Quantity-payment versus two-part tariff contracts in an assembly system with asymmetric cost information

Transportation Research Part E 129 (2019) 60–80

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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre

Quantity-payment versus two-part tariff contracts in an assembly system with asymmetric cost information

T

Fei Lva, Lei Xiaob, , Minghui Xuc, , Xu Guand ⁎

⁎

a

School of Business Administration, Zhongnan University of Economics and Law, Wuhan 430073, China Business School, Hubei University, Wuhan 430062, China Economics and Management School, Wuhan University, Wuhan 430072, China d School of Management, Huazhong University of Science and Technology, Wuhan 430074, China b c

ARTICLE INFO

ABSTRACT

Keywords: Assembly system Asymmetric cost information Quantity-payment contract Two-part tariff contract

This paper investigates an assembly system that consists of one assembler and two suppliers wherein one supplier possesses private cost information. We explore how in such a setting, the contract type (quantity-payment versus two-part tariff) and contracting sequence (simultaneous versus sequential) between the assembler and its suppliers influence the channel and individual firms’ performances. Our results for the basic model show the following: (1) Coordinating the purchase quantities from both suppliers does not always increase the channel’s and the assembler’s profits. (2) The assembler obtains the highest profit under a quantity-payment contract with sequential contracting. (3) The supplier with private information and the channel both prefer a two-part tariff contract over a quantity-payment contract. We also extend our basic model to a case where the assembler contracts with one supplier under a two-part tariff contract and with the other under a quantity-payment contract. We identify the firms’ equilibrium decisions and preferences over different contract types and contracting sequences.

1. Introduction Today, most global supply chains are characterized by decentralized assembly systems wherein the assemblers outsource component production to several suppliers located in low-cost regions such as Southeast Asia and only undertake the final assembly work. The primary factor driving assemblers to this manufacturing strategy is the consideration of cost reduction. However, this manufacturing strategy also has some pitfalls, and the major one is efficiency loss due to conflicts of interests among different firms. In most assembly systems, the assemblers generally possess dominant market power as the channel leader and can extract most profits out of suppliers (Hu and Qi, 2018). To prevent the assemblers from extracting the total channel profit, some suppliers tend to not share their costs with the assemblers. Thus, information asymmetry arises that often hurts assemblers due to cost overruns (Fang et al., 2014), and there is a pressing need for the assembler to design an optimal procurement mechanism to induce truthful information sharing in an assembly system. Intuitively, the relationship among the purchase quantities from different suppliers should be an essential part of the manufacturer’s mechanism design process. Under the full information setting, if the assembler assembles one unit of each component into its final product, it is certain to purchase the same quantity of each component (Gerchak and Wang, 2004). However, should the assembler still coordinate purchase quantities of different inputs when one or more suppliers possess private cost information? Such a

⁎

Corresponding authors. E-mail addresses: [email protected] (L. Xiao), [email protected] (M. Xu).

https://doi.org/10.1016/j.tre.2019.07.010 Received 25 February 2019; Received in revised form 20 June 2019; Accepted 22 July 2019 1366-5545/ © 2019 Elsevier Ltd. All rights reserved.

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question has been of concern to some scholars recently. For example, Hu and Qi (2018) and Li et al. (2019) insist that with information asymmetry in supplier cost, ordering a different number of inputs from each supplier leads to potential inefficiencies for the assembler. However, Fang et al. (2014) design a quantity-payment contract menu to reveal all the suppliers’ costs in an assembly system, and the assembler orders different quantities from different suppliers under this contract. Motivated by this observation, our first research question follows: In an assembly system with information asymmetry in supplier cost, does the assembler always benefit from coordinating purchase quantities of different inputs? Moreover, in an assembly system, the assembler typically acts as the principal since it has substantial bargaining power offering a take-it-or-leave-it contract to suppliers (Nagarajan and Bassok, 2008). Furthermore, the assembler can contract with suppliers either simultaneously or sequentially. It is worth noting that different contracting sequences result in different asymmetric information structures (Hu and Qi, 2018). Under simultaneous contracting, the assembler does not know the costs of the suppliers with asymmetric information during the contracting process. However, under sequential contracting, the assembler is able to learn the costs of the suppliers with asymmetric information before contracting with the suppliers with full information. The above discussion indicates that the contracting sequence highly influences the assembler’s optimal mechanisms and further influences individual firms’ profits and the channel’s profit. This leads to our second research question: How do different contracting sequences affect individual firms’ and the channel’s performances in an assembly system with information asymmetry in supplier cost? Finally, whether the assembler coordinates purchase quantities of different inputs depends on the contractual agreement between it and the suppliers. Moreover, the assembler contracts with different suppliers either simultaneously or sequentially. Furthermore, in a sequential game, two scenarios can arise: the assembler first contracts with the suppliers with full information and then with the suppliers with asymmetric information or the reverse. Therefore, there are many contracting scenarios that are the combinations of different contract types and different contracting sequences emerging in an assembly system. However, it is unclear beforehand how these scenarios are preferred by the firms and the channel. This leads to our third research question: How do contract type interact with contracting sequence to determine individual firms’ and the channel’s performances in an assembly system with information asymmetry in supplier cost? To answer the above questions, we consider an assembly system that consists of one assembler and two suppliers, wherein one supplier possesses private cost information. We study and compare two basic types of contracts: a quantity-payment contract and a two-part tariff contract. We consider these contracts for two reasons. First, the quantity-payment contract is one of the most wellstudied supply chain contracts in the literature on mechanism design problem (e.g., Kalkanci and Erhun, 2012; Fang et al., 2014; Dong et al., 2018). Under this contract, due to information asymmetry, the assembler may be unable to purchase coordinated quantities from the suppliers. Second, the two-part tariff contract is commonly adopted in business-to-business practice (Yang and Ma, 2017) and is widely used to coordinate the decentralized supply chain (Biswas et al., 2018; Yang and Tang, 2019). Under this contract, the assembler always coordinates its purchase quantities from the suppliers (Hu and Qi, 2018). Moreover, the assembler’s contracting sequence can be either simultaneous or sequential under each contract. Therefore, for the assembly system, we study three scenarios under each contract: (i) in qp-sim (tpt-sim) the assembler contracts with two suppliers simultaneously; (ii) in qp-seq1 (tpt-seq1) the assembler first contracts with the supplier with full information and then with the supplier with private information; (iii) in qp-seq2 (tpt-seq2) the assembler first contracts with the supplier with private information and then with the supplier with full information. Here, qp-sim, qp-seq1 and qp-seq2 are scenarios under a quantity-payment contract, and tpt-sim, tpt-seq1 and tpt-seq2 are scenarios under a two-part tariff contract. Although Scenario qp-seq2 is governed by a quantity-payment contract, the assembler can also coordinate purchase quantities under this scenario because he can design the contract for the supplier with full information based on the reported cost of the supplier with asymmetric information. With this model setup, we derive our main findings as follows. First, coordinating the purchase quantities of two inputs does not always increase the channel’s and the assembler’s profits. In particular, the assembler may obtain a higher profit under Scenario qpsim than under Scenario tpt-sim. The channel may obtain a higher profit under Scenario qp-sim than under Scenario qp-seq2. The intuition behind this unintended result is as follows. In the absence of information asymmetry, the inefficiency in an assembly system arises from uncoordinated purchase quantities; hence the assembler and the channel both benefit from a coordinated purchasing strategy. However, in the presence of information asymmetry, there are two types of inefficiencies in an assembly system: information asymmetry and uncoordinated purchase quantities, and the negative effects of the two types of inefficiencies on the assembler and the channel vary under different contracting scenarios. Therefore, the contracting scenario preferences of the assembler and the channel are determined by the total effects of the two types of inefficiencies. Second, the assembler obtains the highest profit when it first contracts with the supplier with private information and then with the supplier with full information under a quantity-payment contract. Intuitively, under Scenarios qp-sim and qp-seq1, the optimal procurement mechanisms are equivalent, and the assembler suffers from information rent, downward quantity distortion and uncoordinated purchase quantities. Under Scenario qp-seq2, the assembler suffers from information rent and downward quantity distortion. Under Scenarios tpt-sim, tpt-seq1 and tpt-seq2, the optimal procurement mechanisms are equivalent, and the optimal purchase quantity is the same as that under the full information setting; it follows that the assembler only suffers from information rent. Looking into the comparison between Scenarios qp-sim and qp-seq2, the negative effect of uncoordinated purchase quantities under Scenario qp-sim makes the assembler prefer Scenario qp-seq2. On the other hand, under Scenario tpt-sim, the information rent is very high since no quantity distortion emerges; it follows that the assembler prefers Scenario qp-seq2 over this scenario. Finally, the supplier with private information and the channel both prefer a two-part tariff contract over a quantity-payment contract. This is because under a two-part tariff contract, regardless of the contracting sequence, the assembler’s optimal purchase quantity is always determined by its total marginal sourcing cost, which equals the sum of both suppliers’ costs. Therefore, no quantity distortion emerges under a two-part tariff contract, and the channel generates the same profit as that under a full 61

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information setting. We also study cases where the assembler adopts hybrid contract. We consider two types of hybrid contracts: a hybrid two-part tariff and quantity-payment contract and a hybrid quantity-payment and two-part tariff contract. Under the first hybrid contract, the assembler contracts with the supplier with full (private) information under a two-part tariff (quantity-payment) contract, and the following three scenarios emerge: tpt-qp-sim, tpt-qp-seq1 and tpt-qp-seq2. Under the second hybrid contract, the assembler contracts with the supplier with full (private) information under a quantity-payment (two-part tariff) contract, and the following three scenarios emerge: qp-tpt-sim, qp-tpt-seq1 and qp-tpt-seq2. We find that the application of hybrid contract improves each firm’s and the channel’s performances. In particular, the supplier with private information obtains the highest profit under a hybrid quantity-payment and two-part tariff contract. Both a quantity-payment contract with sequential contracting and a hybrid two-part tariff and quantity-payment contract lead to the highest assembler’s profit. Moreover, both a hybrid quantity-payment and two-part tariff contract with sequential contracting and a two-part tariff contract lead to the highest channel profit. The main contributions of this paper are summarized as follows. First, we explore how contract type interacts with contracting sequence to affect individual firms’ and the channel’s performances in an assembly system. To our knowledge, most previous literatures on assembly systems consider either contract type or contracting sequence, there are few papers taking into account both these two factors. Second, compared with the existing literatures, we demonstrate that under asymmetric cost information, coordinating the purchase quantities of different inputs in an assembly system does not always benefit the assembler and the channel. Third, we extend our analysis to the case where the assembler offers different contracts to the suppliers, and prove that individual firms and the channel are better off under a hybrid contract. The rest of this paper is organized as follows. Section 2 reviews related literature. Section 3 introduces the model parameters and assumptions. In Sections 4 and 5, we analyze the assembler’s optimal procurement mechanisms under quantity-payment and two-part tariff contracts, respectively. We compare the assembler’s optimal mechanisms under the above two contracts in Section 6. We extend our study to cases where the assembler adopts hybrid contract in Section 7. Section 8 concludes this paper. All proofs are shown in Appendix A. 2. Literature review Our article is closely related to the stream of literature that studies optimal contracting problem under asymmetric cost information. For example, Corbett and De Groote (2000) investigate the supplier’s optimal quantity discount contract when the buyer possesses private cost information. Ha (2001) characterizes the supplier’s optimal contract and cutoff level policy when the buyer has private cost information. Shen and Willems (2012) propose a contract consisting of wholesale and buyback prices to coordinate a channel with private retail cost information. Li et al. (2012) consider a similar setting and propose a side payment contract to coordinate the channel. Zhu (2016) studies the effect of information risk on outsourcing management in a supply chain consisting of a supplier and a buyer with private cost information. In these papers, the downstream party has asymmetric cost information. Some papers study a case where the upstream party has asymmetric cost information. For example, Kim and Netessine (2013) study how the manufacturer can incentivize a supplier with private cost information to collaborate to reduce the unit cost of a critical component. Shi et al. (2014) investigate how the presence of a backup supplier with private cost information affects the manufacturer’s procurement decisions. Lei et al. (2015) propose a revenue sharing contract to coordinate a supply chain consisting of one retailer and one supplier with private cost information. Bolandifar et al. (2017) investigate the buyer’s optimal capacity reservation contract when the supplier has private capacity cost information. Dong et al. (2018) study the OEM’s optimal procurement contract when the upstream CM has private discount ability information. However, all these papers study a simplified supply chain with an upstream supplier and a downstream retailer or manufacturer. The contractual agreement between a single supplier and a manufacturer under asymmetric cost information cannot immediately be applicable to assembly systems due to interactions among suppliers (Kalkanci and Erhun, 2012). Another related stream of literature focuses on assembly systems. Gerchak and Wang (2004) study and compare revenue sharing and wholesale price contracts in an assembly system with random demand. Wang (2006) compares simultaneous and sequential suppliers’ decision settings in terms of each supplier’s and the channel’s profits in an assembly system. Granot and Yin (2008) consider the cooperative behavior of suppliers in an assembly system; they study how system inefficiency due to suppliers’ decentralization can be reduced by allowing them to freely form alliances. Chen et al. (2014) study the channel’s as well as each firm’s preferences of power structure in an assembly system. Guan et al. (2015) compare three different contracts: push, pull and hybrid push-pull, in terms of each firm’s and the channel’s performances in an assembly system. Hnaien et al. (2016) develop a Branch and Bound algorithm to solve the replenishment problem of an assembly system with stochastic lead time and demand. Yin et al. (2017) study how the assembly system can use a flexible local sourcing policy to mitigate the global sourcing disruption risks. Yin et al. (2018) examine the value of expediting service for an assembly system in coordinating overseas and local sourcing. However, all these papers study assembly systems under a full information setting. In contrast, our paper focuses on the issue of information asymmetry in an assembly system. To our knowledge, this issue has not been adequately addressed. Kalkanci and Erhun’s (2012) paper is the first to study an assembly system under asymmetric information, however, they focus on demand information asymmetry. Fang et al. (2014) consider an assembly system that consists of one manufacturer and n suppliers with private cost information. To disclose all the suppliers’ costs, they derive a mechanism in which the quantity-payment contract offered to each supplier depends only on that supplier’s cost. Li et al. (2019) consider a similar setting and design a mechanism in which the contract parameters depend on all suppliers’ reported costs. Hu and Qi (2018) consider an assembly system that consists of one manufacturer and two suppliers with private cost information. They derive a two-part tariff contract to achieve truthful information sharing and 62

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compare the optimal mechanisms under simultaneous and sequential contracting. However, our work differs from the above three papers in the following two aspects. First, we focus on how do contract type interact with contracting sequence to affect individual firms’ and the channel’s performances, which is absent in those papers. Second, our work also touches upon the application of hybrid contract, which is also absent in those papers. Our article also falls within the stream of literature that studies the comparison of optimal simultaneous and sequential mechanisms. Gal-Or et al. (2007) consider a supply chain that consists of multiple suppliers and one buyer with private information about the fitness of each supplier. The buyer can search among the suppliers either sequentially or simultaneously. They study which searching scheme is best for the buyer and under what conditions. Kalkanci and Erhun (2012) consider an assembly system that consists of two suppliers and one manufacturer with private demand information. The two suppliers offer contracts to the manufacturer either sequentially or simultaneously. They compare the optimal simultaneous and sequential mechanisms in terms of the suppliers’ total profit and the manufacturer’s profit. Hu and Wang (2017) compare the simultaneous and sequential contest mechanisms in crowdsourcing contests. Li et al. (2018) consider a supply chain that consists of one manufacturer and two competing retailers who have private demand information. They study whether the manufacturer should offer subsidies simultaneously or sequentially to retailers to induce their sharing of demand information. Hu and Qi (2018) find that the optimal simultaneous and sequential procurement mechanisms in an assembly system with asymmetric cost information are revenue-equivalent. We differ from these papers in that our paper considers the effect of contract type on the optimal mechanism. 3. Model In line with the vast literature that studies the assembly system (e.g., Kalkanci and Erhun, 2012; Chen et al., 2014; Guan et al., 2015; Hu and Qi, 2018), we consider an assembly system that consists of one assembler (he) and two individual suppliers (indexed 1 and 2, she). The suppliers produce complementary components and incur unit production costs, c1 and c2 , respectively. The exact value of c2 is supplier 2's private information, but it is common knowledge that c2 follows a two-point distribution with P (c2 = c2L )= , P (c2 = c2H )= 1 . Thus, c2H and c2L correspond to the high and low cost types of supplier 2, respectively, where c2H > c2L > 0 . For ease of exposition, we denote supplier 2 with c2 = c2L (c2 = c2H ) as the low-cost (high-cost) supplier. Similar to Hu and Qi (2018), when the assembler outputs q units of final product, he sells the product at price m q , where m denotes the market size, and it is sufficiently large to guarantee a positive output. In line with the vast literature of assembly systems (e.g., Kalkanci and Erhun, 2012; Fang et al., 2014; Hu and Qi, 2018), we assume that the assembler assembles one unit of each component into his final product at zero assembly cost. Moreover, in line with the substantial literature on optimal mechanism design problem (e.g., Yang and Babich, 2015; Huang et al., 2018; Ma et al., 2018), the reservation profit of supplier 2 is normalized to zero. To induce information sharing in this assembly system, the assembler will design a contract for supplier 2. We first study and compare two basic types of contracts: a quantity-payment contract and a two-part tariff contract. Note that in line with Hu and Qi (2018), under the two-part tariff contract, the assembler decides his purchase quantity after the supplier with private information has chosen a contract from the contract menu offered by him. Later, the analysis is extended to the cases where the assembler adopts a hybrid contract. We study and compare two types of hybrid contracts: a hybrid two-part tariff and quantity-payment contract and a hybrid quantity-payment and two-part tariff contract. Under the former contract, the assembler contracts with supplier 1 (supplier 2) under a two-part tariff (quantity-payment) contract. Under the latter, the assembler contracts with supplier 1 (supplier 2) under a quantity-payment (two-part tariff) contract. The assembler’s contracting sequence can be either simultaneous or sequential under the above four contracts. Moreover, there are two scenarios under sequential contracting: (i) the assembler first contracts with supplier 1 and then with supplier 2; (ii) the assembler first contracts with supplier 2 and then with supplier 1. According to the foregoing discussion, we have twelve contracting scenarios that are the combinations of four contracts and three contracting sequences, and Table 1 summarizes the twelve scenarios. Throughout the paper, we use the following notations. Let qiz1 and Tiz1 be the purchase quantity and payment offered by the assembler to supplier i under Scenario z1. Let wiz2 and fiz2 be the unit purchasing price and payment offered by the assembler to supplier i under Scenario z2 . Similarly, az , 2z and cz represent the assembler’s, supplier 2′s and the channel’s profits under Scenario z , respectively, i = 1, 2 , z = z1 z 2 , z1

{qp - sim , qp - seq1, qp - seq2, tpt - qp - sim , tpt - qp - seq1, tpt - qp - seq2, qp - tpt - sim , qp - tpt - seq1, qp - tpt - seq2},

z2

{ tpt - sim, tpt - seq1, tpt - seq2, tpt - qp - sim , tpt - qp - seq1, tpt - qp - seq2, qp - tpt - sim , qp - tpt - seq1, qp - tpt - seq2}.

Table 1 The twelve contracting scenarios. Contracting sequence Contract type

Simultaneous

First with Supplier 1 and then with Supplier 2

First with Supplier 2 and then with Supplier 1

Quantity-payment Two-part tariff Hybrid two-part tariff and quantity-payment Hybrid quantity-payment and two-part tariff

qp-sim tpt-sim tpt-qp-sim qp-tpt-sim

qp-seq1 tpt -seq1 tpt-qp-seq1 qp-tpt-seq1

qp-seq2 tpt -seq2 tpt-qp-seq2 qp-tpt-seq2

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4. Quantity-payment contract Under a quantity-payment contract, denoted by (qi , Ti ), the assembler commits to purchasing qi units of component from supplier i and pays a transfer payment Ti to the assembler, i = 1, 2 . 4.1. Full information case under a quantity-payment contract We first analyze the quantity-payment contract under the full information benchmark. Since the assembler knows supplier 2's cost type c2t , he orders the same quantity qt from both suppliers and pays payments T1t and T2t to suppliers 1 and 2, respectively, t = H , L . The problem of the assembler can be formulated as follows:

max

= (m

a

qt , T1t , T2t

s.t. T1t

T2t

qt ) qt

c1 qt

c2t qt

T1t

T2t

0,

0.

In such a setting, the assembler captures the total channel profit and orders the first-best quantity, q¯t = Moreover, the assembler pays T¯1t = c1 q¯t and T¯2t = c2t q¯t to suppliers 1 and 2, respectively.

m

c1 2

c 2t

, from each supplier.

4.2. Scenario qp-sim: The assembler contracts with two suppliers simultaneously under a quantity-payment contract Under this scenario, the sequence of events is as follows. (1) The assembler simultaneously offers a contract (q1, T1) and a contract menu (q2L , T2L, q2H , T2H ) to suppliers 1 and 2, respectively. (2) With its cost state, supplier 2 chooses a contract (q2t , T2t ) from the menu, t = H , L . (3) The suppliers deliver components and obtain payments according to their contracts with the assembler. (4) The assembler produces min (q1, q2t ) units of product and realizes his profit in the market. The problem of the assembler can be formulated as follows:

max

+ (1

s.t. T1

c1 q1

T2L

c2L q2L

T2H

c2H q2H

T2L

c2L q2L

T2H

c2H q2H

=

a

q1, T1, q2L, T2L, q2H , T2H

{[m

){[m

min(q1, q2L )] min(q1, q2L)

min(q1, q2H )] min(q1, q2H )

T1

T1

T2L}

T2H },

(1) (2)

0,

(3)

0,

(4)

0, T2H T2L

(5)

c2L q2H ,

(6)

c2H q2L .

Constraint (2) guarantees the participation of supplier 1. Constraints (3) and (4) are the individual rationality (IR) constraints, which guarantee the participation of supplier 2. The last two constraints are the incentive compatibility (IC) constraints, which guarantee truth telling of supplier 2. We are now ready to present an important lemma as follows: Lemma 1.. The optimal solution to the assembler’s problem satisfies: q1 = q2L q2H , T1 = c1 q1, T2L = c2L q2L + (c2H c2L ) q2H , T2H = c2H q2H . Lemma 1 shows that the assembler always purchases the same number of components from supplier 1 and the low-cost supplier 2. To better understand this result, we consider the case where q1 > q2L . In this case, the assembler’s revenue is independent of q1 because his final output is limited by the minimum component purchase quantity, whereas his payment to supplier 1 increases in q1. Thus, the assembler’s profit decreases in q1 for all q1 > q2L . Therefore, we have q1 q2L and q2L q1, which imply that q1 = q2L . On the basis of Lemma 1, we characterize the assembler’s optimal purchase quantities in the following proposition. Proposition 1.. The assembler’s optimal purchase quantities under Scenario qp-sim are as follows: (c 2 H c 2 L ) (a) If c1 , then 1

q1qp

sim

m 2

= q2qp L

sim

=

(b) If c1 >

(c 2H 1

c 2 L)

, then

q1qp

= q2qp L

sim

= q2qp H

sim

c1 + c2L , q2qp H 2

sim

=

m

c1 2

sim

c2H

=

m 2

c2H 2(1

c2L . )

.

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This proposition shows that with information asymmetry, when c1

c 2L )

, the assembler distorts downward his purchase

sim sim < q¯H and q2qp < q¯L . Otherwise, the assembler does not quantities from both the low-cost and high-cost supplier 2, i.e., q2qp H L sim = q¯H and distort his purchase quantity from the high-cost supplier 2 but downward from the low-cost supplier 2, i.e., q2qp H qp sim q2L < q¯L , and its reason is as follows. Under Scenario qp-sim, the assembler always purchases the same quantity from supplier 1 sim sim ) c1 q2qp ) c1 q2qp and the low-cost supplier 2. It can be viewed as the assembler paying more (less) given by (1 ((1 ) to L H supplier 1 when facing a low-cost (high-cost) supplier 2, compared with the case under full information. On the other hand, the sim assembler pays an information rent given by (c2H c2L ) q2qp to the low-cost supplier 2 to prevent her from mimicking the highH qp sim cost supplier 2. The extra payment to supplier 1 distorts q2L downward. The information rent paid to supplier 2 and the reduced

sim sim sim 2H 2L (1 ) c1 q2qp payment to supplier 1 influence q2qp in opposite directions. If c1 , then (c2H c2L ) q2qp . That is, H H H 1 the decreasing effect of information rent dominates the increasing effect of reduced payment, and thus yields the downward dis(c c ) sim tortion of q2qp . If c1 > 21H 2L , the increasing effect of the reduced payment outplays the decreasing effect of the information H (c

rent, therefore, the assembler does not distort q2qp H

sim

c

)

.

4.3. Scenario qp-seq1: The assembler first contracts with supplier 1 and then with supplier 2 Under this scenario, the sequence of events is similar to that in Section 4.2 except that the assembler first contracts with supplier 1 and then with supplier 2. We adopt backward induction to solve the optimal sequential contracting problem of the assembler. We first find the assembler’s optimal mechanism for supplier 2. Then, we derive the assembler’s optimal mechanism for supplier 1. Given the assembler’s optimal contract (q1, T1) for supplier 1, his optimal mechanism design problem for supplier 2 can be formulated as follows:

max

q2L, T2L, q2H , T2H

a

=

+ (1

{[m

){[m

min(q1, q2L)] min(q1, q2L) min(q1, q2H )] min(q1, q2H )

T1 T1

T2L} (7)

T2H },

s.t. (3), (4), (5), (6). Lemma 2.. The assembler’s optimal mechanism for supplier 2 satisfies: q1 q2L q2H , T2L = c2L q2L + (c2H c2L ) q2H , T2H = c2H q2H . On the basis of Lemma 2, we characterize the assembler’s optimal purchase quantity from supplier 2 in the following proposition. Proposition 2.. The assembler’s optimal purchase quantity from supplier 2 is as follows: (1) If q1 <

(2) If

m 2

(3) If q1

m c 2H c 2L , q2L = q2H = q1. 2 2(1 ) c 2H c 2L m c 2L q , q2L = q1, q2H 1 < 2(1 ) 2 m c 2H m c 2L m c 2L , q2L = 2 , q2H = 2 2(1 2

=

m 2

c 2L . )

c 2H 2(1

c 2L . )

Once the assembler’s optimal mechanism for supplier 2 is characterized, his optimal contracting problem for supplier 1 can be formulated as follows:

max q1, T1

a

=

[(m

q2L ) q2L

T1

T2L] + (1

)[(m

q2H ) q2H

T1

T2H ],

(8)

s.t. (2). Proposition 3.. The optimal mechanism under Scenario qp-seq1 is equivalent to that under Scenario qp-sim. Intuitively, under Scenario qp-seq1, the asymmetric information structure is the same as that under Scenario qp-sim. The assembler does not know supplier 2’s cost during the contracting process under both scenarios. On the other hand, although the timing issues seq1 seq1 q2qp under the two scenarios are different, similar to Scenario qp-sim, the assembler still sets q1qp seq1 = q2qp under Scenario L H qp seq1 qp seq1 qp seq1 > q2L q2H qp-seq1. To better understand the result, we consider the special case where q1 . If the assembler lowers seq1 seq1 q1qp seq1 to q2qp ) , doing so changes neither the supplier’s profit nor the assembler’s and lowers T1qp seq1 by c1 (q1qp seq1 q2qp L L revenue but reduces the assembler’s payment to supplier 1. To summarize, the same asymmetric information structure and relationship among the assembler’s optimal purchase quantities result in the equivalent optimal mechanism under the two scenarios. 4.4. Scenario qp-seq2: The assembler first contracts with supplier 2 and then with supplier 1 Under this scenario, the sequence of events differs from those in Sections 4.2 and 4.3, not only on timing but also on the asymmetric information structure. In Sections 4.2 and 4.3, supplier 2's cost is unknown to the assembler during the contracting process, however, in this section, the assembler has learned supplier 2's cost before contracting with supplier 1. Thus, no information asymmetry exists when the assembler designs the contract for supplier 1. Intuitively, the assembler is certain to purchase the same quantity from both suppliers. Suppose that supplier 2's cost is c2t , given her chosen contract, (q2t , T2t ) , the assembler offers a contract (q2t , T1t ) to supplier 1, t = H , L . The assembler’s optimal contracting problem can be formulated as follows: 65

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max

a

q2L, T2L, q2H , T2H , T1L, T1H

s.t. T1L T1H

c1 q2L

c1 q2H

=

[(m

q2L) q2L

T1L

T2L] + (1

)[(m

q2H ) q2H

T1H

T2H ],

(9)

0,

0,

(3), (4), (5), (6). Proposition 4.. The assembler’s optimal mechanism under Scenario qp-seq2 is q2qp L seq2 c2H q2qp , H

T2qpH seq2

T2qpL seq2

seq2 c2L) q2qp , H

seq2 c2L q2qp L

seq2 T1qp L

seq2 c1 q2qp , L

seq2

=

seq2 T1qp H

c1 c 2L seq2 , q2qp H 2 qp seq2 c1 q2H

m

=

m

c1 2

c 2H 2(1

c 2L , )

= = = = + (c2H This proposition suggests that with information asymmetry, the assembler does not distort his purchase quantity from the lowseq2 seq2 = q¯L and q2qp < q¯H . The reason is as follows: cost supplier 2 but distorts downward from the high-cost supplier 2, i.e., q2qp L H Under Scenario qp-seq2, the assembler always purchases the same quantity from both suppliers, which is similar to the full inseq2 formation case. However, the assembler should pay an information rent given by (c2H c2L ) q2qp to the low-cost supplier 2 to H seq2 prevent her from mimicking the high-cost supplier 2, which leads to the downward distortion of q2qp . H 5. Two-part tariff contract Under a two-part tariff contract, denoted by (wi, fi ), the assembler purchases qi units of component from supplier i and is obliged to pay the supplier wi qi + fi , i = 1, 2 . This contract differs from the quantity-payment contract in that the assembler determines his purchase quantity from each supplier after supplier 2 has chosen a contract from the contract menu offered by him. Intuitively, under this contract with all contracting sequences, the assembler is certain to purchase the same quantity from each supplier. 5.1. Full information case under a two-part tariff contract

Similar to Section 4, we first study the full information benchmark under a two-part tariff contract. Under this case, the assembler offers contracts (w1, f1) and (w2t , f2t ) to suppliers 1 and 2, respectively, t = H , L . Moreover, the assembler orders the same quantity qt from both suppliers. The problem of the assembler can be formulated as follows:

max

a

qt , w1, f1 , w 2t , f2t

s.t. f1 + (w1

f2t + (w2t

= (m

qt ) qt

c1) qt

(w1 qt + f1 )

(w2t qt + f2t ),

(10)

0,

c2t ) qt

0.

In such a setting, the assembler also orders the first-best quantity, q¯t = f¯1 = f¯2t = 0 .

m

c1 2

c 2t

, from each supplier and sets w¯1 = c1, w¯2t = c2t ,

5.2. Scenario tpt-sim: The assembler contracts with two suppliers simultaneously under a two-part tariff contract Under this scenario, the sequence of events is as follows. (1) The assembler simultaneously offers a contract (w1, f1) and a contract menu (w2L, f2L , w2H , f2H ) to suppliers 1 and 2, respectively. (2) With its cost state, supplier 2 chooses a contract (w2t , f2t ) from the menu, t = H , L . (3) The assembler decides to purchase qt units of component from each supplier. (4) The suppliers deliver components and obtain payments according to their contracts with the assembler. (5) The assembler produces qt units of product and realizes his profit in the market. Following the backward analysis, we start with the assembler’s optimal purchase quantity decision qt . Given the contracts for suppliers 1 and 2, i.e., (w1, f1) and (w2t , f2t ) , the assembler solves the following problem to determine his optimal purchase quantity.

max qt

Obviously,

qt =

a

= (m

a

is concave in qt , and the assembler’s optimal purchase quantity is

m

qt

w1 2

w1

w2t ) qt

f1

f2t

w2t

Thus, the assembler’s mechanism design problem can be formulated as follows:

max

w1, f1 , w 2L, f2L , w 2H , f2H

s. t . f1 + (w1

f1 + (w1

c1)

a

c1)

m

=

m

w1 2

[

(m

w1 2

w2H

w1 4

w2L

w2L) 2

f2L ] + (1

)[

(m

0

w1 4

w2H )2

f2H ]

f1 ,

(11) (12)

0

(13) 66

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F. Lv, et al.

f2L + (w2L

c2L)

f2H + (w2H

m

c2H )

f2L + (w2L

c2L)

f2H + (w2H

m

m

c2H )

m

w1 2

w2L

w1 2 w1 2

0

w2H w2L

w1 2

(14)

0

(15)

f2H + (w2H

w 2H

c2L)

f2L + (w2L

m

c2H )

w1 2

m

w2H

w1 2

(16)

w2L

(17)

sim = c2L , Proposition 5.. The assembler’s optimal mechanism under Scenario tpt-sim is w1tpt sim = c1, f1tpt sim = 0 , w2tpt L m c 1 c 2H tpt sim tpt sim tpt sim f = 0 f2L = (c2H c2L ) w = c , , . The assembler’s optimal purchase quantities under Scenario tpt-sim are 2 H 2 H 2 H 2 m c1 c 2L m c1 c 2H tpt sim qLtpt sim = q = , . H 2 2 Proposition 5 shows that under Scenario tpt-sim, the assembler purchases components from each supplier at her cost price. c1. If w1 < c1, then f1 > 0 , the assembler must Intuitively, if w1 > c1 , then f1 = 0 and a decreases in w1, therefore, we have w1 )2

)2

1 2L 1 2H f2L ] + (1 )[ f2H ] denote the sum of the first compensate supplier 1 by a fixed payment. Let a1 = [ 4 4 two terms in a , which can be interpreted as the assembler’s revenue less his payment to supplier 2. Obviously, a1 decreases in w1 1 since a higher w1 implies a lower purchase quantity from each supplier, and a term 2 (m w1 w2L) + 2 (m w1 w2H ) measures m w1 w 2L c1 w1 + 2 . The assembler should set the decrease rate of a1 in w1. The last term in a , i.e., f1, decreases in w1 at the rate of 2 w2H w2L to guarantee truth telling of supplier 2, therefore, f1 decreases in w1 more significantly than a1. It follows that a increases c1; thus, w1 = c1. With regard to supplier 2, she is not concerned about this unfair purchasing price because she in w1 for all w1 would be compensated by the assembler. Moreover, this proposition shows that although the assembler has to pay the information rent to supplier 2, there is no quantity distortion under this scenario, and the intuition is as follows. On one hand, the assembler’s purchase quantities from both suppliers are identical. On the other hand, under a two-part tariff contract, the assembler’s purchase quantity is only determined by his purchasing prices of different inputs and is irrelevant to his payments to the suppliers.

(m

w

w

(m

w

w

5.3. Scenarios tpt-seq1 and tpt-seq2: The assembler contracts with two suppliers sequentially under a two-part tariff contract Under Scenario tpt-seq1 (tpt-seq2), the sequence of events is similar to that under Section 5.2 except that the assembler first contracts with supplier 1 (supplier 2) and then with the other supplier. It is worth noting that under Scenario tpt-seq2, the assembler has learned supplier 2's cost before contracting with supplier 1. The asymmetric information structure under Scenario tpt-seq2 is different from those under Scenarios tpt-sim and tpt-seq1. Proposition 6.. The optimal mechanisms under Scenarios tpt-seq1 and tpt-seq2 are equivalent to that under Scenario tpt-sim. The proposition is similar to Proposition 4 in Hu and Qi (2018) and can be explained following their logic. Although the asymmetric information structures are different among Scenarios tpt-seq1, tpt-seq2 and tpt-sim, the knowledge of supplier 2’s cost is irrelevant to the assembler’s contract design with supplier 1. This is because under a two-part tariff contract, the assembler’s optimal purchase quantity is only determined by his purchasing prices of different inputs and is irrelevant to his payments to the suppliers. Moreover, the assembler purchases components from each supplier at her cost price under this contract. 6. Performance comparison between quantity-payment and two-part tariff contracts The above discussions in Sections 4 and 5 have derived the assembler’s optimal mechanisms under different contracting scenarios. Building upon this, we now compare each firm’s and the channel’s profits under different scenarios to understand their preferences over different contract types and different contracting sequences. Table 2 presents each firm’s and the channel’s profits under different scenarios. As preparation, we first compare the assembler’s optimal purchase quantities from supplier 2 under different scenarios. In presenting the results, we discard Scenario qp-seq1 because the optimal mechanism under this scenario is the same as that under Scenario qp-sim. Similarly, we discard Scenarios tpt-seq1 and tpt-seq Corollary 1.. q2qp L

sim

< q2qp L

seq2

= qLtpt

sim

. If c1 >

(c 2H 1

c 2 L)

, thenqHtpt

sim

= q2qp H

sim

> q2qp H

seq2

; otherwise, qHtpt

sim

> q2qp H

sim

> q2qp H

seq2

Table 2 The firms’ and the channel’s profits under quantity-payment and two-part tariff contracts. Scenario

Conditions

qp-sim

c1 >

c1 qp-seq2

/

tpt-sim

/

1

1

(c2H

(c2H

Assembler

c2L)

c2L)

Supplier 2

q12

Channel

q12 +

(c2H

c2L) q1

)2

(c2H

c2L) q2H

(q2L )2 + (1

)(q2H ) 2 +

(c2H

c2L) q2H

(q2L )2 + (1

)(q2H ) 2

(c2H

c2L) q2H

(q2L )2 + (1

)(q2H ) 2 +

(c2H

c2L) q2H

(qL ) 2 + (1

)(qH ) 2

(c2H

c2L) qH

(qL ) 2 + (1

)(qH ) 2

(q2L

)2

+ (1

)(q2H

(c2H

c2L) qH

67

(c2H

c2L) q1

.

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This corollary suggests that the assembler’s purchase quantity from supplier 2 is no lower under a two-part tariff contract than under a quantity-payment contract. This is because no quantity distortion happens under the former contract. Moreover, this corollary also suggests that given a quantity-payment contract, the assembler’s purchase quantity from the high-cost (low-cost) supplier 2 is higher under simultaneous (sequential) contracting than under sequential (simultaneous) contracting. The reason is as follows. sim seq2 Recall from Propositions 1 and 4, the assembler does not distort q2qp but downward distorts q2qp , it follows that L L (c 2 H c 2 L ) sim , the assembler does not distort q2qp , otherwise, in H 1 (c 2 H c 2 L ) qp seq2 qp sim qp seq2 q > q contrast. However, q2H is always distorted downward. As a result, if c1 , then ; otherwise, the relative 2H 2H 1 sim qp sim qp seq2 q q distortion level of q2qp and determines the relation between them. Since is affected by the decreasing effect of the H 2H 2H seq2 information rent and the increasing effect of the reduced payment, whereas q2qp is only affected by the first effect. Therefore, the H seq2 distortion level is greater in q2qp . H

q2qp L

sim

< q2qp L

seq2

. Further recall from Proposition 1 that if c1

Proposition 7.. Supplier 2’s profits under the quantity-payment and two-part tariff contracts satisfy the following: If sim sim seq2 sim sim seq2 = qp > qp > qp > qp c1 > 1 (c2H c2L) , then tpt ; otherwise, tpt . 2 2 2 2 2 2 This proposition shows that supplier 2 prefers a two-part tariff contract over a quantity-payment contract, and its reason is as follows. Under both contracts, supplier 2 only earns an information rent given by (c2H c2L ) q2H when she is a low-cost supplier. sim seq2 > q2qp Recall from Corollary 1 that qHtpt sim q2qp ; therefore, supplier 2 obtains a higher profit under the two-part tariff conH H tract. Proposition 8.. The assembler’s profits under the quantity-payment and two-part tariff contracts satisfy the following: (a)

qp seq2 a

(b) If c1

qp sim , a 1 )

> (1

qp seq2 a

(c2H

1

tpt sim . a sim then qp a

>

c2L) ,

>

tpt sim , a

otherwise, in contrast.

Part (a) shows that the assembler obtains the highest profit when he first contracts with supplier 2 and then with supplier 1 under a quantity-payment contract. Intuitively, under Scenario qp-seq2, the assembler suffers from information rent and quantity distortion. Under Scenario qp-sim, when c1 > 1 (c2H c2L) , the assembler suffers from information rent and quantity distortion; when c1 1 (c2H c2L) , besides information rent and quantity distortion, uncoordinated purchase quantities also hurts the assembler. Recall from Proposition 7 that the information rent is higher under Scenario qp-sim than under Scenario qp-seq2. Furthermore, only sim sim seq2 q2qp is downward distorted under Scenario qp-seq2, however, both q2qp and q2qp may be downward distorted under Scenario L H H qp seq2 qp sim qp seq2 tpt sim > a > a qp-sim. Therefore, a . We now explain the reason why a . Under Scenario tpt-sim, the assembler only suffers from information rent, however, the information rent is sufficiently high because no quantity distortion emerges. This is the immediate reason why the assembler prefers Scenario qp-seq2 over Scenario tpt-sim. Part (b) implies that coordinating purchase quantities of different inputs does not always drive up the assembler’s profit, since (1 1 ) qp sim sim > tpt (c2H c2L) . This result seems to go against conventional wisdom. Note that such an holds whenc1 a a 1 unintended result arises only under simultaneous contracting and when supplier 2’s cost uncertainty, i.e., c2H c2L , is higher than the 1 threshold, (1 1 ) c1, and its reason is as follows. When c1 > 1 (c2H c2L) , the information rents under Scenarios tpt-sim and qpsim are equivalent, whereas the negative effect of quantity distortion makes the assembler perform worse under Scenario qp-sim. sim sim When c1 1 (c2H c2L) , qp and tpt decrease in supplier 2’s cost uncertainty, i.e., c2H c2L . The relative decrease rate of a a

(

qp sim a

c2H

and

c2L <

tpt sim determines the assembler’s a (1 ) c1 sim , qp decreases in c2H a (1 1 )

)

preference between Scenarios tpt-sim and qp-sim. When c2H

c2L more significantly than

tpt sim , a

c2L is small

otherwise, in contrast.

Proposition 9.. The channel’s profits under the quantity-payment and two-part tariff contracts satisfy the following: sim > (a) tpt c 1 (b) If < 2 or

sim qp seq2 ; tpt c c 1 c and 1 2

> qp c 2 (c2H

sim

. c2L) , then

qp seq2 c

qp sim , c

>

otherwise,

qp sim c

>

qp seq2 . c

Part (a) shows that the channel is always better off under a two-part tariff contract. Intuitively, no quantity distortion emerges under a two-part tariff contract; thus the channel’s performance is the same as that under a full information case. However, the channel suffers from information asymmetry under a quantity-payment contract. Consequently, the assembly system benefits from a two-part tariff contract. Part (b) shows that coordinating the purchase quantities from both suppliers does not always increase the channel’s profit, since 1 qp sim seq2 > qp holds when and c1 > 2 (c2H c2L) . This can be explained with different types of inefficiencies in the asc c 2 sembly system under Scenarios qp-sim and qp-seq2. Under Scenario qp-seq2, information asymmetry is the only source of inefficiency in the assembly system. However, under Scenario qp-sim, in addition to information asymmetry, uncoordinated purchase quantities may be another source of inefficiency in the assembly system when c1 < 1 (c2H c2L) . From the proof of Corollary 1, we observe sim seq2 sim seq2 q2qp > |q2qp q2qp | holds when that q2qp . That is, under a quantity-payment contract, when , if switching H H L L 2 2 the assembler’s contracting sequence from sequential to simultaneous, the increase in his purchase quantity from the high-cost supplier 2 outplays the decrease in his purchase quantity from the low-cost supplier 2. Therefore, the negative effect of asymmetric 1 sim seq2 > qp information on the channel is lower under Scenario qp-sim than under Scenario qp-seq2 when . As a result, qp c c 2

1

holds when

1 2

and c1

1

(c2H

c2L) . Moreover, when

1

1 2

and 2 (c2H

68

c2L ) < c1 <

1

(c2H

c2L ) , the gap between q1qp

sim

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F. Lv, et al.

and q2qp H

sim

, i.e.,

(c 2H

c 2L) 2 (1

(1 )

) c1

, is small. That is, under Scenario qp-sim, the negative effect of uncoordinated purchase quantities

on the channel is relatively low. Therefore, 1 when and c1 > 2 (c2H c2L) . 2

qp sim c

qp seq2 c

>

still holds under this setting. To summarize,

qp sim c

>

qp seq2 c

holds

7. Hybrid contract In the previous sections, we have treated the contract types of the assembler as exogenous. We have covered two extremes in terms of contract types (quantity-payment versus two-part tariff). In this section, we analyze situations where the assembler adopts a hybrid contract. We study two types of hybrid contracts: a hybrid two-part tariff and quantity-payment contract and a hybrid quantity-payment and two-part tariff contract. The results of this analysis allow us to capture contract preference of the assembler in situation where such decisions are endogenous. 7.1. Hybrid two-part tariff and quantity-payment contract Similar to the two basic types of contracts, we consider three scenarios, tpt-qp-sim, tpt-qp-seq1 and tpt-qp-seq2, under a hybrid twopart tariff and quantity-payment contract. Intuitively, under the above three scenarios, the assembler determines his purchase quantity from supplier 1, q1, after supplier 2 has chosen a contract (q2t , T2t ), t = H , L . Hence, q1 = q2t , the problem of the assembler can be formulated as follows:

max

+ (1 s.t. f1 + (w1 f1 + (w1

=

a

q2L, T2L, q2H , T2H , w1, f1

)[(m

q2L

w1) q2L

w1) q2H

q2H

c1) q2L

c1) q2H

[(m

T2L]

T2H ]

(18)

f1 ,

(19)

0,

(20)

0,

(3), (4), (5), (6). Proposition 10.. The assembler’s optimal mechanisms under Scenarios tpt-qp-sim, tpt-qp-seq1 and tpt-qp-seq2 are equivalent, and the m c1 c 2 L qp sim qp sim qp sim f1tpt qp sim = 0 , w1tpt qp sim = c1, T2tpt = c2H q2tpt q2tpt = optimal mechanism is , , H H L 2 m

c

c

c

qp sim qp sim qp sim 2H 2L = c2L q2tpt + (c2H c2L) q2tpt = 2 1 , T2tpt . L L H 2(1 ) Proposition 10 shows that although the asymmetric information structures are different under Scenarios tpt-qp-sim, tpt-qp-seq1 and tpt-qp-seq2, the knowledge of supplier 2’s cost is irrelevant to the assembler’s contract design with supplier 1. The intuition behind this result is as follows. On one hand, supplier 1 always makes zero expected profit. On the other hand, the assembler coordinates purchase quantities of the two inputs under the above three scenarios; thus these three scenarios are equivalent to Scenario qp-seq2. Therefore, we can obtain the following corollary.

q2tpt H

qp sim

Corollary 2 (.).

tpt qp sim a

=

qp seq2 ; a

tpt qp sim 2

qp seq2 ; 2

=

tpt qp sim c

=

qp seq2 . c

7.2. Hybrid quantity-payment and two-part tariff contract Similar to Section 7.1, we consider three scenarios, qp-tpt-sim, qp-tpt-seq1 and qp-tpt-seq2, under a hybrid quantity-payment and two-part tariff contract. We first analyze Scenario qp-tpt-sim where the assembler contracts with two suppliers simultaneously. Following the backward analysis, we start with the assembler’s optimal purchase quantity decision. Given the contracts for suppliers 1 and 2, i.e., (q1, T1) and (w2t , f2t ), t = H , L , the assembler solves the following problem to determine his optimal purchase quantity from supplier 2.

max

a

= [m

Obviously,

a

is a concave function of q2t ; thus the assembler’s optimal purchase quantity as a function of q1 is

q2t

min(q1, q2t )] min(q1, q2t )

q2t (q1) = min q1,

m

w2t 2

T1

w2t q2t

f2t .

.

The assembler optimally sets T1 = c1 q1; thus the assembler’s mechanism design problem can be formulated as follows:

max

q1, w 2L, w 2H , f2L , f2H

+ (1 s. t . f2L + (w2L

a

=

){[m

{[m

min q1,

(

w2H

c2L) min q1,

(

w2L

min q1, m

w2L 2

m

) ] min (q , ) ) ] min (q , ) f

m

w 2H 2

w 2L 2

1

1

m

0

m

w 2H 2

w 2L 2

2H }

f2L } c1 q1,

(21) (22)

69

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F. Lv, et al.

f2H + (w2H

c2H ) min q1,

f2L + (w2L

c2L) min q1,

f2H + (w2H

m

w2H

0

2

m

w2L

f2H + (w2H

2

c2H ) min q1,

m

(23)

w2H

f2L + (w2L

2

m

c2L) min q1,

w 2H

(24)

2

c2H ) min q1,

m

w2L

(25)

2

Proposition 11.. The assembler’s optimal mechanism under Scenario qp-tpt-sim is (a) If

c1 <

c2L) ,

(c2H

1+ tpt sim w2qp = L

|

(b) If c1

c c 2L + 1 ,

f2qp L

tpt sim

= (c2H

w2L) q1qp tpt sim ,

(c2H

c1 tpt sim tpt sim = c2H , f2qp = 0, , T1qp tpt sim = c1 q1qp tpt sim , w2qp H H 2 qp tpt sim tpt sim qp tpt sim c2L) q2qp ( w c ) q , where ξ is any positive number. 2L 1 2L H m c1 c 2H tpt sim qp tpt sim qp tpt sim qp tpt sim sim = 2 | 2 1, w = 1 | 1 c2H , w2qp T = c q = , , 1 1 L 2H 1 2 tpt sim qp tpt sim = (c2H w2H ) q1 , where 1 and 2 are any positive numbers.

q1qp

c2L) , then q1qp

(c2H

1+ tpt sim f2qp = L

then

tpt

f2qp H

tpt sim

=

m

c 2L

2

The assembler’s optimal purchase quantities under Scenario qp-tpt-sim are (a) If c1 <

(b) If c1

1+

(c2H

1+

(c2H

c2L) , then q2qp L

c2L) , then

Proposition 11 shows that q2qp H

tpt sim

= q1qp

tpt sim q2qp L

tpt sim

=

tpt sim

tpt sim q2qp H

, q2qp H

=

tpt sim

=

c 2H . 2

m

q1qp tpt sim .

is upward distorted when c1 <

1+

(c2H

c2L) , and its reason is as follows. On one hand, to

2H 2H , then q2H = q1; otherwise, q2H = . On screen supplier 2’s type, the assembler should set w2H > w2L , it follows that if q1 < 2 2 m w 2H the other hand, when c1 is small, the assembler can purchase a large number of components, i.e., q1 = q2L > from supplier 1 to 2 m c 2H tpt sim = achieve a high profit. Therefore, q2qp . H 2 Proposition 11 also shows that under Scenario qp-tpt-sim, the assembler may purchase components from supplier 2 at any price that is lower than her cost. Intuitively, to reduce the profit loss due to uncoordinated purchase quantities, the assembler may set q2t = q1, t = H , L . Thus, the assembler’s purchase quantity from supplier 2 is irrelevant to his unit purchasing price w2t . It follows that the assembler’s expected profit is also irrelevant to his unit purchasing price paid to supplier 2. Next, we analyze Scenario qp-tpt-seq1 where the assembler first contracts with supplier 1 and then with supplier 2. The following proposition characterizes the assembler’s optimal solution under this scenario.

m

m

w

w

Proposition 12.. The optimal mechanism under Scenario qp-tpt-seq1 is equivalent to that under Scenario qp-tpt-sim. The intuition behind this proposition is similar to that under Proposition 3. Thus, it is omitted for brevity. Finally, we analyze Scenario qp-tpt-seq2 where the assembler first contracts with supplier 2 and then with supplier 1. Intuitively, the assembler will offer a contract (q2t , T1t ) to supplier 1 after he has determined his purchase quantity q2t from supplier 2. We follow the backward analysis. We start with the assembler’s optimal purchase quantity decision for supplier 2. Given supplier 2’s chosen contract (w2t , f2t ) , the assembler solves the following problem to determine q2t .

max q2t

a

s.t. T1t

= (m

q2t

c1 q2t

0.

We have q2t =

max

m

w 2L, w 2H , f2L , f2H

a

f2L + (w2L f2H + (w2H

c2H ) c2L)

m

m

m

c2H )

T1t ,

c 1 w 2L ) 2 4

(m

=

c2L )

f2t

and T1t = c1 q2t . Thus, the assembler’s problem under Scenario qp-tpt-seq2 can be formulated as follows:

c1 w 2t 2

s.t. f2L + (w2L

f2H + (w2H

w2t ) q2t

m

c1 2

c1 2 c1 2 c1 2

w 2L

w 2H w2L w2H

f2L

+

(m

c1 w2H ) 2 4

f2H ,

(26)

0,

0, f2H + (w2H f2L + (w2L

c2L)

m

c2H )

m

c1 2

w2H

c1 2

,

w2L

.

The problem in (26) is similar to that in (11), but w1 is replaced by c1. From Proposition 5, we can immediately obtain the following proposition. Proposition 13.. The assembler’s optimal mechanism under Scenario qp-tpt-seq2 is q2qp t

tpt seq2 w2qp t

= c2t ,

tpt seq2 f2qp H

= 0,

tpt seq2 f2qp L

= (c2H

tpt seq2 c2L ) q2qp , H

70

t = H , L.

tpt seq2

=

m

c1 2

c 2t

, T1qp t

tpt seq2

= c1 q2qp t

tpt seq2

,

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Table 3 The firms’ and the channel’s profits under the hybrid contracts. Scenario

Conditions

tpt-qp-sim

/

Assembler

Supplier 2

(q2L) 2 + (1

)(q2H ) 2

(c2H

c2L) q2H

Channel

(q2L)2 + (1 +

qp-tpt-sim

c1 < c1

qp-tpt-seq2

/

1+

1+

(c2H

q12 + (1

c2L)

(c2H (c2H

)(q2H ) 2

[qL2

c2L) q2H

(c2H

c2L) q1

(c2H

c2L) qH

q12 + (1

)(q2H )2 c2L) q2H )(q2H )2

c2L) q2H

q12

c2L)

(c2H

(c2H

(c2H

c2L) qH ] + (1

) qH2

q12 +

(c2H

qL2 + (1

c2L) q1 2 ) qH

By comparing Propositions 5 and 13, we can immediately obtain the following corollary. tpt seq2 sim tpt seq2 sim tpt seq2 sim = tpt = tpt = tpt Corollary 3 (.). qp ; qp ; qp .The intuition behind this corollary is as follows. On a a c c 2 2 one hand, the assembler coordinates his purchase quantities from different suppliers under Scenarios tpt-sim and qp-tpt-seq2. On the other hand, the asymmetric information structures are the same under the above two scenarios.

7.3. Performance comparison among different contracts In Sections 7.1 and 7.2, we have analyzed two hybrid contracts, and Table 3 presents each firm’s and the channel’s profits under these hybrid contracts. Propositions 7–9 in Section 6 have showed the channel’s and individual firms’ performances under two pure contracts. Building upon these propositions, we now compare individual firms’ and the channel’s performances under these four contracts. In presenting the results, we discard Scenario qp-tpt-seq1 because the optimal mechanism under this scenario is the same as that under Scenario qp-tpt-seq1. Similarly, we discard Scenarios tpt-qp-seq1 and tpt-qp-seq2. Proposition 14.. The following holds for individual firms’ and the channel’s profit comparisons among different contracts: (a) If (b) (c)

c1 < 1 + (c2H tpt sim tpt seq2 = qp > 2 2 qp seq2 tpt qp sim = > a a tpt sim tpt seq2 = qp > c c

c2L) ,

then

tpt qp sim . 2 qp tpt seq2 a qp tpt sim ; c

>

qp tpt sim 2

>

qp tpt sim . a tpt sim tpt seq2 = qp c c

>

tpt sim 2

=

qp tpt seq2 2

>

tpt qp sim ; 2

otherwise,

qp tpt sim 2

=

tpt qp sim . c

tpt sim = 2qp tpt seq1; part (a) shows that supplier 2 obtains the highest profit under a hybrid Recall from Proposition 12 that qp 2 quantity-payment and two-part tariff contract when the assembler contracts with the two suppliers simultaneously or first contracts with supplier 1 and then with supplier 2. It is because the assembler does not distort or upward distorts q2H under these scenarios.

qp sim qp seq1 qp seq2 = tpt = tpt Recall from Proposition 10 that tpt ; part (b) shows that the assembler obtains the highest profit a a a when either one of the following two conditions holds: (i) he contracts with two suppliers under a hybrid two-part tariff and quantitypayment contract; (ii) he first contracts with supplier 2 and then with supplier 1 under a quantity-payment contract. Intuitively, under Scenarios qp-tpt-seq2 and tpt-qp-sim, the assembler only suffers from information rent. Under Scenario qp-tpt-sim, besides information rent and quantity distortion, the assembler may suffer from uncoordinated purchase quantities. Since the information rent is highest under Scenario qp-tpt-sim, the assembler obtains the highest profit under either Scenario qp-tpt-seq2 or Scenario tpt-qpsim. Moreover, since the information rent is higher under Scenario qp-tpt-seq2 than under Scenario tpt-qp-sim, the assembler performs better under the latter scenario. sim seq1 seq2 = tpt = tpt Recall from Proposition 6 that tpt ; part (c) shows that the channel generates the highest profit when c c c either one of the following two conditions holds: (i) the assembler contracts with the two suppliers under a two-part tariff contract; (ii) the assembler first contracts with supplier 2 and then with supplier 1 under a hybrid quantity-payment and two-part tariff contract because no quantity distortion emerges under these two scenarios.

8. Conclusion In this paper, we study an assembly system with one assembler and two suppliers wherein one supplier possesses private cost information. To induce credible information sharing in this assembly system, we study two basic types of contracts: a quantitypayment contract and a two-part tariff contract. With regard to the contracting sequence among the firms, we study three different scenarios, qp-sim, qp-seq1 and qp-seq2, under the quantity-payment contract, as well as another three scenarios, tpt-sim, tpt-seq1 and tpt-seq2, under the two-part tariff contract. We first propose the assembler’s optimal contracts under each scenario and then explore how the choice of contractual agreements and contracting sequences determine channel’s and individual firms’ performances. There are several major findings in this paper. First, we find that coordinating the purchase quantities from both suppliers does not always increase the channel’s and the assembler’s profits. Second, we find that the assembler obtains the highest profit under a quantitypayment contract with sequential contracting. Finally, we find that the supplier with private information and the channel prefer a 71

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F. Lv, et al.

two-part tariff contract over a quantity-payment contract. We extend our basic model to analyze cases where the assembler contracts with two suppliers under a hybrid contract. We consider two types of hybrid contracts: a hybrid two-part tariff and quantity-payment contract and a hybrid quantity-payment and two-part tariff contract. The assembler’s contracting sequence can be either simultaneous or sequential under the two hybrid contracts. Comparing the aforementioned four contracts, we find that the application of hybrid contract improves each firm’s and the channel’s performances. In particular, the supplier with private information obtains the highest profit under a hybrid quantitypayment and two-part tariff contract. Both a quantity-payment contract with sequential contracting and a hybrid two-part tariff and quantity-payment contract lead to the highest assembler’s profit. Moreover, both a hybrid quantity-payment and two-part tariff contract with sequential contracting and a two-part tariff contract lead to the highest channel’s profit. The above research findings have some useful implications. In practice, they can help firms to determine how to cooperate with their suppliers in a global supply chain. Nowadays, many firms have target Southeast Asia as their procurement destination due to lower wage and sufficient labor supply in this area. However, the true cost of purchasing from suppliers in Southeast Asia may be uncertain due to many reasons such as customs clearance fee and inland transportation cost. If the supplier in Southeast Asia is judged as a short-term partner, the firm can cooperate with it under a pure quantity-payment or a hybrid two-part tariff and quantitypayment contract. However, if the supplier in Southeast Asia is judged as a long-term partner, the firm can cooperate with it under a pure two-part tariff or a hybrid quantity-payment and two-part tariff contract. There are three potential directions for the future research on this issue. First, although we believe that the system we study with one assembler and two suppliers suffices to generate necessary insights, it may be worthwhile looking into a system with more than two suppliers. Second, we consider the case where the supplier possesses private cost information, it would be interesting to study the case where the supplier has private information about quality or corporate social responsibility. Finally, although the contract menus proposed in this paper are theoretically optimal, they are difficult to implement in practice due to their complex nature. Note that simple contracts such as price-only are popularly used in practice. Hence, it would be interesting to compare the performances of contract menus and simple contracts in an assembly system. However, since all these issues require substantial effort, we reserve them for future exploration. Acknowledgements The authors sincerely thank the Editor in Chief Tsan-Ming Choi and two anonymous reviewers for their constructive comments and suggestions to improve the paper. The authors have also benefited from the support from National Natural Science Foundation of China (grant numbers 71602189, 71871167, 71821001) and Humanities and Social Science Fund of Ministry of Education of China (grant numbers 16YJC630083, 19YJA630095). Funding This work was supported by National Natural Science Foundation of China (grant numbers 71602189, 71871167, 71821001) and Humanities and Social Science Fund of Ministry of Education of China (grant numbers 16YJC630083, 19YJA630095). Appendix A Proof of Lemma 1.. Obviously, a decreases in T1; thus, by (2), we have T1 = c1 q1. We now prove q2L q2H . Using (4) and (5), it follows that (3) is redundant. (4) binds, otherwise decrease T2L and T2H by > 0 , which will keep the other constraints satisfied, and the assembler is better off. (5) also binds, otherwise decrease T2L by > 0 , which will keep the other constraints satisfied, and the assembler is better off. Therefore, by (4) and (5), we have

T2H = c2H q2H ,

(A1)

T2L = (c2H

(A2)

c2L) q2H + c2L q2L .

Plugging (A1) and (A2) into (6), we have q2L q2H . We then prove q1 = q2L . If q1 > q2L , the assembler’s expected profit given by (1) can be rearranged as follows, [(m q2L ) q2L c1 q1 c2L q2L (c2H c2L ) q2H ] + (1 ){(m q2H ) q2H c1 q1 c2H q2H },Obviously, a decreases in q1 for a = all q1 > q2L ; thus we have q1 q2L . Following the similar logic, we obtain q2L q1, therefore, we have q1 = q2L . Proof of Proposition 1.. From Lemma 1, the assembler’s problem in Eq. (1) can be rewritten as follows:

max

a

q1, q2H

s.t. q1

=

[(m

q1

c2L) q1

(c2H

c2L ) q2H ] + (1

)(m

q2H

c2H ) q2H

c1 q1,

(A3)

q2H .

We adopt Karush–Kuhn–Tucker (KKT) conditions to solve Eq. (A3) and obtain a

q1

=

(m

2q1

c2L)

c1 +

1

=0

(A4)

72

Transportation Research Part E 129 (2019) 60–80

F. Lv, et al. a

= (1

q2H 1 (q1

)(m

2q2H

c2H )

(c2H

c2L )

1

= 0,

(A5) (A6)

q2H ) = 0.

where 1 is the Lagrange multiplier and 1 0 . Solving (A4)–(A6) yields the order quantities in equilibrium. When 1

= (1

) c1

(c2H

m 2

c1 + c 2L , 2

q1 = q2L =

c2L) ,

since

since q1

> 0,

1

we

q2H , we have c1

have c1 > (c 2 H 1

c 2L )

(c 2H 1

.

c 2 L)

.

1

> 0 , we have q1 = q2L = q2H =

When

1

= 0,

we

have q2H =

m

m 2

and

c1 c 2 H 2 c 2L )

c 2H 2(1

and

Proof of Lemma 2.. Given q1, the proof is similar to that of Lemma 1, thus is omitted for brevity. Proof of Proposition 2.. Given q1 and T1, from Lemma 2, the assembler’s optimal mechanism design problem for supplier 2 can be rewritten as

max

=

a

q2L, q2H

[(m

q2L

+ (1

s.t. q1

q2L

c2L) q2L

)(m

q2H

(c2H

c2L) q2H ]

c2H ) q2H

(A7)

T1,

q2H .

We adopt Karush–Kuhn–Tucker (KKT) conditions to solve Equation (A7) and obtain a

q2L

=

a

(m

=

q2H 1 (q1

(c2H

+

1

c2L) + (1

2

= 0,

)(m

(A8)

2q2H

c2H )

2

= 0,

(A9) (A10)

2

1

(2) When

(m

(A11)

q2H ) = 0.

where 1 and (1) When 2q1 1 = m

=

c2L)

q2L) = 0,

2 (q2L

1

2q2L

are the Lagrange multipliers and 1 0 , 2 0 . We divide our analysis into four cases as follows: > 0 and 2 > 0 , from (A10) and (A11), we have q1 = q2L = q2H . Plugging this equation into (A8) and (A9), we have m c 2H c 2L c2H , 2 = (1 )(m 2q1) (c2H c2L) , it follows that q1 < 2 . 2(1 ) 1

> 0 and

2

= 0 , we have q1 = q2L

c2L ), it follows that

2q1

m 2

c 2H 2(1

c 2L )

q1 <

q2H . Jointly solving (A8) and (A9), we have q2H =

m

c 2L

2

.

m 2

c 2H 2(1

c 2L , )

2H (3) When 1 = 0 and 2 > 0 , we have q1 q2L = q2H . Jointly solving (A8) and (A9), we have q2L = q2H = , 2 (c2H c2L) < 0 , which is contradictory with the condition that 2 > 0 , therefore, this case (c2H c2L) . Obviously, 2 = 2 = never holds. m c 2H c 2L m c m c 2L (4) When 1 = 2 = 0 , we have q2L = 2 2L , q2H = 2 , it follows that q1 . 2(1 ) 2

m

c

Proof of Proposition 3.. Since a decreases in T1, by (2), we have T1 = c1 q1. The assembler’s objective can be divided into three regions as follows: m c 2H c 2L (1) When q1 < 2 , plugging q2L = q2H = q1 and T1 = c1 q1 into (8), we rewrite the assembler’s objective as 2(1 )

max q1

that

a a

= (m

q1

c2H ) q1, which we can show is concave in q1. Thus, we have q1 =

c1

is continuous on q1, together with q1 < m

q1 =

c 1 c 2H 2 c 2H c 2L 2(1 )

m 2

m 2

(2) When

max

a

q1

c 2H 2(1

=

if c1 >

c 2L )

c 2H 2(1

c 2L , )

a

=

2q1

m c + 2H 2 2(1

1 (q1

2

(m

m

c2L 2

From Proposition 2, we know

we have

,

q1 <

q1

m

c 2L 2

c2L) q1

, together with T1 = c1 q1 and q2L = q1, we can rewrite the assembler’s problem in (8) as

(c2H

c2L ) q2H ] + (1

)[(m

q2H ) q2H

c2H q2H ]

KKT conditions imply that the optimal solutions of q1L and q1H can be found by solving

q1

c1 + c 2 H . 2

otherwise.

c 2L )

[(m

(c 2 H 1

m 2

m 2

c2L)

c1 +

1

2

= 0,

c2L ) = 0, )

q1 = 0,

73

c1 q1.

Transportation Research Part E 129 (2019) 60–80

F. Lv, et al. 1,

0.

2

Following the standard KKT conditions problem approach, q1 is as follows: m 2 m 2

q1 =

c 2H c 2L 2(1 ) c1 + c 2L 2

if c1 >

(c 2 H 1

c 2L )

,

otherwise.

2L (3) When q1 . Since T1 = c1 q1, thus, a decreases in q1, we have q1 = 2 Together with Proposition 2, q1, q2L and q2H have the structure below:

m

c 2H c 2L . 2(1 ) m c 2H c 2L q = , . 2H 2 2(1 ) 2 m c1 c 2H q2H = , when c1 > 2 m c 2H m c1 + c 2L q , 2H = 2 2 2 2(1

(1) q1 = q2L = q2H =

(2) q1 = q2L = (3) q1 = q2L = (4) q1 = q2L =

c

m

c 2L

m

c 2L 2

.

m 2

(c 2H c2L) . 1 c 2L , when )

(c 2 H 1

c1

c 2L )

.

By simply comparing the assembler’s profits under the above four cases, we can obtain his optimal purchase quantities as shown in this proposition. We omit the tedious calculations here. Then the assembler’s expected profit follows. Proof of Proposition 4.. Similar to Lemma 1, we can easily obtain that q2L q2H , T2L = c2L q2L + (c2H c2L ) q2H , T2H = c2H q2H , T1L = c1 q2L and T1H = c1 q2H . Plugging the above expressions into (9), we obtain that a is separable and concave in q2L and q2H . Thus, we derive the assembler’s optimal purchase quantity from supplier 2 as follows,

m

q2L =

c1 2

c2L

,

q2H =

m

c1 2

c2H 2(1

c2L . )

Proof of Proposition 5.. Similar to Lemma 1, we can easily obtain that (15), (16) must bind and (14) is redundant, therefore, m w 1 w 2H m w1 w 2H m w 1 w 2L f2H = (c2H w2H ) (w2L c2L) , f2L = (c2H c2L) . Plugging the expressions of f2L and f2H into (17), we 2 2 2 0 , thus c2H w2H , it follows that c2H w2H w2L . Focusing on (12) and (13), we find that if w1 c1, have w2H w2L . Since f2H m w1 w 2L then f1 = 0 ; otherwise, f1 = (c1 w1) . Therefore, we solve the assembler’s mechanism design problem in two regions: 2 w1 c1 and w1 < c1. (1) w1 c1. The assembler’s objective function in (11) can be rewritten as

max

a

w1, w 2L, w 2H

w1 w 2L)2 4

(m

=

+ (1

s.t. c2H w1

w2H

)

(c2H

(m

c2L )

w1 w 2H )2 4

m

w 1 w 2H 2

(c2H

+ (w2L

w2H )

m

c2L)

w 1 w 2H 2

m

w1 2

w 2L

.

w2L,

c1.

1 2H < 0 , since w1 Taking the first derivative of a with respect to w1, we have wa = 2 1 KKT conditions imply that the optimal solutions of w2L and w2H can be found by solving

m

a

=

w2L a

w2H

2

=

c2L) +

(c2H

c2L)

2

1 (c2H

2

c

c1, therefore, w1 = c1.

= 0,

1 2

(w2H

c2H )

1

+

2

= 0,

w2H ) = 0,

2 (w2H 1,

(w2L

w

w2L) = 0, 0.

2

Following the standard KKT conditions problem approach, we have w2L = c2L and w2H = c2H . (2) w1 < c1. The assembler’s objective function in (11) can be rewritten as

max

w1, w 2L, w 2H

a

=

+ (1

s.t. c2H

w2H

)

(m

w1 w 2L)2 4

(c2H

c2L)

(m

w1 w 2H )2 4

(c2H

w2H )

m

m

w1 w 2H 2

+ (w2L

w1 w 2H 2

(c1

w2L,

w1 < c1 74

c2L)

m

w1)

m

w1 2 w1 2

w 2L w 2L

,

(A12)

Transportation Research Part E 129 (2019) 60–80

F. Lv, et al.

Taking the first derivative of a

a

with respect to w1 and w2H , respectively, we have

a

w1

= > 0 . Therefore, w1 = c1 and w2H = c2H . 2 2 2 KKT conditions imply that the optimal solutions of w2L can be found by solving c 2H

c 2L

1

w 2H

c 2H

=

c 2H

w 2L 2

w1

c1 2

> 0,

w 2H

a

=

w2L

2

1 (w 2H

(w2L

c2L) +

1

= 0,

w2L ) = 0,

0.

1

Following the standard KKT conditions problem approach, we have w2L = c2L . In summary, the assembler’s optimal solutions are w1 = c1, f1 = 0 , w2L = c2L , f2L = (c2H

c2L)

m

c1 c 2H , 2

w2H = c2H , f2H = 0 .

sim Proof of Proposition 6.. From the proof of Proposition 5, we observe that tpt first increases in w1 for all w1 < c1 then decreases a c1. Therefore, the assembler’s optimal solution under Scenario tpt-seq2 is the same as that under Scenario tpt-sim. in w1 for all w1 We only need to focus on Scenario tpt-seq1. When w1 c1, a and a under Scenario tpt-seq1 are the same as those under Scenario tpt-sim. Therefore, the assembler’s w 2L w 2H optimal solutions under the above two scenarios are identical. When w1 c1, the assembler’s objective function under Scenario tpt-seq1 is the same as that in Eq. (A12). We adopt backward induction to solve the optimal sequential contracting problem. We first derive the optimal values of w2L and w2H , and then derive the optimal value of w1. Taking the first order derivatives of a with respect to w2L and w2H , respectively, we have c 2H c2L 1 w 2H c 2H (w 2L c 2L) w1 c1 a a = > 0 = , . Therefore, we have w 2 2 2 w 2 2 2H

2L

c2H

w2L =

a

w1

w1 c1

c2L +

w1

c1

c1

(c2H

(c2H

c2L)

c2L ), w1

c1.

, w2H = c2H

(c2H c2L) . Together with w2L = w2H = c2H , taking the first order derivative of (1) w1 c1 c w (c2H c2L) . It follows that = 1 2 1 > 0 , thus, w1 = c1 tpt seq1 |w1= c1 a

(2) c1

=

(c 2H c2L)

(c2H

c2L )

spect to w1, we have tpt seq1 |w1= c1 a

w1

=

c 2H

=

c 2L 2

(m

w1 4

c2H ) 2

(c2H

c2L)

c1. Together with w2L = c2L +

w1

a

(m

c1 4

+

1

c2L) 2

(w 1

2

+ (1

c1) , thus, )

(m

a

w1

m

w1 2

with respect to w1, we have

.

and w2H = c2H , taking the first order derivative of a with re(c2H c 2L) , therefore, we havew1 = c1. It follows that w2L = c2L and 2

c1

w1

>

c2H ) 2

c1 4

c2H

a

(c2H

c2L)

m

c1 2

c2H

.

(1 + ) seq1 tpt seq1 |w1= c1 |w1= c1 (c 2H c2L) = (c2H c2L) 2 > 0 , thus, when w1 c1, the assembler’s optimal solution under Since tpt a a 4 Scenario tpt-seq1 is w1 = c1, w2L = c2L and w2H = c2H , which is the same as that under Scenario tpt-sim.

Proof of Corollary 1.. Together with Propositions 1, 4 and 5, we have c 2L

q2qp L

seq2

= qLtpt

sim

- sim , q2qp L

c 2H 2 1

- seq2 q2qp = L 2

- sim q2qp H

- seq2 q2qp = H

qHtpt

q2qp H

sim

sim

=

(c 2 H 2(1

c 2L ) )

if c1 >

c1 2

(c 2H 1

c 2L )

,

otherwise.

,

otherwise.

0

i f c1 > (c 2 H 2(1

c1 c 2L)

(c 2 H 1

if c1 >

c 2L ) )

c1 2

(c 2H 1

c 2L)

,

otherwise.

Proof of Proposition 7.. Table 2 shows that supplier 2’s profit is determined by the assembler’s purchase quantity from the high-cost supplier 2. Together with Corollary 1, we can easily prove this proposition, thus it is omitted for brevity. Proof of Proposition 8.. For Part (a), we first prove qp seq2 a

(c 2H 1

>

qp sim . a

qp sim a

=

When c1

qp seq2 a

c 2L)

When c1 >

(c 2 H 1

seq2 2 seq2 2 = [(q2qp ) (q1qp sim )2] + (1 )[(q2qp ) (q1qp sim )2] L H (c 2 H c 2 L ) (c 2 H c 2 L ) qp seq2 qp sim qp seq2 qp sim (q2L + q1 ) (q2H + q1 ) 2 2 (c 2H c2L) qp seq2 qp seq2 = (q2L q2H ) > 0. 2

, we have 75

c 2L)

, we have

Transportation Research Part E 129 (2019) 60–80

F. Lv, et al. qp seq2 a

qp sim a

1

= We now prove

qp seq2 a

(q2qp L

seq2 c1 (q2qp L

2

tpt sim , a

>

qp seq2 a

tpt sim a

seq2 2 )

[(q2qp L

=

since qLtpt

[(qLtpt sim ) 2

=

sim 2 )]

seq2 q2qp H

sim

+

= q2qp L

sim q2qp L

seq2

seq2 2 )

)[(q2qp H

+ (1

sim q2qp ) H

(q2qp H

> 0.

, we have

(c2H c2L ) qHtpt sim] + (1 )(qHtpt sim )2 tpt sim 2 qp seq2 2 )[(qH ) (q2H )] (c2H (c 2H c2L) qp seq2 tpt sim = (q2H qH ) 2

= (1

sim 2 )]

c 2L ) 2 )

(c 2 H 4(1

=

seq2 2 )

[ (q2qp L

seq2 2 )]

)(q2qp H

+ (1

c2L) qHtpt sim

< 0.

For Part (b), our analysis can be divided into two regions as follows: (c 2H c 2L) (1) c1 , we have 1 tpt sim a

qp sim a

[(qLtpt

=

= 1

=

c1 (qLtpt sim

2

sim 2 )

=

1

sim 2 (q2qp )] L

sim ) q2qp L

+

c1 (qLtpt

2

c2L) qHtpt

(c2H

[(qLtpt sim ) 2

sim

1

DefineF (c1) = F (c1) c1

=

c 2H

c 2L

Note that

F (0) = c1 <

4(1 1

(1

) )

2

c1

4(1

1

1

tpt sim a

)

c2L

qp sim a

(c2H

)2

(c 2 H 1

c2L) > and

<0

[(qLtpt

=

c 2L )

q2qp H

c 2L

c1

2

sim 2 )

(qLtpt

sim

+ q1qp

(c 2 H

c 2L )

(qLtpt

c 2L 2

(1

c2L) qHtpt

(c2H

c 2H

1

)

(q1qp

2

sim

sim 2 )

tpt sim c

When c1 > qp sim c

sim

)

qHtpt

=

1

qp sim c

(qLtpt

=

sim 2 )

1

=

A (c1) =

havec1 =

>

(c 2 H 1

c 2L )

(c 2 H 1

c 2L )

(c2H

(1 ± 1 1

hence

c2L)

)

sim

)

c2L) .

(c2H

1

)

1

have

c2L) . Since

(c2H

tpt sim a

<

for

qp sim a

.

sim 2 )

]

sim

otherwise,

qp sim . a

tpt sim a

The Proof of Proposition 8 is

qp seq2 . c seq2 2

From Table 2, we have seq2 2 ) + (1 )(q2qp ) + (c2H H

(c2H

c2L )(qHtpt

sim

c2L) q1qp

(c2H

c2L) q1qp

q2qp H

seq2

c2L) q2qp H

seq2

] We

)

now

prove

have

(q1qp sim ) 2 (c2H

(q1qp

c 2L)

(1

) > 0.

qp sim , a

)+

sim

we (c 2H 1

c1 <

sim 2 )

c2L) qHtpt

(c 2H c2L)2 = 4(1 > 0. ) q1qp sim = qHtpt sim , we

4

1 4

[1

c12 +

±

(1

c12 +

)(c2H 2

(1

c 2L )

)2

)(qHtpt

+ (1

= Define

qHtpt

sim

sim

sim

]

> 0.

c2L) , we have

(c2H

tpt sim c

> 0;

)2

c2L) qHtpt

sim

<

sim

)(qHtpt sim ) 2 (q1qp sim ) 2 (c c2L) 2 4 2H

= When c1

qHtpt

c2L) , since

(c2H

(qLtpt sim ) 2 + (1 = [(qLtpt sim ) 2

c2L) qHtpt

(c2H

c2L) 2 .

(c2H

)(qHtpt

(c2H

tpt sim a

(c 2H c 2L) seq2 (q2qp H 2

1

c2L )

] + (1

(c2H

sim

c2L ) , then

(c2H

1

= qp sim . c

)

c1 2

we have

sim > Proof of Proposition 9.. For Part (a), we first prove tpt c tpt sim qp seq2 tpt sim 2 tpt sim 2 = (qL ) + (1 )(qH ) [ (q2qp c c L

>

4(1

sim 2 )]

c2L ) qHtpt sim

(c2H

c 2L) )

)(q2qp H

+ (1

(c 2 H c 2 L ) sim (q2qp H 2

)+ 2

(c2H c 2L) ) = 4 (c2H 1 (1 1 qp sim for a 1

F(

sim 2 )

=

tpt sim c

sim

sim 2 )

, therefore, the only feasible solution is c1 =

= [(qLtpt =

To summarize, if c1 < completed.

sim

c 2H

(c 2 H 2(1

sim ) q2qp H

+

[ (q2qp L

sim 2 (q2qp )] H

c2L)2 , taking the first derivative of F (c1) with respect to c1, we have

(c2H

)

qHtpt c12 +

4

sim (c2H c2L ), whereas tpt a (c2H c 2L) qp sim tpt sim = qH , since q1 , 1

(2) c1 >

sim

sim 2 )

0 . Thus, F (c1) increases in c1. Let F (c1) = 0 , we have c1 =

c

(c2H

c 2L 2

1 2 (1 + 1 1

2 2

c 2H

c12 +

4 1

)[(qHtpt sim ) 2

)(qHtpt sim

+ q2qp L

)(qHtpt

] + (1

+ (1

+ (1

=

sim

+

c 2H

c 2L 2 4

c1 +

4(1

c12 + 2

4(1

]. Since

, therefore, A (c1) > 0 for all c1 when c1

[ (q2qp L

2

c1

1

sim 2 )

)

(1

(c2H

)

+ (1

c2L )2 +

(c2H

)(c2H 2

sim 2 )

c 2L )

c2L) 2 .

(c2H c 2L) [1 1 (c 2 H c 2 L ) . 1

c1 +

(c2H 2

4(1

)

sim 2 )

+

c2L)

(c 2 H 2(1

c 2L ) )

(1 It follows that

is

A (c1)

)2

c2L) q2qp H

(c2H

sim

]

c1 2

c2L) 2 .

(c2H

Obviously,

76

)(q2qp H

+

concave

] < 0 and

tpt sim c

>

qp sim . c

in

(c 2 H 1

c 1. c 2L)

[1

Let

A (c1) = 0 ,

+

(1

)2

we

+

]

Transportation Research Part E 129 (2019) 60–80

F. Lv, et al.

For Part (b), our analysis can be divided into two cases. (1) c1 > 1 (c2H c2L) , we have qp seq2 c

qp sim c

seq2 2 seq2 2 ) + (1 )(q2qp ) H (c2H c 2L) qp seq2 qp seq2 ( q q ) + 2L 2H 2 (c2H c 2L) c2H c 2L = 2 2(1 )

(q2qp L

= =

Thus, if (2) c1

1 , 2 1

then (c2H

qp seq2 c

qp sim , c

qp seq2 c

c2L) , we have

qp sim c

[(q2qp L

=

seq2 2 )

+ =

1

=

1

c1

2

c1

2 (c2H

1 , 2

then

(

1 2

seq2

c 2H 1

1 2

, otherwise, seq2 < qp c2L) , whereas qp c c 1

qp seq2 c

sim 2 )]

+ (1

seq2 c2L)(q2qp H

q2qp H

= When

(q2qp L

(c2H

seq2 c1 (q2qp L

2

otherwise,

+ q2qp L

c 2L

(

1

< sim

sim

(c2H

seq2

q1qp

sim

q1qp

sim

)

)

c 2L) )

qp sim . c

<

seq2 2 sim 2 )[(q2qp ) (q2qp )] H H qp sim q2H ) (c 2H c 2L) sim q2qp ) c1 H 2

c2L

c1 2

).

c2L ) < c1

seq2 qp sim , then qp . If > 2 , then To summarize, if c c 2 for c1 2 (c2H c2L) . The Proof of Proposition 9 is completed.

1

seq2

c2L)(q2qp H

).

(c 2H c 2L) c1 2

for 2 (c2H

(c2H

(c 2H 2(1

c2L )

. Therefore, If

1 2

+

c2L)(q2qp H

(c2H

)

c1 2

c1 c2H

sim 2 )

c 2L)2 1 (2 )

(c 2 H 2(1

=

(q1qp

1

1 , 2

then

1 qp seq2 c

qp seq2 c

qp sim . c

qp sim c

for c1 > 2 (c2H

(c2H <

c2L) .

If

> 2 , then 1

qp seq2 c

c2L) , while

qp sim c

qp seq2 c

for

qp sim c

Proof of Proposition 10.. We first analyze Scenario tpt-qp-sim. Similar to the proof of Lemma 1, we can easily obtain (A1), (A2) and q2L q2H . Plugging (A1) and (A2) into (18), we have

max

=

a

q2L, q2H , w1, f1

[(m

+ (1

q2L

w1

)(m

q2H

c2L) q2L w1

(c2H

c2L ) q2H ]

c2H ) q2H

(A13)

f1 ,

q2L (1 ) q2H < 0 , it follows that If w1 c1, then f1 = 0 . Taking the first derivative of a with respect to w1, we have w = 1 w1 = c1. When w1 c1, then f1 = (c1 w1) q2L . Plugging this expression into (A13) and taking the first derivative of a with respect to a

w1,

we

have

m

a

= (1

)(q2L

w1 c 2H c 2L . 2(1 )

c1

0,

q2H )

it

follows

that

w1 = c1. Then we

can

easily

obtain q2L =

m

c1 2

c 2L

and

q2H = 2 c1. Therefore, the From the above analysis, we find that a first increases in w1 for all w1 < c1 then decreases in w1 for all w1 assembler’s optimal mechanism under Scenario tpt-qp-seq2 is the same as that under tpt-qp-sim. We now consider Scenario tpt-qp-seq1. We adopt backward induction to solve the optimal sequential contracting problem. We first derive the optimal values of q2L and q2H , and then derive the optimal value of w1. We divide our following analysis into two regions. c1. Taking the first order derivative of (1) w1 with respect to q2L and q2H , respectively, we have a a

q2L

=

q2H =

(m m

q2L

w1 c 2H c 2L . 2(1 ) m w1 c 2H 2

w1 2

c2L) ,

a

q2H

= (1

)(m

2q2H

w1

c2H )

(c2H

c2L) .

Therefore,

Plugging the expressions of q2L and q2H into (A13) and taking the first derivative of

< 0 , thus, w1 = c1. have wa = 1 (2) w1 c1. Plugging f1 = (c1 w1) q2L into (A13) and taking the first order derivative of spectively, we have

q2H =

m

w1

m

q2L = When c1

c2H 2(1

2 w1 2 w1

m

1

c 2L

+

2

c 2H 2(1

(c2H

c2L)

w1 = c1. When w1 < c1

w1 c 1 2 c 2L )

1

=

m

c1 2

a

haveq2L =

a

m

w1 2

c 2L

,

with respect to w1, we

with respect to q2L and q2H , re-

c2L )

w1

c1

c2L) (c2H

1

w1

c1,

c2L).

c1, taking the first derivative of

a

with respect to w1, we have

c2L ) , taking the first derivative of

(c2H

c2L

(c2H

1

w1 < c1

thus w1 = c1 1 (c2H c2L) . Now we need to compare a |w1= c1 a |w1= c1

we

2

1

+ (1

(c 2H c 2L) and

)

m

c1 2

a |w1= c1

c2H 2(1

a

a

w1

with respect to w1, we have

=

c 2H

c 2L 2

a

w1

=

to determine the optimal value of w1. We have

c2L )

77

2

,

+

1

c1 + c 2 H 2

2

w1 > 0 , thus w1

c 2H 2(1

c 2L , )

Transportation Research Part E 129 (2019) 60–80

F. Lv, et al. a |w1= c1

Since

a |w1= c1

>

= (m

(c2H c 2L)

1

c1 c 2H )2 4

(m

c1

>

c2H

q2H ) q2H . (c2H c 2L) ,

a |w1= c1 1

thus, we have w1 = c1. We thus complete the proof of this proposition.

Proof of Proposition 11.. We divide our analysis into two regions: w2L w2H and w2L w2H . m w 2L m w 2H m w 2H m w 2L q1 < (1) w2L w2H . In this region, we consider three cases: q1 < , and q1 . 2 2 2 2 m w 2H c w w2L . Plugging the ex(1.1) q1 < . By (22)–(25), we have and , thus, f = ( c w ) q f = ( c w ) q 2 H 2 H 2 H 2 L 2 H 2 H 1 1 2 L 2 H 2 m c1 c 2H pressions of f2L and f2H into (21) we have a = (m c1 c2H q1) q1. We can easily obtain that q1 = , w2H = 1 | 1 c2H , 2 w2L = 2 | 2 1, where 1 and 2 are any positive numbers. It follows that a = q12 . m w m w m w 2L m w 2H (w2L c2L) q1 and f2H = (c2H w2H ) 2 2H , thus, q1 < (1.2) . By (22)–(25), we have f2L = (c2H c2L) 2 2H 2 2 f2H c2H w2H w2L . f2L Plugging the expressions of and into (21) we ham w m+w m w )( 2 2H c2H ) 2 2H c1 q1.We can easily obtain that w2H = c2H , ve a = [(m q1 c2L) q1 (c2H c2L ) 2 2H ] + (1 m

c 2H 2 m c 2L 2

q1 =

a

c1 c1 2 q12

=

q12

(1.3) q1 w2H w2L . m

c2H

have

w2L =

(

=

a

a

=

c2L ),

c1 < (c2H c1 q1

c2L ).

+ (1

c1

)

w 2L . 2

m + w 2L 2

c2H c2L +

(c2H

(

m

)

c 2H 2 2

c1

(c2H

c2L),

c1 <

(c2H

c2L).

c1

(c2H

c2L ),

c1 <

(c2H

c2L ).

c 2H

|

c 2L

(c2H

c + 1

c2L)

(

m

c 2H 2

)

By (22)–(25), we have f2H = (c2H Plugging the expressions

w2H ) of

c2L

c2L )

)

m

w 2L 2

+ (1

c1

(c2H

c2L),

c1 <

(c2H

c2L).

q12

|

w2L =

)(

m

m + w 2H 2

c 2H 2 m c 2L 2

q1 =

c1 q1

q12 + (1

)

(c2H

m

c 2H 2 ) 2

(c2H

c1 2

c2L)(

m

w 2H 2

m

w 2H 2

c1

(c2H

c2L ),

c1 <

(c2H c1

c2L ). (c2H

m

c 2H ) 2

c1 <

Where ξ is any positive number. It follows that

and f2L = (c2H f2L and

c1 q1.

We

m

can

c2L) f2H

w 2H 2

easily

(w2L into

obtain

m

w 2L

c2L) 2 (21)

that

, thus, we

w2H = c2H ,

It follows that

c2L),

(c2H

c2L).

2H 2L 2H (2) w2L w2H . In this region, we also consider three cases: q1 < , 2 2L q1 < and q1 . 2 2 2 m w 2L (2.1) q1 < . Obviously, the assembler’s optimal solution is the same as that in Case 1.1. 2 m c 2H m w 2L m w 2H q1 < (2.2) . Similar to Case 1.2, we can obtain that q1 = , w2L = c2H , w2H = | c2H , where ξ is any positive 2 2 2 2 c1 q1. number. It follows that a = q1 m w 2H m c 2H (2.3) q1 . Similar to Case 1.3, We can obtain that q1 = , w2L = w2H = c2H . It follows that a = q12 c1 q1. 2 2 m c c m c (c2H c2L) . Next, we compare a |q1= 1 2H with a |q1= 2H to determine the assembler’s optimal mechanism when c1 2 2 Similarly, we compare a |q1= m c1 c2H with a |q1= m c2L c1 to determine the assembler’s optimal mechanism when c1 < (c2H c2L) .

m

2

2

When c1

c2L) , we have

(c2H

a |q1= m c1 c 2H 2

Define F (c1) = F (c1) c1 1

=

c 2H

(c2H

4

2

c2L) >

c2L)) =

c2L)

a |q1= m c2L 2 1

1

2

(c2H

F ( (c2H 1+

c 2L

2

4

a |q1=

c1

c12 + 0.

c 2H

c 2L 2

Thus,

1

=

c1 2

c1

4 4

2

a |q1=

m c1 c 2H 2

(c2H

c12 +

m c 2H 2

=

c2H

c2L 2

c12 4

w

m

w

> 0 . When c1 <

c1

4

(c2H

F (c1) increases

in

c 1.

Let

F (c1) = 0 ,

(c2H

c2L) 2 . (c 2H 1

c 2 L)

for

F (c1) < 0

c1 <

.

To summarize, the assembler’s optimal solution is as follows. If c1 <

q1 = 2 . If c1 1 + (c2H the proof of this proposition. m

c 2L

c1 2

c2L) , q1 =

m

w

c2L) , we have

c2L)2 , taking the first derivative of F (c1) with respect to c1, we have

c2L), therefore, the only feasible solution is c1 =

c1 <

(c2H

m

w

c2L) 2 .

(c2H

Therefore,

m

c1 c 2H , 2

w2H =

1 | 1 c 2H ,

w2L =

we

have

c1 =

1±

(c2H

1+

(c2H

c2L) . Since F (0) =

1+

(c2H

c2L) ,

(c2H

1+

2| 2

1

whereas

c2L) , then w2L = |

c2L) .

Note

that

(c 4 2H

c2L)2 and

F (c1)

0

for

c c 2L+ 1 ,

w2H = c2H ,

. Then T1, f2L and f2H follow. We thus complete

Proof of Proposition 12.. Under Scenario qp-tpt-seq1, the assembler’s problem is the same as that in (21). Furthermore, in the proof of Proposition 11, we find that a is separable and concave in q1 , w2L and w2H . Thus, the optimal mechanism under Scenario qp-tptseq1 is equivalent to that under Scenario qp-tpt-sim. Proof of Proposition 13.. The proof is straightforward, thus, it is omitted for brevity. Proof of Proposition 14.. We first prove part (a). Supplier 2’s profit is only determined by the assembler’s purchase quantity from the tpt sim tpt seq2 qp q2qp > qHtpt sim = q2qp > q2tpt high-cost supplier 2. Whenc1 < 1 + (c2H c2L) , . Otherwise, H H H

q2qp H

tpt seq2 qp = qHtpt sim = q2qp > q2tpt . H H We now prove part (b). Our analysis can be divided into two cases. (1) c1 < 1 + (c2H c2L) , we have tpt sim

78

Transportation Research Part E 129 (2019) 60–80

F. Lv, et al.

qp tpt sim a

qp tpt seq2 a

1

=

2

tpt qp a

(

1

c1 q1qp

2

=

1 2

qp tpt sim a tpt qp a

c1

)

qp tpt seq2 a

(c2H

= (q2tpt L

=

= To

m

tpt sim

2 2(1

qp tpt sim a

tpt sim 2 )

2

c 2H 2

=

(q2tpt L

+ (1

(

(c2H

qp 2 )

)

+ q2tpt L c1 2

c2L )

4

+

When c1

1+ qp tpt sim c

Therefore, tpt sim = c

qp tpt seq2 c

(c2H

=

2

c2H

)(q2tpt H

+ (1

c 2H 2

q2tpt H

)

2

qp

(c2H

)(q2tpt H

+ (1

qp tpt seq2 tpt sim > qp summarize, and a a qp qp tpt seq2 qp tpt sim = tpt > > . a a a We finally prove part (c). When c1 < 1 + (c2H qp tpt sim c

2

c2L) c1 < 0,

(c2H

)+

c2L)

2

c2L )

(c2H

2(1

qp 2 )

)

m

c2L)

(

(c2H

c 2H 2 m

c 2H 2

c2L) +

q2tpt H c1 2

qp

)

c1

1+

(c2H

c2L)

> 0.

c2L) 2

qp seq2 a

1

m

(c2H

2

qp 2 )

qp

qp c2L )(q2tpt L

(c2H

c1 2

c2L

qp tpt sim a

(q1qp =

c2H

c2L

qp 2 )

qp q2tpt ) H

(q1qp

tpt sim 2 )

> 0.

tpt qp a

qp tpt sim . a

>

Together

with

Corollaries

1

and

3,

we

have

c2L) , we have c1 2

< 0.

c2L) , we have qp tpt seq2 c

qp tpt sim c qp tpt seq2 > c

=

4

(c2H

c2L) 2 .

qp tpt seq2 . Together with c sim tpt seq2 qp tpt sim = qp > , tpt c c c

<

Proposition

tpt qp . c

9

and

Corollaries

2

and

3,

we

obtain

Appendix B. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.tre.2019.07.010.

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