in quality under a RSC than under a WPC over time. This paper extends knowledge of the situation in which a supplier is serving two price-competing manufacturers. To do so, we use another essential and complementary dimension of quality, that is, design quality, which refers to the set of product attributes or features that enhance the match with the customer’s needs (Garvin, 1988). In a setup consisting of interdependent, competing chains we investigate how the compensation scheme adopted (WPC or RSC) can affect design quality competition at the manufacturing level via the way the players allocate their quality improvement efforts. In doing so, we analyze how any mitigation of the double marginalization effect through a RSC affects supply quality management in the context of horizontal competition at the manufacturing level. The paper is organized as follows. Section 2 presents a dynamic game model where a single supplier collaborates with two competing manufacturers in improving the quality of their respective products. Section 3 analyzes the three players’ differential game in a decentralized setting and compares the outcomes obtained under WPC and RSC. To gain additional insights into the optimal dynamic behavior of the players, a numerical study is conducted in Section 4. Section 5 draws the key managerial implications from the analytical and numerical analyses. 2. A stylized non-cooperative differential game model We consider a situation with three players: two duopolistic manufacturers and a monopolist supplier. This configuration belongs to the category of interdependent, competing supply chains (El Ouardighi et al., 2009), that is, supply chains which share common members and compete for market demand. Both manufacturers purchase a similar part from their supplier, which is used in their respective finished products. The manufacturers’ products are in competition for the final demand market. Each manufacturer invests in quality improvement for her product, and the monopolist supplier is supposed to collaborate in each manufacturer’s quality improvement activity. Quality here means design quality, i.e., the set of product attributes or features that enhance the match with the customer’s needs (Garvin, 1988). In this sense, quality improvement activity aims to increase the desirability of the product, notably through the implementation and use of dedicated tools such as quality function deployment (QFD), Taguchi arrays, Ishikawa diagrams, etc. Although the adoption of such tools is generally costly in terms of instantaneous effort, it does not necessarily incur an extra cost per unit produced. Let Q i ðtÞ > 0 represent the design quality level at time t for manufacturer i’s product, for which quality improvement is given by:
Q_ 1 ðtÞ ¼ u1 ðtÞ þ v 1 ðtÞ Q 1 ð0Þ ¼ Q 10 > 0; Q_ 2 ðtÞ ¼ u2 ðtÞ þ v 2 ðtÞ Q 2 ð0Þ ¼ Q 20 > 0;
ð1Þ ð2Þ
where ui ðtÞ P 0 and v i ðtÞ P 0 denote the respective effort input by manufacturer i and the monopolist supplier to manufacturer i’s product quality, i ¼ 1; 2. Note that a similar representation of the evolution of design quality is used by Mukhopadhyay and Kouvelis (1997) in the context of a dynamic game model of duopolistic competition. We assume that manufacturer i’s final demand, Di ðtÞ; i ¼ 1; 2, is determined both by price and quality competition on the duopoly market, as follows:
Di ðtÞ ¼ a bðpi ðtÞ p3i ðtÞÞ þ dðQ i ðtÞ Q 3i ðtÞÞ;
ð3Þ
where a 0, and b and d are symmetric positive constants for the sake of simplicity. In Eq. (3), manufacturer i’s demand is: - A decreasing function of the differential between her own price, pi ðtÞ P 0, and the rival firm’s price, p3i ðtÞ P 0, and - An increasing function of the differential between her own quality, Q i ðtÞ, and the rival firm’s quality, Q 3i ðtÞ; i ¼ 1; 2. In the case where both manufacturers have constant design quality ðQ_ i ¼ 0; 8 iÞ, Eq. (3) would reduce to a standard Bertrand demand function where manufacturer i’s demand shows a linear decrease in own price and a linear increase in the rival manufacturer’s price. Manufacturer i’s demand equation thus provides an extension to the situation in which both manufacturers also compete on design quality. We assume that each manufacturer enters into either a WPC or a RSC with the supplier. A two-parameter scheme is therefore used, where: The first parameter is the transfer price (supposed constant), ci > 0, paid by manufacturer i to the supplier for each unit purchased, and The second parameter is the supplier’s share (supposed constant) in manufacturer i’s sales revenue, xi ; xi 2 ½0; 1½; i ¼ 1; 2. The parameter values are such that ci > 0 and xi ¼ 0 for a WPC, while ci > 0 and xi 20; 1½ for a RSC. The assumption of fixed contract parameters lies with the fact that, in the real world, when companies sign a contract, it is usual for them to specify the terms of contract, as
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they want to design their contract in a predictable manner as much as possible. Predetermining the transfer price and the revenue-sharing factor (if any) as fixed parameters is part of such effort. On the other hand, note that the adoption of a RSC presupposes that two non-trivial issues are solved, i.e., how the supplier can monitor manufacturer i’s sales, and how the value of the supplier’s share in each manufacturer’s revenue is determined. In the absence of any empirical evidence that design quality improvement positively affects the supplier’s production cost, transfer price and product quality are presumed to be independent for each manufacturer (e.g., Kim and El Ouardighi, 2007). Also, in order to keep the model analytically tractable, the manufacturers’ operating costs are supposed to be both equal to zero. Finally, the assumption that the duopolist manufacturers will potentially have different transfer prices derives from the possible substitutability between the transfer price and the supplier’s share in each manufacturer’s revenue, if any. i 20; 1, manufacturer i’s instantaneous profit can thus be expressed as i 1 xi ; x Denoting the manufacturer i’s revenue share by x follows:
i pi ðtÞ ci Di ðtÞ ½x
bi ui ðtÞ2 ; 2
ð4Þ
where bi ui ðtÞ2 =2 denotes manufacturer i’s quadratic cost function of own improvement effort, bi being a positive constant, i ¼ 1; 2 (e.g., El Ouardighi and Pasin, 2006). Note that the results would remain qualitatively the same if a general cost function fi ðui ðtÞÞ were adopted, with @fi =@ui > 0; @ 2 fi =@u2i > 0; f i ð0Þ ¼ 0. We consider a finite planning horizon and assume that the time value of money is negligible. Manufacturer i’s objective criterion can thus be expressed as follows:
Z
T
(
) bi ui ðtÞ2 i pi ðtÞ ci ½a bðpi ðtÞ p3i ðtÞÞ þ dðQ i ðtÞ Q 3i ðtÞÞ dt þ hi ðQ i ðTÞ Q 3i ðTÞÞ; ½x 2
0
ð5Þ
where hi ðQ i ðTÞ Q 3i ðTÞÞ is the salvage value for the differential between own quality and the rival manufacturer’s quality, i ¼ 1; 2. Manufacturer i’s salvage value is interpreted as a goodwill reward (resp., goodwill penalty) for a larger (resp., lower) own terminal quality level than the rival’s. On the other hand, the monopolist supplier’s objective criterion is:
Z
T
( 2 X
0
i¼1
) 2 2 X X di v i ðtÞ2 dt þ ½xi pi ðtÞ þ ci ½a bðpi ðtÞ p3i ðtÞÞ þ dðQ i ðtÞ Q 3i ðtÞÞ gi Q i ðTÞ; 2 i¼1 i¼1
ð6Þ
where di v i ðtÞ2 =2 is the supplier‘s quadratic cost function of own improvement effort, di being a positive constant, and gi Q i ðTÞ > 0 is the salvage value (i.e., goodwill reward) for manufacturer i’s quality, i ¼ 1; 2. For simplicity, the rest of the paper assumes symmetrical value for the supplier’s improvement effort cost coefficients (i.e., d1 ¼ d2 ¼ dÞ. 3. Supply chain contract and equilibrium strategies We focus on the decentralized setting of the game in which each player aims to maximize her own objective function. The game defined by (5) and (6) under constraints (1) and (2) involves more than one state variable, which precludes the search for closed-loop equilibrium, although it is more attractive from a strategic viewpoint. We confine our interest to the search for an open-loop Nash equilibrium, which implies that all the players precommit to their control actions throughout the game (Dockner et al., 2000). As it fulfils the time-consistency requirement, analysis of open-loop Nash equilibrium can serve as a benchmark for the assessment of more complex strategies. First, we characterize manufacturer i’s optimal control problem of determining his strategies pi ðtÞ (sales price) and ui ðtÞ (improvement effort). Skipping the time index for convenience, manufacturer i’s Hamiltonian is:
i pi ci ½a bðpi p3i Þ þ dðQ i Q 3i Þ H i ¼ ½x
bi u2i þ li1 ðui þ v i Þ þ li2 ðu3i þ v 3i Þ; 2
ð7Þ
where lij lij ðtÞ; i ¼ 1; 2; j ¼ 1; 2, are manufacturer i’s costate variables. Necessary conditions for optimality are:
i ½a bðpi p3i Þ þ dðQ i Q 3i Þ bðx i pi ci Þ ¼ 0; Hipi ¼ x
ð8Þ
Hiui
ð9Þ
i 1
¼ bi ui þ l ¼ 0:
According to (8) and (9), manufacturer i’s Hamiltonian is strictly concave with respect to manufacturer i’s control variables. From (8), it can be shown that @p@pi ¼ 1=2 > 0, that is, manufacturer i should increase her price in proportion to an increase in the rival’s price. In other words, 3i there should be a positive (horizontal) interdependency between the manufacturers’ sales prices. Lemma 1. Manufacturer i’s optimal non-cooperative price, i ¼ 1; 2, satisfies:
pi ¼
a 1 2ci c3i d þ þ ðQ Q 3i Þ : þ i x 3i b i b 3 x
ð10Þ
Proof. See the Appendix. h This rule positively relates manufacturer i’s price to the differential between own quality and the rival’s quality, i.e., a manufacturer with a larger (resp., lower) quality level than her rival should set a higher (resp., lower) sale price. Manufacturer i’s pricing policy is implicitly
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dynamical, i.e., the rate of change of manufacturer i’s sales price should vary proportionally with the relative instantaneous efforts to manufacturer i’s product quality, as:
p_ i ¼
d ðui þ v i u3i v 3i Þ: 3b
Finally, for similar quality levels, manufacturer i’s sales price is positively influenced not only by her own contract parameter values, but also to a certain extent by the rival’s. Lemma 2. Assuming symmetric manufacturers, it holds that:
pij0
cijx
cij0
i ¼0
ð11Þ ; i ¼ 1; 2.
Proof. See the Appendix. h According to (11), the optimal sales price should be lower under a RSC than under a WPC whenever the supplier’s share in manufacturer i’s sales revenue is not larger than the rate of decrease of the transfer price from a WPC to a RSC. This result is in line with the existing literature (e.g., Kogan and Tapiero, 2007), since it shows that the RSC effectively mitigates the double marginalization effect. Plugging (10) into (3) gives manufacturer i’s optimal demand:
Di ¼ a
1 ci c3i dðQ i Q 3i Þ ; b i x 3i 3 x
ð12Þ
i ¼ 1; 2, that is expected to be positive. If the manufacturers’ contract parameters are similar, a manufacturer with a larger (resp., lower) quality level than her rival should benefit from larger (resp., lower) sales. On the other hand, for similar quality levels, manufacturer i’s demand is negatively (resp., positively) influenced by her own (resp., the rival’s) contract parameter values. Note that the rate of change of manufacturer i’s demand should also evolve as a linear function of the relative instantaneous efforts to manufacturer i’s product quality, that is:
d _ ¼ ðui þ v i u3i v 3i Þ i ¼ 1; 2: D_ i ðtÞ ¼ bpðtÞ 3 Lemma 3. The dynamics of the non-cooperative manufacturer i’s improvement policy, i ¼ 1; 2, is:
u_ i ¼
i i 2dx 1 c c 2dx a b i 3i dðQ i Q 3i Þ ¼ Di : 3 3bbi xi x3i 3bbi
ð13Þ
Proof. See the Appendix. h Whatever the nature of the contract between manufacturer i and the supplier (either a WPC or a RSC), the non-cooperative manufacturer i’s improvement effort should be initially high and decreasing over time until ui ðTÞ ¼ hi =bi is reached, i ¼ 1; 2. If one manufacturer has a larger value on both contract parameters than the rival, or a larger value than the rival on one parameter and a similar value on the other parameter, her improvement effort should start from a lower initial value than the rival. In other words, the larger the compensation offered by a manufacturer to the supplier, the lower the manufacturer’s improvement effort. Comparing the specific case where one manufacturer has a WPC and the other a RSC (e.g., c1 > c2 and 0 ¼ x1 < x2 ), all other things being equal (i.e., b1 ¼ b2 \ Q 10 ¼ Q 20 \ h1 ¼ h2 ), it can be shown that the improvement effort decline rate for the manufacturer with a WPC should be larger than that for the manufacturer with a RSC (i.e., u_ 1 < u_ 2 ), that is, a manufacturer using a WPC should make a greater quality improvement effort than a manufacturer using a RSC. We now turn to the determination of optimal improvement effort strategies v 1 ðtÞ and v 2 ðtÞ for the supplier. Substituting (10) for optimal sales prices into (6), the supplier’s Hamiltonian is:
Hs ¼
2 2 2 X v 2i X 1 X ð3 xi Þci c3i ci c3i xi 3a þ b þ wi ðui þ v i Þ; þ þ dðQ i Q 3i Þ 3a b þ dðQ i Q 3i Þ d i 3i i x 3i 9b i¼1 xi x x x 2 i¼1 i¼1
ð14Þ
where wi wi ðtÞ; i ¼ 1; 2, are the supplier’s costate variables. The supplier’s necessary conditions for optimality are:
Hsv i ¼ dv i þ wi ¼ 0 i ¼ 1; 2:
ð15Þ
The supplier’s Hamiltonian is strictly concave with respect to the supplier’s control variables. Lemma 4. The dynamics of the non-cooperative supplier’s contribution policy to manufacturer i’s quality improvement, i ¼ 1; 2, is:
v_ i ¼
( ! ) X d 2ðxi x3i Þ ci c3i ½3a þ dðQ i Q 3i Þ þ 3 2 : xi 9d b xi x3i i
ð16Þ
Proof. See the Appendix. h At the end of the planning horizon, the supplier’s contribution effort to manufacturer i’s quality improvement should equal i ¼ 1; 2. The supplier’s contribution policy depends on the value of the contract parameters with the manufacturers (Table 1). If the supplier has similar contract parameter values with both manufacturers, i.e., both c1 ¼ c2 and x1 ¼ x2 , her contribution to both manufacturers’ quality improvement activity should be constant over the entire planning horizon, i.e., v i ðtÞ ¼ gi =d; 8t 2 ½0; T; i ¼ 1; 2. This
v i ðTÞ ¼ gi =d;
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F. El Ouardighi, B. Kim / European Journal of Operational Research 206 (2010) 329–340 Table 1 Supplier’s effort allocation over time.
c1 < c2 c1 ¼ c2 c1 > c2
x1 < x2
x1 ¼ x2
x1 > x2
v_ 1 P 0; v_ 2 6 0j06xi 63=4 v_ 1 P 0; v_ 2 6 0j06xi 63=4 v_ 1jx1 ¼0 > 0; v_ 2jc2 >0 < 0
v_ 1 P 0; v_ 2 6 0j06xi 63=4 v_ i ¼ 0 v_ 1 6 0; v_ 2 P 0j06xi 63=4
v_ 1jc >0 < 0; v_ 2jx ¼0 > 0 v_ 1 6 0; v_ 2 P 0j06x 63=4 v_ 1 6 0; v_ 2 P 0j06x 63=4 1
2
i i
result prevails under any symmetric configuration (either WPC or RSC). A symmetric compensation scheme should lead the monopolist supplier to provide each of the competing manufacturers with a constant contribution which does not depend on the contract parameters, but rather on her own marginal salvage value for each manufacturer’s terminal quality. Given this symmetry effect, whenever the manufacturers are in a position to offer similar contractual conditions to the monopolist supplier, their only lever to ensure that the supplier will make a positive contribution to their respective quality improvement activities is to provide her with a maximum incentive to achieve their respective terminal design quality level. On the other hand, if the supplier has either a larger value on both contract parameters with one manufacturer than with the other, or a larger value on one contract parameter with one manufacturer and a similar value on the other contract parameter with both manufacturers, her contribution effort to this manufacturer’s quality improvement activity should be initially high and decreasing over time, while her contribution effort to the other manufacturer’s quality improvement activity should be initially low and increasing over time. Finally, if the supplier has a larger (resp., lower) transfer price and a lower (resp., larger) revenue share with one manufacturer than with the other, her contribution effort path to both manufacturers’ quality improvement activity is ambiguous. However, in the specific case where one manufacturer has a WPC and the other manufacturer has a RSC (e.g., c1 > c2 and 0 ¼ x1 < x2 ), it can be shown, for similar manufacturers (i.e., Q 10 ¼ Q 20 \ h1 ¼ h2 ) that the rates of change of the supplier’s contribution to the competing manufacturers have opposite signs, i.e., v_ 1jWPC > 0; v_ 2jRSC < 0Þ. As a consequence, a manufacturer using a RSC should receive a larger contribution from the monopolist supplier than a rival manufacturer having a WPC. To summarize our results from the qualitative analysis, the monopolist supplier should contribute less to the quality improvement activity of the manufacturer offering the less rewarding contractual conditions, and vice versa. Meanwhile, the less rewarding manufacturer should make a greater quality improvement effort to compensate for the supplier’s smaller effort, in order to maintain its competitive position with respect to the rival manufacturer. Clearly, depending on the contractual conditions, there should be a direct substitutability between the monopolist supplier’s respective contributions to the competing manufacturers, as well as between each manufacturer’s quality improvement effort and the monopolist supplier’s contribution effort. In this sense, a manufacturer’s improvement effort path is indirectly determined by the rival manufacturer’s contractual conditions for the monopolist supplier, since she has to compensate for the supplier’s shortfall. This result is notably valid when one manufacturer has a WPC and the other has a RSC. Proposition 1 below characterizes the explicit behavior of the three players over time for any feasible contract parameter value. Proposition 1. The non-cooperative quality improvement policies of the duopolist manufacturers and the monopolist supplier are:
u1 ðtÞ ¼ ðk1 C h1 ÞðT tÞ þ k1 fA½sinðwTÞ sinðwtÞ B½cosðwTÞ cosðwtÞg=w þ h1 =b1 ;
ð17Þ
u2 ðtÞ ¼ ðh2 þ k2 CÞðT tÞ k2 fA½sinðwTÞ sinðwtÞ B½cosðwTÞ cosðwtÞg=w þ h2 =b2 ;
ð18Þ
v 1 ðtÞ ¼ ðm þ nCÞðT tÞ þ nfA½sinðwTÞ sinðwtÞ B½cosðwTÞ cosðwtÞg=w þ g1 =d; v 2 ðtÞ ¼ ðnC mÞðT tÞ þ nfA½sinðwTÞ sinðwtÞ B½cosðwTÞ cosðwtÞg=w þ g2 =d;
ð19Þ
xi d h1 1 with A ¼ Q 10 Q 20 C; B ¼ w cosðwTÞ bh22 þ g1 d g2 þ A tanðwTÞ; C ¼ h1 hw222m ; hi ¼ 23b b1 i P ffiffiffiffiffiffiffiffiffiffi p P ð32 i xi Þ c1 2ðx1 x2 Þd2 c2 ; w ¼ k ; n ¼ ; i ¼ 1; 2. i 1 2 i 3 x x 9db
h
ð20Þ a1 b
3
ci
i x
i xc3i ; ki ¼ 3i
i d2 2x 9bbi
d ; m ¼ 3d
h
2ðx1 x2 Þa þ b
Proof. See the Appendix. According to Proposition 1, under any symmetric configuration (either WPC or RSC) and for perfectly identical manufacturers (i.e., b1 ¼ b2 \ Q 10 ¼ Q 20 \ h1 ¼ h2 \ g1 ¼ g2 ), manufacturer i’s improvement effort, that is:
ui ðtÞ ¼
i daðT tÞ 1 2x þ hi ; bi 3b
ð21Þ
i ¼ 1; 2; should be initially high and monotonically decreasing over time until its terminal value ui ðTÞ ¼ hi =bi , while the supplier’s contribution should remain time-invariant at v i ðtÞ ¼ gi =d. Accordingly, the total efforts dedicated to manufacturer i’s quality improvement activity should be strictly lower under symmetric RSC than under symmetric WPC. Whatever the compensation scheme, the manufacturer i’s improvement effort is positively influenced by the marginal sensitivity of demand for quality, but negatively impacted by the marginal sensitivity of demand with respect to price. Also, the longer the planning horizon, the greater the manufacturer i’s improvement effort. Under similar conditions but asymmetric manufacturers’ marginal salvage values (i.e.,b1 ¼ b2 \ Q 10 ¼ Q 20 \ h1 – h2 \ g1 ¼ g2 Þ, manufacturer i’s improvement effort, that is:
ui ðtÞ ¼
i adðT tÞ 1 4x cosðwtÞ þ hi þ h3i þ ðhi h3i Þ ; 2bi 3b cosðwTÞ
ð22Þ
i ¼ 1; 2; should be initially high and periodically instead of monotonically decreasing over time. If the planning horizon is not too large, i.e., qffiffiffiffiffi i such that T < 34dp bb i , the manufacturer with the larger marginal salvage value should make a larger improvement effort than the rival over x h i 3i Þ cosðwtÞ Note that the requirement of not too large planning horizon is in line with our time, with a differential of magnitude ðhi h 2b cosðwTÞ . i
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assumption of negligible time value of money. For the case of larger planning horizon, the effect of discounting should be taken into account, which is not the case in our model. In this respect, the case of larger planning horizon is disregarded hereafter. The partial derivative of manufacturer i’s improvement effort with respect to her own revenue share is given by:
uix i ¼
1 4adðT tÞ ðhi h3i Þ sin ½wðT tÞ : þ 2bi 3b cos2 ðwTÞ
The manufacturer with the larger marginal salvage value should make a larger improvement effort under symmetric WPC over time than under symmetric RSC. Under such conditions, the rival manufacturer, i.e., with the lower marginal salvage value, may nevertheless have a larger improvement effort under symmetric RSC than under symmetric WPC. The previous results generalize to any symmetric compensation scheme (either WPC or RSC) with both asymmetric manufacturers’ marginal salvage values and supplier’s marginal salvage values (i.e., b1 ¼ b2 \ Q 10 ¼ Q 20 \ h1 – h2 \ g1 –g2 ), whenever the asymmetry in the manufacturers’ marginal salvage values is consistently reflected in that of the supplier’s marginal salvage values, i.e., gi >jhi >h3i g3i . This is a plausible assumption since the manufacturer with a larger terminal goodwill reward is more likely to provide the supplier with a larger incentive on the terminal design quality level than the rival manufacturer. Manufacturer i’s improvement effort, i ¼ 1; 2, then writes:
ui ðtÞ ¼
i adðT tÞ 1 2x 1 hi h3i gi g3i cosðwtÞ : 1 þ þ hi bi 3b 2 cosðwTÞ bi d
ð23Þ
Here also, given that the supplier’s contribution effort to the manufacturers’ quality improvement activity should be constant over time under any symmetric configuration, the manufacturer with the larger marginal salvage value and the larger supplier’s marginal salvage value should hold the leading position for quality improvement effort, as described above. Consider now a symmetric configuration with perfectly identical manufacturers except for their initial quality levels (i.e., b1 ¼ b2 \ Q 10 – Q 20 \ h1 ¼ h2 \ g1 ¼ g2 ). In these conditions, manufacturer i’s improvement effort, that is:
( ) i d 2aðT tÞ ðQ 10 Q 20 Þ sin ½wðT tÞ hi x pffiffiffiffiffiffiffiffiffiffiffiffi ; ui ðtÞ ¼ þ þ 3 bi b bi i bbi cosðwTÞ x
ð24Þ
i ¼ 1; 2; should be initially high and periodically decreasing over time. For not too large planning horizon, the improvement effort of the man pffiffiffiffi x dðQ 10 Q 20 Þ sin ½wðTtÞ ; 8t < T. ffiffiffiffiffi ufacturer with the greater initial design quality level should be larger than the rival’s, with a gap of i p 3 bbi cosðwTÞ On the whole, three effects have been identified, that is: - The main effect, as described in Eq. (21), which combines the influences of the potential demand, the sensitivity of demand with respect to price and quality, and the supplier’s share in the manufacturer i’s revenue, - The effect of the differential between the manufacturers’ marginal salvage values and the supplier’s marginal salvage values, as shown in Eq. (23), and - The effect of the differential between the manufacturers’ initial design quality levels, as given in Eq. (24). These influences add to the asymmetry effect due to the compensation schemes, if any. Note that the length of the planning horizon has a significant influence on the outcome of the quality duopolistic competition game, which requires taking into account the discounting effect for large enough planning horizon. Finally, from (19) and (20), it can be shown that the supplier’s contribution policy to each manufacturer’s quality improvement activity under asymmetric compensation schemes may have a non-monotonic path, notably if the manufacturers’ marginal salvage values and the supplier’s marginal salvage values are different. Proposition 2 below determines the optimal path for the manufacturers’ design quality. Proposition 2. The non-cooperative design quality path for each manufacturer’s product is:
h1 g1 1 n þ k1 n þ k1 þ ½h1 ðn þ k1 ÞC m½2T t þ ½A cosðwtÞ þ B sinðwtÞ A; ð25Þ ½A sinðwTÞ B cosðwTÞ t þ b1 d 2 w w2 h2 g2 1 n k2 n k2 ½A sinðwTÞ B cosðwTÞ t þ ½A cosðwtÞ þ B sinðwtÞ A: ð26Þ Q 2 ðtÞ ¼ Q 20 þ þ ½h2 ðn k2 ÞC þ m½2T t þ b2 d 2 w w2 Q 1 ðtÞ ¼ Q 10 þ
Proof. See the Appendix. Under symmetric WPC and perfectly identical manufacturers (i.e., b1 ¼ b2 \ Q 10 ¼ Q 20 \ h1 ¼ h2 \ g1 ¼ g2 ), manufacturer i’s design quality writes:
Q i ðtÞ ¼ Q i0 þ
i da hi g i x þ þ ð2T tÞ t; bi d 3bi b
ð27Þ
i ¼ 1; 2; that is, manufacturer i’s design quality should be increasing at a decreasing rate over time. Further, design quality under symmetric WPC should be larger than under symmetric RSC at any time period, and whatever the length of the planning horizon. Under similar conditions but asymmetric manufacturers’ marginal salvage values and supplier’s marginal salvage values (i.e.,b1 ¼ b2 \ Q 10 ¼ Q 20 \ h1 – h2 \ g1 – g2 ), manufacturer i’s design quality is given by:
"
Q i ðtÞ ¼ Q i0 þ
# pffiffiffiffiffiffiffi 2 i da 1X hi gi x 3 bbi sinðwtÞ hi h3i gi g3i þ : þ þ ð2T tÞ t þ pffiffiffiffiffiffi i cosðwTÞ 2 i¼1 bi d 3bi b bi d 4d x
ð28Þ
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Under not too large planning horizon, the quality level of the manufacturer with the larger marginal salvage value and/or the larger supplier’s marginal salvage value should be larger than that of the rival, 8t 6 T. Under symmetric manufacturers except for initial quality levels (i.e., b1 ¼ b2 \ Q 10 – Q 20 \ h1 ¼ h2 \ g1 ¼ g2 ), manufacturer i’s design quality is given by:
Q i ðtÞ ¼ Q i0 þ
i dað2T tÞ hi gi x 1 þ þ t þ ðQ i0 Q 3i0 Þ½wt sinðwTÞ þ cosðwtÞ 1: 3bbi 2 bi d
ð29Þ
Here also, for not too large planning horizon, the quality level of the manufacturer with the greater initial design quality level should be larger than the rival’s at any time period. For each manufacturer, the design quality path may then follow either monotonous or oscillatory increasing trend until:
Q i ðTÞ ¼ Q i0 þ
hi gi 1 n þ oj ki n þ oj k1 þ hi ðn þ oj ki ÞC oj m T þ ½A cosðwTÞ þ B sinðwTÞ A ½AsinðwTÞ B cosðwTÞ T þ bi d 2 w w2
ð30Þ
is reached at the end of the planning horizon, i ¼ 1; 2; j ¼ 1; 2; o1 ¼ 1 and o2 ¼ 1. Lemma 5. Under perfectly symmetric configurations, it holds that:
Q i ðTÞjxi ¼0 > Q i ðTÞj0
i ¼ 1; 2:
ð31Þ
Proof. See the Appendix. According to (31), the terminal design quality under a symmetric WPC configuration is greater than under a symmetric RSC configuration. This result is due to the fact that both manufacturers make less quality improvement effort under a RSC than under a WPC, while the monopolist supplier’s contribution effort to both manufacturers’ improvement activities remains similar under any symmetric configuration. On the whole, by mitigating the double marginalization effect, a compensation scheme based on RSC can indeed lead to lower sales prices than under a WPC, but this price reduction should be associated with a lower terminal design quality level. This result contradicts those obtained for a different dimension of quality, namely conformance quality (El Ouardighi et al., 2008), where it is shown that a RSC gives rise to a higher pricing policy than a WPC, but is associated with a higher conformance quality policy. This suggests that the way players are compensated may have conflicting impacts on different quality dimensions of a product. 4. Numerical illustrations Numerical examples are provided to illustrate the players’ non-cooperative policies in quality improvement under WPC and RSC. The preference between the two contract arrangements is also evaluated for both the supplier and the manufacturers. The parameter values of the game are given in Table 2. For a sensitivity analysis, a broad range of values is used for all parameters, as well as for the initial values of state variables. In order to test the robustness of our results with respect to the length of the planning horizon, the duration of the game isffi set at a sufficiently large extent, that is, 20 time units, t 2 ½0; 20, which for the whole range of parameters pffiffiffiffiffiffi pffiffiffiffiffiffi i. values is such that T > 3p bbi =4d x The evaluation of the impact of the compensation structure on players’ competitive behavior leads us to consider three potential configurations, that is: - Both manufacturers agree on a WPC with the supplier, i.e., c1 > 0; c2 > 0 and x1 ¼ x2 ¼ 0 (configuration 1, or C1), - Both manufacturers agree on a RSC with the supplier, i.e., c1 > 0; c2 > 0; x1 > 0 and x2 > 0 (C2), - Manufacturer 1 agrees on a RSC and manufacturer 2 agrees on a WPC with the supplier, i.e., c1 > 0; c2 > 0; x1 > 0 and x2 ¼ 0 (C3). The following four figures depict the typical dynamics of control and state variables under each configuration, within the ranges of values defined above. Fig. 1 shows that both manufacturers have a monotonically decreasing quality improvement effort policy over time. Both manufacturers make a greater improvement effort over time under symmetric WPCs than under symmetric RSCs. However, the manufacturer with an asymmetric WPC makes more effort than under a symmetric WPC, while the manufacturer with an asymmetric RSC makes less effort than under the symmetric RSC. The asymmetric configuration cannot be seen as a simple combination of the two symmetric configurations. In Fig. 2, due to a symmetry effect, the path of the supplier’s contribution effort to both manufacturers’ quality improvement activity is unaffected by the compensation structure, and constant paths are observed in both symmetric configurations. In the asymmetric configuration, the supplier’s contribution paths take opposite directions: the supplier’s contribution effort for the manufacturer with an asymmetric WPC shows a monotonic increase from a low initial level, and the supplier’s contribution effort for the manufacturer with an asymmetric RSC shows a monotonic decrease from a high initial level. The asymmetric configuration thus gives rise to unequal treatment of the competing manufacturers by the monopolist supplier, which generates substitutability between each manufacturer’s quality improvement effort and the monopolist supplier’s contribution effort. This effect is ultimately reflected in a negative mutual dependency between the duopolist manufacturers’ respective quality improvement efforts, as described in the previous figure. Table 2 Parameter values. Parameter
ci
xi
a
b
d
bi
d
hi
gi
Q i0
Low value High value
50 80
0 0.2
100 1000
0.5 2
0.5 2
10 20
10 20
100 200
50 100
1 10
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70 u1|C1
60
ui(t)
u2|C1
50
u1|C2
40
u1|C3
u2|C2 u2|C3
30 20 10 0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
t Fig. 1. Manufacturers’ quality improvement effort over time.
16 14 12 10
vi(t)
8
v1|C1
6
v2|C1 v1|C2
4
v2|C2
2
v1|C3 v2|C3
0 0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
t Fig. 2. Supplier’s contribution efforts over time.
According to Fig. 3, each manufacturer should apply a constant pricing policy over time under both symmetric configurations. Furthermore, the pricing policy should be higher under a symmetric WPC than under a symmetric RSC. Under the asymmetric configuration, the manufacturers’ pricing policies should start from close initial values and then evolve in opposite directions. Under an asymmetric WPC, the sales price should increase to reach a significantly higher level than under a symmetric WPC, while under asymmetric RSC, it should decrease to a lower level than under a symmetric RSC. According to Fig. 4, the design quality should follow an increasing path in all the configurations. With symmetric WPCs, quality should be larger than under symmetric RSCs. All other things being equal, this result is valid whatever the marginal sensitivity of sales to price or quality. Duopolist manufacturers should therefore adopt a price-reduction strategic orientation under a symmetric RSC, and a qualityimprovement strategic orientation under a symmetric WPC. In the asymmetric configuration, the manufacturer with a WPC also performs better than her rival with a RSC. This suggests that the enhanced contribution effort of the monopolist supplier to the manufacturer with a RSC is outweighed by the improvement effort of the manufacturer with a WPC. Hence, the manufacturer with a WPC sets a higher sale price than the manufacturer with a RSC not because of a lack of coordination with the monopolist supplier, but rather because she holds a competitive advantage on quality.
250 200 150 p1|C1
pi(t)
p2|C1
100
p1|C2 p2|C2
50
p1|C3 p2|C3
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
t Fig. 3. Sales price policies over time.
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900
Qi(t)
800
Q1|C1
700
Q2|C1
600
Q1|C2 Q2|C2
500
Q1|C3
400
Q2|C3
300 200 100 0 0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
t Fig. 4. Design quality paths over time. Table 3 Players’ total payoff under WPC and RSC. Manufacturer 2
Manufacturer 1
WPC ðc1 ¼ 80; x1 ¼ 0Þ RSC ðc1 ¼ 50; x1 ¼ 0:2Þ
WPC ðc2 ¼ 80; x2 ¼ 0Þ
RSC ðc2 ¼ 50; x2 ¼ 0:2Þ
J1 ¼ 291969; J 2 ¼ 291969; JS ¼ 405294 J1 ¼ 255078; J2 ¼ 272328; J S ¼ 445464
J 1 ¼ 272328; J2 ¼ 255078; J S ¼ 445464 J 1 ¼ 242813; J2 ¼ 242813; J S ¼ 490905
In Table 3, the players’ total payoff is determined for each of the three configurations under specific values of the contract parameters. It is noteworthy that in all configurations, the greater the sensitivity of demand with respect to quality, the lower the manufacturers’ profits, and the larger the supplier’s profit. Furthermore, the greater the sensitivity of demand with respect to price, the lower the profits for both manufacturer and supplier. On the other hand, the symmetric WPC configuration is the least profitable to the supplier, while the symmetric RSC configuration is the most beneficial. In other words, the symmetric WPC configuration represents an economic equilibrium for the manufacturers, while the symmetric RSC configuration represents an economic equilibrium for the supplier. Assuming that both manufacturers have a WPC, there is a disincentive for both of them to adopt a RSC, so that both the asymmetric configuration and the symmetric RSC configuration appear unfeasible. Table 3 implies that there might be a potential conflict between the manufacturers and their supplier. If collaboration between the manufacturers is permitted, each manufacturer has a great incentive to set up a WPC with the supplier (cell WPC-WPC in Table 3) so as to maximize her profit. On the other hand, the supplier will try to sign a RSC with each manufacturer separately so as to maximize her own profit (cell RSC-RSC in Table 3). Even if explicit collaboration between the manufacturers were not allowed, a rational decision-maker, i.e., manufacturer, would be able to see the potential gain in aligning her strategy with her counterpart’s. Similarly, a rational supplier would be aware of the possibility of such implicit collaboration between the manufacturers (or at least, of each manufacturer’s implicit assumption that collaboration would be possible). As such, our analysis enables the players, both the manufacturers and the supplier, to assess the limitations as well as the potential of their strategic choices, and to use compromise to find a better solution giving all players profits larger than in extreme situations (i.e., either WPC-WPC or RSC-RSC). 5. Conclusions This paper has adopted the perspective of interdependent, competing supply chains, where a monopolist supplier dynamically collaborates with duopolist manufacturers in improving the design quality of their respective products. It was assumed that the manufacturers compete for the final demand both on sale price and design quality. The transactions between each manufacturer and the supplier are determined under a two-parameter approach. The objective of the paper was twofold: - First, to determine the three players’ optimal effort allocation for quality improvement in a decentralized context; - Second, to evaluate how the compensation structure adopted influences the competitive dynamics of the three players’ game. Using analytical means, it has been shown that a supplier should take advantage of her monopolistic position by applying a discriminatory contribution policy to the respective quality improvement activities of the competing manufacturers. It is thus not in the best interests of rival manufacturers to offer similar contractual conditions to the monopolist supplier, since it gives rise to a symmetry effect which leads the supplier to determine her contribution effort to each manufacturer’s quality improvement activities as a function not of the contract parameters, but rather of her salvage value for the manufacturer’s terminal design quality level (i.e., goodwill reward). If identical contractual conditions are unavoidable, the manufacturers should negotiate them at their lowest value, and provide the supplier with a maximum incentive to contribute to their respective terminal design quality level. Supply quality competition then emerges as a combination of supply quality management and horizontal market competition, where two forms of interdependencies take place: A negative horizontal direct interdependency between the contribution efforts of the monopolist supplier to the manufacturers’ respective improvement activities, i.e., the better the contractual conditions with one manufacturer relatively to the other manufacturer, the heavier (resp., the lower) the supplier’s contribution effort to the improvement activity of this manufacturer (resp., the other manufacturer).
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A negative vertical direct interdependency between each manufacturer’s improvement effort and the monopolist supplier’s improvement effort, i.e., the heavier (resp., the lower) the supplier’s contribution effort to one manufacturer’s improvement activity, the lower (resp., the heavier) the improvement effort of this manufacturer. These two interdependencies give rise to a negative horizontal indirect mutual dependency between the duopolist manufacturers’ respective improvement efforts, i.e., the manufacturer which offers the poorer contractual conditions should make a greater improvement effort to outweigh the unfavorable supplier’s effort contribution policy, in order to maintain its competitive position relatively to the rival manufacturer. A manufacturer’s improvement effort path should not be determined only by her own contractual conditions with the monopolist supplier, but also by the rival manufacturer’s contractual conditions. Given the contrast in pricing policy between the WPC and the RSC, the presence of a double marginalization effect could have been wrongly diagnosed. Justification for a higher price under a WPC is found in the existence of a competitive advantage on design quality: the WPC entails a quality premium, reflected in a higher sales price. This result contradicts the static analysis of Economides (1999) according to which quality is negatively affected by the presence of a double marginalization effect in a bilateral monopoly setting. Actually, we show that, under dynamic horizontal competition, the double marginalization effect appears as a potentially misleading concept; since it may lead to a price differential between competing manufacturers being mistakenly interpreted as a consequence of a lack of vertical coordination, rather than a competitive advantage on quality. The WPC is more beneficial to the duopolist manufacturers, while the RSC is more profitable for the monopolist supplier. There is a negative incentive for the manufacturers to adopt a RSC, and it is thus not a coordinating contract in the context of a monopolist supplier serving duopolist manufacturers. Although the three players’ joint profits can be significantly increased, turning the supply chain into a revenue chain (Cachon and Lariviere, 2001) may result in lower design quality levels, and be detrimental to the manufacturers’ profits. Our results clearly show that RSC is not necessarily the perfect solution for supply quality management under horizontal competition. Another important implication here is that ‘‘one should not take it for granted that a successful mechanism in a one shot game works in the same manner in a dynamic game” (Zaccour, 2008). Although a different definition of quality was used, the results obtained are similar to those in El Ouardighi et al. (2008) in the sense that this study also finds that the positive interaction between the supplier’s transfer price and the manufacturers’ prices, i.e., the double marginalization effect, is not clearly visible under a WPC. Interestingly, the comparison with El Ouardighi et al. (2008) also suggests that WPC performs better on design quality and worse on conformance quality than RSC and vice versa. The formulation of a general demand function which would include both design quality and conformance quality could therefore open up a promising avenue for future research, for a more accurate comparison of the respective strengths and limitations of WPC and RSC in terms of both supply quality management and profit distribution among the chains’ members. Appendix Proof of Lemma 1. Using (8) to determine pi and substituting the corresponding expression for the rival manufacturer, p3i ; i ¼ 1; 2, gives (10). h Proof of Lemma 2. Obvious from (10). h Proof of Lemma 3. Substituting (10) for optimal sales price into (7) yields:
Hi ¼
2 bi u2i ci c3i 3a b þ li1 ðui þ v i Þ þ li2 ðu3i þ v 3i Þ; þ dðQ i Q 3i Þ i x 3i 9b x 2 i x
i ¼ 1; 2; which reveals the linear quadratic structure of the game. The costate variables are derived as follows:
id 2x ci c3i 3a b þ dðQ i Q 3i Þ i x 3i 9b x id 2x ci c3i ¼ 3a b þ dðQ i Q 3i Þ i x 3i 9b x
l_ i1 ¼ HiQ i ¼
li1 ðTÞ ¼ hi ;
l_ i2 ¼ HiQ 3i
li2 ðTÞ ¼ hi ;
i ¼ 1; 2. Differentiating (9) with respect to time and substituting for l_ i1 and simplifying gives (13).
h
Proof of Lemma 4. The supplier’s costate variables are derived from (14) as follows:
( ! ) X c1 c2 _w1 ¼ Hs ¼ d ðx1 x2 Þ ½6a þ 2dðQ 1 Q 2 Þ þ 3 2 w1 ðTÞ ¼ g1 ; xi Q1 9 b x1 x2 i ( ! ) X c2 c1 _w2 ¼ Hs ¼ d ðx2 x1 Þ ½6a þ 2dðQ 2 Q 1 Þ þ 3 2 w2 ðTÞ ¼ g2 : xi Q2 9 b x2 x1 i Differentiating (15) with respect to time and substituting for w_ i ; i ¼ 1; 2, and simplifying gives (16).
h
Proof of Proposition 1. Subtracting (2) from (1), differentiating the resulting expression with respect to time, and substituting (13) and (16) for u_ i and v_ i ; i ¼ 1; 2, gives:
€ 2 ðtÞ þ ðk1 þ k2 ÞðQ ðtÞ Q ðtÞÞ ¼ h1 h2 2m: € 1 ðtÞ Q Q 1 2
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Given ki > 0; i ¼ 1; 2, this equation has the solution:
Q 1 ðtÞ Q 2 ðtÞ ¼ A cosðwtÞ þ B sinðwtÞ þ where w2 ¼
P
i ki ,
h1 h2 2m ; w2
for which only the positive root, i.e., w > 0, is considered. Using the initial conditions Q i0 > 0; i ¼ 1; 2, we get:
A ¼ Q 10 Q 20
h1 h2 2m : w2
Using the following derivations:
Q_ 1 ðtÞ Q_ 2 ðtÞ ¼ Aw sinðwtÞ þ Bw cosðwtÞ; Q_ 1 ðTÞ Q_ 2 ðTÞ ¼ u1 ðTÞ þ v 1 ðTÞ u2 ðTÞ v 2 ðTÞ ¼ h1 =b1 þ g1 =d h2 =b2 g2 =d; p modp, we obtain: and assuming T – 2w
B¼
h1 =b1 þ g1 =d h2 =b2 g2 =d h1 h2 2m tanðwTÞ: þ Q 10 Q 20 w cosðwTÞ w2
Substituting the expression of Q 1 ðtÞ Q 2 ðtÞ into (13) and (16), i ¼ 1; 2, gives:
u_ 1 ðtÞ ¼ h1 k1 ½C þ A cosðwtÞ þ B sinðwtÞ; u_ 2 ðtÞ ¼ h2 þ k2 ½C þ A cosðwtÞ þ B sinðwtÞ;
v_ 1 ðtÞ ¼ m n½C þ A cosðwtÞ þ B sinðwtÞ; v_ 2 ðtÞ ¼ m n½C þ A cosðwtÞ þ B sinðwtÞ; where:
C¼
h1 h2 2m : w2
These four equations are integrated using their boundary conditions to obtain (17)–(20). Proof of Proposition 2. To solve (1) and (2), we first compute U i ðtÞ ¼ ui0 þ yields:
Rt 0
h
ui ðsÞds and V i ðtÞ ¼ v i0 þ
Rt 0
v i ðsÞds, respectively, i ¼ 1; 2, which
1 k1 h1 k1 þ ½A sinðwTÞ B cosðwTÞ t þ K 1 ; ðh1 k1 CÞðt TÞ2 þ 2 ½A cosðwtÞ þ B sinðwtÞ þ 2 w b1 w 1 k2 h2 k2 2 ½A sinðwTÞ B cosðwTÞ t þ K 2 ; U 2 ðtÞ ¼ ðh2 þ k2 CÞðt TÞ 2 ½A cosðwtÞ þ B sinðwtÞ þ 2 w b2 w ng o 1 n n 2 V 1 ðtÞ ¼ ðm þ nCÞðt TÞ þ 2 ½A cosðwtÞ þ B sinðwtÞ þ 1 þ ½A sinðwTÞ B cosðwTÞ t þ K 3 ; 2 w d w ng o 1 n n 2 2 þ ½A sinðwTÞ B cosðwTÞ t þ K 4 ; V 2 ðtÞ ¼ ðm nCÞðt TÞ þ 2 ½A cosðwtÞ þ B sinðwtÞ þ 2 w d w
U 1 ðtÞ ¼
where K 1 ; . . . ; K 4 are constants of integration. Substituting the corresponding expressions in (1) and (2), respectively, and solving gives:
1 n þ k1 h1 g1 n þ k1 ½h1 ðn þ k1 ÞC mðt TÞ2 þ ½ A cosðwtÞ þ B sinðwtÞ þ þ þ ½ A sinðwTÞ B cosðwTÞ t þ L1 ; 2 w2 b1 d w 1 n k2 h2 g2 n k2 ½A cosðwtÞ þ B sinðwtÞ þ þ þ ½A sinðwTÞ B cosðwTÞ t þ L2 ; Q 2 ðtÞ ¼ ½h2 ðn k2 ÞC þ mðt TÞ2 þ 2 w2 b2 d w
Q 1 ðtÞ ¼
where L1 and L2 are constants of integration. Using the initial values Q i0 ; i ¼ 1; 2, to determine L1 and L2 leads to (25) and (26).
h
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