# Manufacturer’s R&D Investment Strategy and Pricing Decisions in a Decentralized Supply Chain

Authors: Hengyun Zhang; Dingjun Hong
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

Consider that a manufacturer Stackelberg supply chain consists of an upstream supplier and a downstream manufacturer. The manufacturer purchases a component from the supplier and then transforms it into a final product which is sold in a price and quality sensitive market. The manufacturer considers to make R&D investment to improve the product quality and reduce the production cost. We first investigate and derive the optimal investment strategy and pricing decisions by establishing a three-stage game model. We show that the optimal investment strategy and pricing decisions in the decentralized model may deviate from those in the centralized model. We then propose a mechanism to coordinate the decentralized supply chain, by introducing a profit sharing policy, a production cost sharing policy, and an investment cost sharing policy. Finally, we show that both the supplier and the manufacturer can benefit from participating in the proposed coordination mechanism.


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## Body

## 1. Introduction

With the rapid development of economic globalization and the increasingly fierce market competition, the business environment of a firm becomes more changeable and complex. Under such a background, many firms have to maintain research and development (R&D) activities in quality improvement, cost reduction, and so on, to increase competitive edge [1]. For example, recent studies show that automakers spend more than $100 billion annually in R&D activities (please see http://www.autoalliance.org/auto-innovation/randd-investments).This paper is related to two streams of OR/MS literature. One stream is on quality improvement investment in supply chains. For example, Wan and Xu [2] investigate inspection policies in a decentralized supply chain where manufacturer uses an inspection policy and a damage cost sharing contract to encourage the supplier to invests in quality improvement. They show that if the supplier’s share of the damage cost exceeds a threshold, then all-or-none inspection policies are optimal for the manufacturer. Seifbarghy et al. [3] study the problem of the supplier’s quality improvement investment in a supply chain, where the market demand is divided into two categories. Xie et al. [4] consider quality improvement in a given segment of the market which is shared by two supply chains and provide valuable insights into the selection of supply chain structure and quality improvement investment strategy by two competing supply chains. Shi et al. [5] consider quality improvement problem in a supply chain with two suppliers and one manufacturer, where one is an urgent supplier with low quality and short lead time while the other is a strategic supplier with high quality and long lead time. Gao et al. [6] consider quality improvement investment strategy in a supply chain with failure root analysis and develop a contract mechanism to coordinate the supply chain. Zhu et al. [7] consider a supply chain where both the supplier and the manufacturer conduct investment to improve product quality and find that the manufacturer’s investment has a significant impact on the profits of each player and the entire supply chain. Please refer to Baiman et al. [8], Balachandran and Radhakrishnan [9], and Hwang et al. [10] for more discussions on quality improvement investment in supply chains. Although the above literature has investigated the problem of quality improvement investment from different perspectives, it supposes that all players in the supply chain do not make cost reduction investment.The other stream of literature is on cost reduction investment in supply chains. For example, Iyer et al. [11], Ge et al. [12], and Huang et al. [13] consider the problem of cost reduction investment in a supply chain with one upstream supplier and one downstream manufacturer. Bernstein et al. [14] explore the problem of cost reduction investment in a supply chain consisting of multiple suppliers and one manufacturer. Banerjee and Lin [15] investigate the problem of cost reduction investment in a supply chain consisting of one supplier and multiple manufacturers. The above literature has investigated the problem of cost reduction investment from different perspectives; it assumes that each player in the supply chain does not conduct quality improvement investment.The existing literature either supposes that players in the supply chain make investment to improve the product quality or assumes that players conduct investment to reduce the production cost. No study considers the problem of investment in both quality improvement and cost reduction. In this paper, we investigate a manufacturer Stackelberg supply chain, where the downstream manufacturer is the Stackelberg leader and the upstream supplier is the follower (see, e.g., [16]). The manufacturer buys a component from the supplier and then transforms it to the final product, which is sold in a price and quality sensitive market. We suppose that the manufacturer can make R&D investment not only to improve his product quality, but also to reduce his production cost. We first characterize the optimal pricing decisions of the two players under any given R&D investment strategy. We then derive the optimal investment strategy for the manufacturer. Finally, we propose a mechanism by introducing a profit sharing policy (see, e.g., [17]), a production cost sharing policy, and an investment cost sharing policy, to coordinate the decentralized supply chain. We also show that the proposed mechanism can result in a Pareto improvement.The contributions of our paper to the OR/MS literature can be summarized as follows.(1)
We establish a model to investigate the manufacturer’s R&D investment in both quality improvement activity and cost reduction activity.(2)
We characterize the optimal R&D investment strategy for the manufacturer and corresponding pricing decisions for both the supplier and the manufacturer in a decentralized supply chain.(3)
We develop a mechanism to coordinate the decentralized supply chain and show that the developed contract mechanism can achieve a win-win solution.The rest of the paper is structured as follows. We introduce the model in Section2. Section 3 investigates the decentralized supply chain, where the investment strategy and the pricing decisions are derived. Section 4 analyzes the problem of supply chain coordination. Some concluding remarks and further research directions are discussed in Section 5.


## 2. The Model

In this section, we describe the model. Consider a supply chain where an upstream supplier sells a component to a downstream manufacturer, who transforms the component into a final product and sells it into a price and quality sensitive market. The production cost and the wholesale price of the component arecs and w, respectively; the retail price of the final product is p. Without loss of generality, we assume similar to Xie et al. [4] that the market demand faced by the manufacturer is given by(1)Dp,α=a-bp+θα,where a in (1) represents the potential intrinsic market demand, b represents the sensitivity of the market demand to the retail price, α corresponds to the quality level of the product, and θ corresponds to the sensitivity of the market demand to the quality level.The manufacturer can make R&D investment to improve the product quality and/or reduce the production cost. We denote the initial quality and the initial production cost of the product to beαm and cm, and the quality improvement level and cost reduction level of the product caused by R&D investment are αΔ and cΔ, respectively. We require that a-b(cm+cs)+θαm≥0, in the sense that when the retail price of the product is set at its initial production cost (i.e., cm+cs), the market demand is nonnegative (see, e.g., [18]). After the manufacturer’s investment, the product quality and the production cost are αm+αΔ and cm-cΔ, respectively. We assume similar to Xie et al. [4] that the investment cost of quality improvement is f(αΔ), which is convexly increasing in αΔ and is given by(2)fαΔ=AαΔ2,where A in (2) is a parameter related to the marginal cost of quality improvement investment and A>0. Moreover, similar to Ge et al. [12], we assume that the investment cost of cost reduction is g(cΔ), which is convexly increasing in cΔ and is given by(3)gcΔ=BcΔ2,where B in (3) is a parameter related to the marginal cost of cost reduction investment and B>0. Furthermore, we suppose that the manufacturer has capital constraint and thus his investment activity is constrained by f(αΔ)+g(cΔ)≤K; that is, the manufacturer can invest at most K to achieve quality improvement and/or cost reduction. To ensure that the production cost of the manufacturer is always positive, that is, cm-cΔ>0, we require that cm-K/B1/2>0; that is, B>K/cm2.We consider a manufacturer Stackelberg supply chain where the manufacturer has larger influence on the market than the supplier and endows the manufacturer with the first-mover advantage [16]. The two players engage in a three-stage game to determine their optimal decisions, specifically as follows.(i)
In the first stage, the manufacturer chooses his investment strategy, that is,f(αΔ) and g(cΔ). Because there is a one-to-one correspondence between f(αΔ) and αΔ and there exists a one-to-one relationship between g(cΔ) and cΔ, choosing values for f(αΔ) and g(cΔ) is equal to choosing the corresponding values for αΔ and cΔ, respectively.(ii)
In the second stage, the manufacturer sets a markupm above his procurement cost, that is, the upstream firm’s wholesale price w, and sells the product to the market at the price of p=w+m (see, e.g., [19]).(iii)
In the last stage, the supplier determines a wholesale pricew for the component to be charged to the manufacturer.


## 3. Decentralized Supply Chain

In a decentralized supply chain, both the supplier and the manufacturer are independent decision makers. They make decisions to maximize their own profits. Using backward induction, we first solve the problem of the supplier’s wholesale price which is chosen in the third stage.


### 3.1. The Third Stage

Based upon the description provided in Section2, we know that for any investment strategy (αΔ, cΔ) and markup m decision chosen by the manufacturer, the profit function of the supplier can be expressed as(4)Πsw∣m,αΔ,cΔ=w-csa-bp+θα=w-csa-bw+m+θαm+αΔ.From (4), we can obtain the following result.Proposition 1.
For any investment strategy(αΔ,cΔ) and markup m decision chosen by the manufacturer, the supplier’s profit function Πs(w∣m,αΔ,cΔ) of (4) is strictly concave in its wholesale price w and reaches its maximum at(5)w∗m,αΔ,cΔ=a+θαm+αΔ-bm-cs2b.Proof of Proposition1.
Taking the first and the second derivatives ofΠsw∣m,αΔ,cΔ of (4) with respect to w, we have (6)dΠsw∣m,αΔ,cΔdw=a-2bw-bm+θαm+αΔ+bcs,d2Πsw∣m,αΔ,cΔdw2=-2b<0.That is, the profit function of the supplier is strictly concave in the wholesale price. Thus, the optimal wholesale price satisfies the first-order condition, that is, dΠsw∣m,αΔ,cΔ/dw=0. Solving the first-order condition for w, we get (5). We thus complete the proof of Proposition 1.Proposition1 gives the supplier’s optimal response wholesale price. From Proposition 1, we can see that the supplier’s optimal response wholesale price w∗(m,αΔ,cΔ) increases as the quality improvement level αΔ increases. This is because when the quality improvement level increases, the quality sensitive demand will increase and the supplier will set a higher wholesale price to earn more profit. We can also see from Proposition 1 that the supplier’s optimal response wholesale price w∗(m,αΔ,cΔ) decreases as the markup m increases. The reason is that a higher markup is chosen by the manufacturer, a higher retail price will be set for the product, and the price sensitive demand will decrease. Thus, the supplier will decrease his wholesale price. In addition, Proposition 1 states that the supplier’s optimal response wholesale price w∗(m,αΔ,cΔ) does not directly depend on the reduction in the manufacturer’s production cost cΔ. In our subsequent analyses of the manufacturer’s markup decision, we will show that cost reduction cΔ influences the supplier’s wholesale price decision via the manufacturer’s markup decision.


### 3.2. The Second Stage

Knowing that the supplier chooses the wholesale price according to (5), the manufacturer sets its markup in the second stage to maximize his profit. Based upon the description provided in Section 2, the profit function of manufacturer can be written as(7)Πmm∣w,αΔ,cΔ=m-cm-cΔa-bw+m+θαm+αΔ-fαΔ-gcΔ.Then, by substitutingw=w∗(m,αΔ,cΔ) of (5) into Πmm∣w,αΔ,cΔ of (7), the manufacturer’s profit can be rewritten as(8)Πmm∣αΔ,cΔ=m-cm-cΔa+θαm+αΔ-bcs-mb2-fαΔ-gcΔ.This enables us to derive the following result.Proposition 2.
For any given investment strategy(αΔ,cΔ), the manufacturer’s profit function Πmm∣αΔ,cΔ of (8) is strictly concave in its markup m and achieves its maximum at(9)m∗αΔ,cΔ=a+θαm+αΔ-bcs+bcm-cΔ2b.Consequently, the corresponding wholesale price of the supplier is given by(10)w∗αΔ,cΔ=a+θαm+αΔ+3bcs-bcm-cΔ4b,and the corresponding retail price and the supply chain output are as follows:(11)p∗αΔ,cΔ=3a+3θαm+αΔ+bcs+bcm-cΔ4b,(12)q∗αΔ,cΔ=a+θαm+αΔ-bcs-bcm-cΔ4.Proof of Proposition2.
Taking the first and the second derivatives ofΠmm∣αΔ,cΔ of (8) with respect to m, we have (13)dΠmm∣αΔ,cΔdm=a+θαm+αΔ-bcs-2mb+bcm-cΔ2,d2Πmm∣αΔ,cΔdm2=-b<0.That is, the profit function of the manufacturer is strictly concave in its markup. Thus, the optimal markup satisfies the first-order condition, that is, dΠmm∣αΔ,cΔ/dm=0. Solving the first-order condition for m, we get (9).
Second, substitutingm=m∗(αΔ,cΔ) of (9) into w∗(m,αΔ,cΔ) of (5), we get w∗(αΔ,cΔ) of (10). Substituting m=m∗(αΔ,cΔ) of (9) and w=w∗(αΔ,cΔ) of (10) into p=w+m, we get p∗(αΔ,cΔ) of (11). Substituting p=p∗(αΔ,cΔ) of (11) into (1), we get q∗(αΔ,cΔ) of (12). We thus complete the proof of Proposition 2.Proposition2 characterizes the manufacturer’s optimal markup under any given investment strategy and the corresponding wholesale price, retail price, and the supply chain output. Firstly, from (9), one can see that the manufacturer’s optimal markup m∗(αΔ,cΔ) increases as the quality improvement level αΔ increases but decreases as the cost reduction level cΔ increases. That is, the manufacturer with a high quality product or a higher production cost will choose a higher markup above the supplier’s wholesale price. This is rather intuitive. Secondly, (10) confirms the discussion following Proposition 1 that the cost reduction influences the supplier’s wholesale price choice via the manufacturer’s markup choice. That is, the supplier’s wholesale price w∗(αΔ,cΔ) increases as the cost reduction level cΔ increases. Thirdly, (11) shows that the retail price p∗(αΔ,cΔ) increases as the quality improvement level αΔ increases; this is because when the quality improvement level αΔ increases, both the wholesale price and the markup increase, and hence the retail price p=w+m increases; the retail price p∗(αΔ,cΔ) decreases as the cost reduction level cΔ increases; this is because the impacts of the cost reduction on the wholesale price and the markup are opposite, and the impact of the cost reduction on the markup outweighs the impact of the cost reduction on the wholesale price. Finally, (12) indicates that the supply chain output q∗(αΔ,cΔ) increases not only with the quality improvement level αΔ but also with the cost reduction level cΔ. The reason for this result can be explained as follows. When either the product quality is improved or the production cost is reduced, the supply chain becomes more efficient, and hence, more product will be produced to meet the market demand.Now, substitutingm=m∗(αΔ,cΔ) of (9) and w=w∗(αΔ,cΔ) of (10) into Πsw∣m,αΔ,cΔ of (4) and Πmm∣αΔ,cΔ of (8), we can obtain the profits of the two players as follows, respectively,(14)Πs∗αΔ,cΔ=a+θαm+αΔ-bcs-bcm-cΔ216b,(15)Πm∗αΔ,cΔ=a+θαm+αΔ-bcs-bcm-cΔ28b-fαΔ-gcΔ.From (14), we can show that the supplier’s profit increases not only with the quality improvement level αΔ but also with the cost reduction level cΔ. This implies that the supplier can always benefit from the investment. The result is not surprising, because the supplier enjoys a more efficient supply chain but does not undertake any cost for improving the supply chain efficiency. However, the manufacturer may not benefit from his investment, because he should balance the investment cost and the corresponding benefit.


### 3.3. The First Stage

In this subsection, we consider the investment strategy of the manufacturer. Recall that we have assumed that the manufacturer’s R&D investment activity is constrained byf(αΔ)+g(cΔ)≤K, f(αΔ)=AαΔ2 and g(cΔ)=BcΔ2. Then, the manufacturer’s investment problem can be formulated as the following two-dimensional constrained optimization problem:(16)maxx,yΠm∗αΔ,cΔ=a+θαm+αΔ-bcs-bcm-cΔ28b-AαΔ2-BcΔ2,s.t.αΔ≥0,cΔ≥0,0≤AαΔ2+BcΔ2≤K.This enables us to get the following result.Proposition 3.
Letk denote the total capital invested by the manufacturer in both the quality improvement activity and the cost reduction activity, that is, k=AαΔ2+BcΔ2, 0≤k≤K, then Πm∗αΔ,cΔ of (16) achieves its maximum at(17)cΔ∗k=Ab2θ2B2+ABb21/2·k1/2,(18)αΔ∗k=A-1B2θ2θ2B2+ABb21/2·k1/2.Proof of Proposition3.
For any givenk=AαΔ2+BcΔ2, we have αΔ=k-B(cΔ)2/A1/2, and then, the manufacturer’s profit function Πm∗αΔ,cΔ of (16) can be rewritten as (19)Πm∗cΔ=a+θαm+k-BcΔ2/A1/2-bcs-bcm-cΔ28b-k. Obviously, for any given k=AαΔ2+BcΔ2, choosing a value to maximize Πm∗cΔ is equal to choosing a value to maximize (20)TcΔ=θαm+k-BcΔ2A1/2-bcm-cΔ.Taking the first derivative of TcΔ with respect to cΔ, we have(21)dTcΔdcΔ=b-θBAk/cΔ2-AB1/2.It is easy to verify that dT(cΔ)/dcΔ decreases as cΔ increases, implying that d2TcΔ/dcΔ2<0. That is, TcΔ is strictly concave in cΔ, and hence the optimal cost reduction level satisfies the first-order condition; that is, dTcΔ/dcΔ=0. Solving the first-order condition for cΔ, we get (17). Second, substituting cΔ=cΔ∗(k) of (17) into k=AαΔ2+BcΔ2, we have αΔ∗(k) of (18). We thus complete the proof of Proposition 3.Proposition3 implies that the optimal investment strategy requires that the capital should be invested not only in the quality improvement activity but also in the cost reduction activity. Moreover, one can see from Proposition 3 that if the total investment cost, that is, k=A(αΔ)2+B(cΔ)2, is given, then the investment costs for achieving quality improvement and cost reduction, respectively, are(22)AαΔ∗k2=B2θ2θ2B2+ABb2·k,(23)BcΔ∗k2=ABb2θ2B2+ABb2·k.Accordingly, the ratio of capital invested by the manufacturer in quality improvement activity and cost reduction activity in the decentralized supply chain is(24)AαΔ∗k2BcΔ∗k2=Bθ2Ab2.Equation (24) reveals the following important insights: when the total capital invested by the manufacturer is fixed, that is, k=A(αΔ)2+B(cΔ)2 is fixed, if the parameter A is relatively small or the parameter θ is relatively large, then a relatively more capital will be invested in quality improvement activity. This is because a smaller value of A implies that the marginal cost of quality improvement investment is lower and a larger value of θ indicates that the market demand is more sensitive to the quality improvement. Of course, the manufacturer is likely to invest a relatively more capital in quality improvement activity. Similar analysis can be conducted for the parameters B and b.Because the manufacturer will allocate its capitalk=A(αΔ)2+B(cΔ)2 in quality improvement activity and cost reduction activity according to (24), we can reduce the two-dimensional constrained optimization problem of (16) to the following one-dimensional constrained optimization problem:(25)maxkΠm∗k=a+θαm-bcs-bcm+A-1B2θ4/θ2B2+ABb21/2·k1/2+Ab4/θ2B2+ABb21/2·k1/228b-k,s.t.0≤k≤K.That is, the manufacturer can first determines the total capital to be invested in quality improvement activity and cost reduction activity and then allocates the total capital between the two activities according to (24). The results are summarized below in Proposition 4.Proposition 4.
The manufacturer’s profit functionΠm∗(k) of (25) is strictly concave in k and achieves its maximum at(26)k∗=a+θαm-bcs-bcmA-1/2Bθ2+A1/2b2θ2B2+ABb21/28bθ2B2+ABb2-A-1/2Bθ2+A1/2b222.Accordingly, the optimal total capital invested by the manufacturer is(27)k¯∗=k∗,ifk∗<K,K,ifk∗≥K,and the optimal capital invested by the manufacturer in quality improvement activity and cost reduction activity, respectively, is(28)AαΔ∗k¯∗2=B2θ2θ2B2+ABb2·k¯∗,(29)BcΔ∗k¯∗2=ABb2θ2B2+ABb2·k¯∗.Proof of Proposition4.
Taking the first derivative ofΠm∗(k) of (25) with respect to k, we have, after some algebra, that (30)dΠm∗kdk=a+θαm-bcs-bcm/k1/2+A-1B2θ4/θ2B2+ABb21/2+Ab4/θ2B2+ABb21/2A-1B2θ4/θ2B2+ABb21/2+Ab4/θ2B2+ABb21/28b-1.
It is easy to verify thatdΠm∗(k)/dk decreases as k increases, implying that d2Πm∗(k)/dk2<0. That is, Πm∗(k) is strictly concave in k, and hence the maximizer of profit function Πm∗(k) is uniquely determined by the first-order condition; that is, dΠm∗(k)/dk=0. Solving the first-order condition for k, we can get k∗ of (26). This, together with the fact that 0≤k≤K, indicates that the optimal total capital for the manufacturer is given by k¯∗ of (27). Second, substituting k=k¯∗ of (27) into (22) and (23), respectively, we can get (28) and (29), respectively. We thus complete the proof of Proposition 4.Proposition4 characterizes the optimal investment strategy for the manufacturer in the decentralized supply chain, which depends on supply chain parameters a, b, β, I, J, K. Substituting p∗(αΔ,cΔ) of (11), q∗(αΔ,cΔ) of (12), AαΔ∗k¯∗2 of (28), and BcΔ∗k¯∗2 of (29) into Πsw∣m,αΔ,cΔ of (4) and Πmm∣w,αΔ,cΔ of (7), we can get the optimal profits of the supplier and the manufacturer in the decentralized supply chain, denoted by Πs∗ and Πm∗, respectively.


## 3.1. The Third Stage

Based upon the description provided in Section2, we know that for any investment strategy (αΔ, cΔ) and markup m decision chosen by the manufacturer, the profit function of the supplier can be expressed as(4)Πsw∣m,αΔ,cΔ=w-csa-bp+θα=w-csa-bw+m+θαm+αΔ.From (4), we can obtain the following result.Proposition 1.
For any investment strategy(αΔ,cΔ) and markup m decision chosen by the manufacturer, the supplier’s profit function Πs(w∣m,αΔ,cΔ) of (4) is strictly concave in its wholesale price w and reaches its maximum at(5)w∗m,αΔ,cΔ=a+θαm+αΔ-bm-cs2b.Proof of Proposition1.
Taking the first and the second derivatives ofΠsw∣m,αΔ,cΔ of (4) with respect to w, we have (6)dΠsw∣m,αΔ,cΔdw=a-2bw-bm+θαm+αΔ+bcs,d2Πsw∣m,αΔ,cΔdw2=-2b<0.That is, the profit function of the supplier is strictly concave in the wholesale price. Thus, the optimal wholesale price satisfies the first-order condition, that is, dΠsw∣m,αΔ,cΔ/dw=0. Solving the first-order condition for w, we get (5). We thus complete the proof of Proposition 1.Proposition1 gives the supplier’s optimal response wholesale price. From Proposition 1, we can see that the supplier’s optimal response wholesale price w∗(m,αΔ,cΔ) increases as the quality improvement level αΔ increases. This is because when the quality improvement level increases, the quality sensitive demand will increase and the supplier will set a higher wholesale price to earn more profit. We can also see from Proposition 1 that the supplier’s optimal response wholesale price w∗(m,αΔ,cΔ) decreases as the markup m increases. The reason is that a higher markup is chosen by the manufacturer, a higher retail price will be set for the product, and the price sensitive demand will decrease. Thus, the supplier will decrease his wholesale price. In addition, Proposition 1 states that the supplier’s optimal response wholesale price w∗(m,αΔ,cΔ) does not directly depend on the reduction in the manufacturer’s production cost cΔ. In our subsequent analyses of the manufacturer’s markup decision, we will show that cost reduction cΔ influences the supplier’s wholesale price decision via the manufacturer’s markup decision.


## 3.2. The Second Stage

Knowing that the supplier chooses the wholesale price according to (5), the manufacturer sets its markup in the second stage to maximize his profit. Based upon the description provided in Section 2, the profit function of manufacturer can be written as(7)Πmm∣w,αΔ,cΔ=m-cm-cΔa-bw+m+θαm+αΔ-fαΔ-gcΔ.Then, by substitutingw=w∗(m,αΔ,cΔ) of (5) into Πmm∣w,αΔ,cΔ of (7), the manufacturer’s profit can be rewritten as(8)Πmm∣αΔ,cΔ=m-cm-cΔa+θαm+αΔ-bcs-mb2-fαΔ-gcΔ.This enables us to derive the following result.Proposition 2.
For any given investment strategy(αΔ,cΔ), the manufacturer’s profit function Πmm∣αΔ,cΔ of (8) is strictly concave in its markup m and achieves its maximum at(9)m∗αΔ,cΔ=a+θαm+αΔ-bcs+bcm-cΔ2b.Consequently, the corresponding wholesale price of the supplier is given by(10)w∗αΔ,cΔ=a+θαm+αΔ+3bcs-bcm-cΔ4b,and the corresponding retail price and the supply chain output are as follows:(11)p∗αΔ,cΔ=3a+3θαm+αΔ+bcs+bcm-cΔ4b,(12)q∗αΔ,cΔ=a+θαm+αΔ-bcs-bcm-cΔ4.Proof of Proposition2.
Taking the first and the second derivatives ofΠmm∣αΔ,cΔ of (8) with respect to m, we have (13)dΠmm∣αΔ,cΔdm=a+θαm+αΔ-bcs-2mb+bcm-cΔ2,d2Πmm∣αΔ,cΔdm2=-b<0.That is, the profit function of the manufacturer is strictly concave in its markup. Thus, the optimal markup satisfies the first-order condition, that is, dΠmm∣αΔ,cΔ/dm=0. Solving the first-order condition for m, we get (9).
Second, substitutingm=m∗(αΔ,cΔ) of (9) into w∗(m,αΔ,cΔ) of (5), we get w∗(αΔ,cΔ) of (10). Substituting m=m∗(αΔ,cΔ) of (9) and w=w∗(αΔ,cΔ) of (10) into p=w+m, we get p∗(αΔ,cΔ) of (11). Substituting p=p∗(αΔ,cΔ) of (11) into (1), we get q∗(αΔ,cΔ) of (12). We thus complete the proof of Proposition 2.Proposition2 characterizes the manufacturer’s optimal markup under any given investment strategy and the corresponding wholesale price, retail price, and the supply chain output. Firstly, from (9), one can see that the manufacturer’s optimal markup m∗(αΔ,cΔ) increases as the quality improvement level αΔ increases but decreases as the cost reduction level cΔ increases. That is, the manufacturer with a high quality product or a higher production cost will choose a higher markup above the supplier’s wholesale price. This is rather intuitive. Secondly, (10) confirms the discussion following Proposition 1 that the cost reduction influences the supplier’s wholesale price choice via the manufacturer’s markup choice. That is, the supplier’s wholesale price w∗(αΔ,cΔ) increases as the cost reduction level cΔ increases. Thirdly, (11) shows that the retail price p∗(αΔ,cΔ) increases as the quality improvement level αΔ increases; this is because when the quality improvement level αΔ increases, both the wholesale price and the markup increase, and hence the retail price p=w+m increases; the retail price p∗(αΔ,cΔ) decreases as the cost reduction level cΔ increases; this is because the impacts of the cost reduction on the wholesale price and the markup are opposite, and the impact of the cost reduction on the markup outweighs the impact of the cost reduction on the wholesale price. Finally, (12) indicates that the supply chain output q∗(αΔ,cΔ) increases not only with the quality improvement level αΔ but also with the cost reduction level cΔ. The reason for this result can be explained as follows. When either the product quality is improved or the production cost is reduced, the supply chain becomes more efficient, and hence, more product will be produced to meet the market demand.Now, substitutingm=m∗(αΔ,cΔ) of (9) and w=w∗(αΔ,cΔ) of (10) into Πsw∣m,αΔ,cΔ of (4) and Πmm∣αΔ,cΔ of (8), we can obtain the profits of the two players as follows, respectively,(14)Πs∗αΔ,cΔ=a+θαm+αΔ-bcs-bcm-cΔ216b,(15)Πm∗αΔ,cΔ=a+θαm+αΔ-bcs-bcm-cΔ28b-fαΔ-gcΔ.From (14), we can show that the supplier’s profit increases not only with the quality improvement level αΔ but also with the cost reduction level cΔ. This implies that the supplier can always benefit from the investment. The result is not surprising, because the supplier enjoys a more efficient supply chain but does not undertake any cost for improving the supply chain efficiency. However, the manufacturer may not benefit from his investment, because he should balance the investment cost and the corresponding benefit.


## 3.3. The First Stage

In this subsection, we consider the investment strategy of the manufacturer. Recall that we have assumed that the manufacturer’s R&D investment activity is constrained byf(αΔ)+g(cΔ)≤K, f(αΔ)=AαΔ2 and g(cΔ)=BcΔ2. Then, the manufacturer’s investment problem can be formulated as the following two-dimensional constrained optimization problem:(16)maxx,yΠm∗αΔ,cΔ=a+θαm+αΔ-bcs-bcm-cΔ28b-AαΔ2-BcΔ2,s.t.αΔ≥0,cΔ≥0,0≤AαΔ2+BcΔ2≤K.This enables us to get the following result.Proposition 3.
Letk denote the total capital invested by the manufacturer in both the quality improvement activity and the cost reduction activity, that is, k=AαΔ2+BcΔ2, 0≤k≤K, then Πm∗αΔ,cΔ of (16) achieves its maximum at(17)cΔ∗k=Ab2θ2B2+ABb21/2·k1/2,(18)αΔ∗k=A-1B2θ2θ2B2+ABb21/2·k1/2.Proof of Proposition3.
For any givenk=AαΔ2+BcΔ2, we have αΔ=k-B(cΔ)2/A1/2, and then, the manufacturer’s profit function Πm∗αΔ,cΔ of (16) can be rewritten as (19)Πm∗cΔ=a+θαm+k-BcΔ2/A1/2-bcs-bcm-cΔ28b-k. Obviously, for any given k=AαΔ2+BcΔ2, choosing a value to maximize Πm∗cΔ is equal to choosing a value to maximize (20)TcΔ=θαm+k-BcΔ2A1/2-bcm-cΔ.Taking the first derivative of TcΔ with respect to cΔ, we have(21)dTcΔdcΔ=b-θBAk/cΔ2-AB1/2.It is easy to verify that dT(cΔ)/dcΔ decreases as cΔ increases, implying that d2TcΔ/dcΔ2<0. That is, TcΔ is strictly concave in cΔ, and hence the optimal cost reduction level satisfies the first-order condition; that is, dTcΔ/dcΔ=0. Solving the first-order condition for cΔ, we get (17). Second, substituting cΔ=cΔ∗(k) of (17) into k=AαΔ2+BcΔ2, we have αΔ∗(k) of (18). We thus complete the proof of Proposition 3.Proposition3 implies that the optimal investment strategy requires that the capital should be invested not only in the quality improvement activity but also in the cost reduction activity. Moreover, one can see from Proposition 3 that if the total investment cost, that is, k=A(αΔ)2+B(cΔ)2, is given, then the investment costs for achieving quality improvement and cost reduction, respectively, are(22)AαΔ∗k2=B2θ2θ2B2+ABb2·k,(23)BcΔ∗k2=ABb2θ2B2+ABb2·k.Accordingly, the ratio of capital invested by the manufacturer in quality improvement activity and cost reduction activity in the decentralized supply chain is(24)AαΔ∗k2BcΔ∗k2=Bθ2Ab2.Equation (24) reveals the following important insights: when the total capital invested by the manufacturer is fixed, that is, k=A(αΔ)2+B(cΔ)2 is fixed, if the parameter A is relatively small or the parameter θ is relatively large, then a relatively more capital will be invested in quality improvement activity. This is because a smaller value of A implies that the marginal cost of quality improvement investment is lower and a larger value of θ indicates that the market demand is more sensitive to the quality improvement. Of course, the manufacturer is likely to invest a relatively more capital in quality improvement activity. Similar analysis can be conducted for the parameters B and b.Because the manufacturer will allocate its capitalk=A(αΔ)2+B(cΔ)2 in quality improvement activity and cost reduction activity according to (24), we can reduce the two-dimensional constrained optimization problem of (16) to the following one-dimensional constrained optimization problem:(25)maxkΠm∗k=a+θαm-bcs-bcm+A-1B2θ4/θ2B2+ABb21/2·k1/2+Ab4/θ2B2+ABb21/2·k1/228b-k,s.t.0≤k≤K.That is, the manufacturer can first determines the total capital to be invested in quality improvement activity and cost reduction activity and then allocates the total capital between the two activities according to (24). The results are summarized below in Proposition 4.Proposition 4.
The manufacturer’s profit functionΠm∗(k) of (25) is strictly concave in k and achieves its maximum at(26)k∗=a+θαm-bcs-bcmA-1/2Bθ2+A1/2b2θ2B2+ABb21/28bθ2B2+ABb2-A-1/2Bθ2+A1/2b222.Accordingly, the optimal total capital invested by the manufacturer is(27)k¯∗=k∗,ifk∗<K,K,ifk∗≥K,and the optimal capital invested by the manufacturer in quality improvement activity and cost reduction activity, respectively, is(28)AαΔ∗k¯∗2=B2θ2θ2B2+ABb2·k¯∗,(29)BcΔ∗k¯∗2=ABb2θ2B2+ABb2·k¯∗.Proof of Proposition4.
Taking the first derivative ofΠm∗(k) of (25) with respect to k, we have, after some algebra, that (30)dΠm∗kdk=a+θαm-bcs-bcm/k1/2+A-1B2θ4/θ2B2+ABb21/2+Ab4/θ2B2+ABb21/2A-1B2θ4/θ2B2+ABb21/2+Ab4/θ2B2+ABb21/28b-1.
It is easy to verify thatdΠm∗(k)/dk decreases as k increases, implying that d2Πm∗(k)/dk2<0. That is, Πm∗(k) is strictly concave in k, and hence the maximizer of profit function Πm∗(k) is uniquely determined by the first-order condition; that is, dΠm∗(k)/dk=0. Solving the first-order condition for k, we can get k∗ of (26). This, together with the fact that 0≤k≤K, indicates that the optimal total capital for the manufacturer is given by k¯∗ of (27). Second, substituting k=k¯∗ of (27) into (22) and (23), respectively, we can get (28) and (29), respectively. We thus complete the proof of Proposition 4.Proposition4 characterizes the optimal investment strategy for the manufacturer in the decentralized supply chain, which depends on supply chain parameters a, b, β, I, J, K. Substituting p∗(αΔ,cΔ) of (11), q∗(αΔ,cΔ) of (12), AαΔ∗k¯∗2 of (28), and BcΔ∗k¯∗2 of (29) into Πsw∣m,αΔ,cΔ of (4) and Πmm∣w,αΔ,cΔ of (7), we can get the optimal profits of the supplier and the manufacturer in the decentralized supply chain, denoted by Πs∗ and Πm∗, respectively.


## 4. Supply Chain Coordination

Because double marginalization [20] exists in the decentralized supply chain, the decentralization decisions are usually suboptimal, as compared to the centralized decisions. In this section, we investigate the problem of supply chain coordination. We first consider the centralized supply chain, which serves as a benchmark for supply chain coordination. We then develop a contract mechanism to coordinate the decentralized supply chain.


### 4.1. Centralized Supply Chain

In the centralized supply chain, there is a central planner who optimizes supply chain profit, that is, the total profit of the supplier and the manufacturer. From (4) and (7), we can obtain the supply chain profit(31)Πcw,m∣αΔ,cΔ=Πsw∣m,αΔ,cΔ+Πmm∣w,αΔ,cΔ=w+m-cm-cs+cΔa-bw+m+θαm+αΔ-fαΔ-gcΔ.This, together with the fact that p=w+m, indicates that the supply chain profit can be rewritten as follows:(32)Πcp∣αΔ,cΔ=p-cm-cs+cΔa-bp+θαm+αΔ-fαΔ-gcΔ.As a matter of fact, if the investment strategy (αΔ,cΔ) is given, then, in the decentralized supply chain, the values for wholesale price w and markup m are, respectively, chosen by the supplier and the manufacturer to maximize their own profits. However, in the centralized supply chain, it needs only to choose a value for w+m, i.e., p, to maximize the entire supply chain profit.From (32), we have the following result.Proposition 5.
In a centralized model, for any given investment strategy(αΔ,cΔ), the supply chain’s profit function Πcp∣αΔ,cΔ of (32) is strictly concave in the retail price p and achieves its maximum at(33)pc∗αΔ,cΔ=a+θαm+αΔ+bcs+bcm-cΔ2b.Consequently, the corresponding supply chain output is(34)qc∗αΔ,cΔ=a+θαm+αΔ-bcs-bcm-cΔ2.Proof of Proposition5.
Taking the first and the second derivatives ofΠcp∣αΔ,cΔ of (32) with respect to p, we have (35)dΠcp∣αΔ,cΔdp=a-2bp+θαm+αΔ+bcs+cm-cΔ,d2Πcp∣αΔ,cΔdp2=-2b<0.That is, the supply chain’s profit function Πcp∣αΔ,cΔ of (32) is strictly concave in the retail price p. Thus, the optimal retail price satisfies the first-order condition, that is, dΠcp∣αΔ,cΔ/dp=0. Solving the first-order condition for p, we get (33). Second, substituting p=pc∗(αΔ,cΔ) of (33) into (1), we have (34). We thus complete the proof of Proposition 5.By comparing Propositions5 and 2, we can make the following corollary.Corollary 6.
For any given investment strategy(αΔ,cΔ), the optimal retail price in the centralized model is lower than that in the decentralized model; that is, pc∗(αΔ,cΔ)<p∗(αΔ,cΔ), and the optimal supply chain output in the centralized model is higher than that in the decentralized model; that is, qc∗(αΔ,cΔ)>q∗(αΔ,cΔ).Proof of Corollary6.
First, it follows from (11) and (33) that (36)pc∗αΔ,cΔ-p∗αΔ,cΔ=-a+θαm+αΔ-bcs-bcm-cΔ4b.This, together with the fact a-b(cm+cs)+θαm≥0, indicates that pc∗(αΔ,cΔ)-p∗(αΔ,cΔ)<0. Second, it follows from (12) and (34) that (37)qc∗αΔ,cΔ-q∗αΔ,cΔ=a+θαm+αΔ-bcs-bcm-cΔ4.This, together with the fact a-b(cm+cs)+θαm≥0, indicates that qc∗αΔ,cΔ-q∗(αΔ,cΔ)>0. We thus complete the proof of Corollary 6.Corollary6 states that, for any given investment strategy (αΔ,cΔ), the optimal retail price and supply chain output in the decentralized model deviate from that in the centralized model. Such a deviation will lead to a profit loss for the entire supply chain.Now, substitutingp=pc∗(αΔ,cΔ) of (33) and q=qc∗(αΔ,cΔ) of (34) into Πcp∣αΔ,cΔ of (32), we can get the entire supply chain’s profit:(38)Πc∗αΔ,cΔ=a+θαm+αΔ-bcs-bcm-cΔ24b-fαΔ-gcΔ.This leads to the following result.Proposition 7.
In a centralized model, the optimal total capital invested in the quality improvement activity and cost reduction activity is(39)k¯c∗=kc∗,ifkc∗<K,K,ifkc∗≥K,where(40)kc∗=a+θαm-bcs-bcmA-1/2Bθ2+A1/2b2θ2B2+ABb21/24bθ2B2+ABb2-A-1/2Bθ2+A1/2b222.Moreover, the optimal capital invested in quality improvement activity and cost reduction activity, respectively, is(41)AαΔ∗k¯c∗2=B2θ2θ2B2+ABb2·k¯c∗,(42)BcΔ∗k¯c∗2=ABb2θ2B2+ABb2·k¯c∗.Proof of Proposition7.
This proof is analog to the proof of Proposition4; we omit it here.By comparing Propositions7 and 4, we obtain the following corollary.Corollary 8.
IfK≤k∗, then the optimal investment strategies are identical in the decentralized and centralized models; that is, A(αΔ∗(k¯∗))2=A(αΔ∗(k¯c∗))2 and B(cΔ∗(k¯∗))2=B(cΔ∗(k¯c∗))2; if K>k∗, then the optimal quality improvement investment level and cost reduction investment level in the decentralized model are both lower than those in the decentralized model; that is, AαΔ∗k¯∗2<AαΔ∗k¯c∗2 and BcΔ∗k¯∗2<BcΔ∗k¯c∗2.Proof of Corollary8.
First, ifK≤k∗, then AαΔ∗k¯∗2=B2θ2/θ2B2+ABb2·K=AαΔ∗k¯c∗2 and BcΔ∗k¯∗2=ABb2/θ2B2+ABb2·K=BcΔ∗k¯c∗2. Second, if k∗<K<kc∗, then AαΔ∗k¯∗2=B2θ2/θ2B2+ABb2·k∗<B2θ2/(θ2B2+ABb2)·K=AαΔ∗k¯c∗2 and BcΔ∗k¯∗2=ABb2/(θ2B2+ABb2)·k∗<ABb2/(θ2B2+ABb2)·K=BcΔ∗k¯c∗2. Third, if K≥kc∗, then AαΔ∗k¯∗2=B2θ2/(θ2B2+ABb2)·k∗<B2θ2/(θ2B2+ABb2)·kc∗=AαΔ∗k¯c∗2 and BcΔ∗k¯∗2=ABb2/(θ2B2+ABb2)·k∗<ABb2/(θ2B2+ABb2)·kc∗=BcΔ∗k¯c∗2. We thus complete the proof of Corollary 8.Corollary8 shows that the optimal investment strategy in the decentralized model may deviate from that in the centralized model. Specifically, when K≤k∗, the investment strategies are identical in the two models. This is because when K≤k∗, the whole capital will be invested in each of the two models. However, when k∗<K<kc∗, the capital constraint has effect in the centralized model but has no effect in the decentralized model, and then the whole capital will be invested in the centralized model but will not be invested fully in the decentralized model. When k>kc∗, the capital constraint has no effect in each of the two models and the investment level in the decentralized model is lower than that in the centralized model. Additionally, substituting pc∗(αΔ,cΔ) of (33), qc∗(αΔ,cΔ) of (34), AαΔ∗k¯c∗2 of (41), and BcΔ∗k¯c∗2 of (42) into (32), we can get the optimal profit of the centralized supply chain, denoted by Πc∗.


### 4.2. Coordination Mechanism

In Section4.1, we have shown that the optimal investment strategy and retail price decision in the decentralized model may deviate from that in the centralized model. Thus, the supply chain profit under the decentralized decision making is suboptimal. In this subsection, we develop a contract mechanism to coordinate the decentralized supply chain. Our mechanism suggests that, in addition to the wholesale price policy in the decentralized model, the mechanism has three other policies: a profit sharing policy (see, e.g., [17]), a production cost sharing policy, and an investment cost sharing policy. For convenience, we denote the developed mechanism as {w,x,y,z}, specifically as follows.(a)
The wholesale price policy suggests that the supplier charges the manufacturer a wholesale pricew per unit for its component, and then the manufacturer sets a markup m above wholesale price w and sells the product to the market at the price of p=w+m.(b)
The profit sharing policy suggests that the supplier sharesx percentage of its net profit to the manufacturer.(c)
The production cost sharing policy suggests that the supplier sharesy percentage of the manufacturer’s production cost.(d)
The investment cost sharing policy suggests that the supplier sharesz percentage of the manufacturer’s R&D investment cost.Then, under the above proposed mechanism{w,x,y,z}, the total transfer payment from the manufacturer to the supplier is given by(43)Tw,x,y,z=wa-bw+m+θαm+αΔ-xw-csa-bw+m+θαm+αΔ-ycm-cΔa-bw+m+θαm+αΔ-zfαΔ+gcΔ.The first term in the above equation corresponds to the payment caused by the wholesale price policy, the second term corresponds to the payment caused by the profit sharing policy, the third term corresponds to the payment caused by the production cost sharing policy, and the last term is the payment caused by the investment cost sharing policy.Consequently, the profit functions of the two players in the supply chain under the proposed mechanism{w,x,y,z} are as follows:(44)ΠT,sw,x,y,z=Tw,x,y,z-csa-bw+m+θαm+αΔ,(45)ΠT,mw,x,y,z=p-cm-cΔa-bw+m+θαm+αΔ-fαΔ+gcΔ-Tw,x,y,z.From (44) and (45), we have the following result.Proposition 9.
Under mechanism{w,x,y,z}, if(46)y=1-x,z=1-x,m=0,then, for any given 0<x<1, the optimal pricing decisions and investment strategies are identical in the decentralized and centralized supply chains; that is, pT∗(αΔ,cΔ)=pc∗αΔ,cΔ, AαΔ∗k¯T∗2=AαΔ∗k¯c∗2, and BcΔ∗k¯T∗2=BcΔ∗k¯c∗2, where pT∗(αΔ,cΔ), AαΔ∗k¯T∗2, and BcΔ∗k¯T∗2 represent the optimal retail price, capital invested in quality improvement activity, and capital invested in cost reduction activity under the proposed mechanism, respectively. That is, the decentralized supply chain is coordinated.Proof of Proposition9.
Substituting (43) and (46) into ΠT,s(w,x,y,z) of (44) and ΠT,m(w,x,y,z) of (45), respectively, we have, after some algebra, that (47)ΠT,sx=1-xp-cm-cs+cΔa-bp+θαm+αΔ-fαΔ+gcΔ=1-xΠcp∣αΔ,cΔ,ΠT,mx=xp-cm-cs+cΔa-bp+θαm+αΔ-fαΔ+gcΔ=xΠcp∣αΔ,cΔ.Thus, for any given x, ΠT,s(x), ΠT,m(x), and Πcp∣αΔ,cΔ achieve their maxima at the same point. Accordingly, under the proposed mechanism, the optimal retail price decision and investment strategy in the centralized model will be chosen. We thus complete the proof of Proposition 9.Proposition9 shows that the decentralized supply chain can be coordinated by the proposed mechanism {w,x,y,z} with appropriate parameters. Here we should point out that zero markup chosen by the downstream manufacturer is used to solve the double marginalization problem, while the profit sharing policy is to ensure that the manufacturer can get a positive profit, and thus the manufacturer has incentive to participate in the coordination. As a matter of fact, many downstream firms, such as Amazon (https://www.amazon.com/) and Joybuy (http://www.joybuy.com/), usually set a zero markup but share upstream suppliers’ revenue. This is quite similar to our coordination mechanism, which requires that the downstream manufacturer sets a zero markup above the upstream supplier’s wholesale price w but shares the supplier’s profit.In reality, an effective coordination mechanism should achieve Pareto improvement; otherwise, there exists at least one player having no incentive to participate in the coordination. At the end of this subsection, we will show that both the supplier and the manufacturer can benefit from participating in the proposed coordination mechanism, that is,ΠT,s(x)>Πs∗ and ΠT,m(x)>Πm∗, where Πs∗ and Πm∗ represent the optimal profits of the supplier and the manufacturer in the decentralized supply chain without coordination mechanism. Note from the proof of Proposition 9 that, under the proposed mechanism {w,x,y,z}, the profits earned by the supplier and the manufacturer are ΠT,s(x)=(1-x)Πc∗ and ΠT,m(x)=xΠc∗, respectively, where Πc∗ represents the optimal profit of the centralized supply chain. This together with the fact Πc∗>Πs∗+Πm∗, that there must exist a set of values for parameter x, such that (1-x)Πc∗>Πs∗ and xΠc∗>Πm∗ holds.


## 4.1. Centralized Supply Chain

In the centralized supply chain, there is a central planner who optimizes supply chain profit, that is, the total profit of the supplier and the manufacturer. From (4) and (7), we can obtain the supply chain profit(31)Πcw,m∣αΔ,cΔ=Πsw∣m,αΔ,cΔ+Πmm∣w,αΔ,cΔ=w+m-cm-cs+cΔa-bw+m+θαm+αΔ-fαΔ-gcΔ.This, together with the fact that p=w+m, indicates that the supply chain profit can be rewritten as follows:(32)Πcp∣αΔ,cΔ=p-cm-cs+cΔa-bp+θαm+αΔ-fαΔ-gcΔ.As a matter of fact, if the investment strategy (αΔ,cΔ) is given, then, in the decentralized supply chain, the values for wholesale price w and markup m are, respectively, chosen by the supplier and the manufacturer to maximize their own profits. However, in the centralized supply chain, it needs only to choose a value for w+m, i.e., p, to maximize the entire supply chain profit.From (32), we have the following result.Proposition 5.
In a centralized model, for any given investment strategy(αΔ,cΔ), the supply chain’s profit function Πcp∣αΔ,cΔ of (32) is strictly concave in the retail price p and achieves its maximum at(33)pc∗αΔ,cΔ=a+θαm+αΔ+bcs+bcm-cΔ2b.Consequently, the corresponding supply chain output is(34)qc∗αΔ,cΔ=a+θαm+αΔ-bcs-bcm-cΔ2.Proof of Proposition5.
Taking the first and the second derivatives ofΠcp∣αΔ,cΔ of (32) with respect to p, we have (35)dΠcp∣αΔ,cΔdp=a-2bp+θαm+αΔ+bcs+cm-cΔ,d2Πcp∣αΔ,cΔdp2=-2b<0.That is, the supply chain’s profit function Πcp∣αΔ,cΔ of (32) is strictly concave in the retail price p. Thus, the optimal retail price satisfies the first-order condition, that is, dΠcp∣αΔ,cΔ/dp=0. Solving the first-order condition for p, we get (33). Second, substituting p=pc∗(αΔ,cΔ) of (33) into (1), we have (34). We thus complete the proof of Proposition 5.By comparing Propositions5 and 2, we can make the following corollary.Corollary 6.
For any given investment strategy(αΔ,cΔ), the optimal retail price in the centralized model is lower than that in the decentralized model; that is, pc∗(αΔ,cΔ)<p∗(αΔ,cΔ), and the optimal supply chain output in the centralized model is higher than that in the decentralized model; that is, qc∗(αΔ,cΔ)>q∗(αΔ,cΔ).Proof of Corollary6.
First, it follows from (11) and (33) that (36)pc∗αΔ,cΔ-p∗αΔ,cΔ=-a+θαm+αΔ-bcs-bcm-cΔ4b.This, together with the fact a-b(cm+cs)+θαm≥0, indicates that pc∗(αΔ,cΔ)-p∗(αΔ,cΔ)<0. Second, it follows from (12) and (34) that (37)qc∗αΔ,cΔ-q∗αΔ,cΔ=a+θαm+αΔ-bcs-bcm-cΔ4.This, together with the fact a-b(cm+cs)+θαm≥0, indicates that qc∗αΔ,cΔ-q∗(αΔ,cΔ)>0. We thus complete the proof of Corollary 6.Corollary6 states that, for any given investment strategy (αΔ,cΔ), the optimal retail price and supply chain output in the decentralized model deviate from that in the centralized model. Such a deviation will lead to a profit loss for the entire supply chain.Now, substitutingp=pc∗(αΔ,cΔ) of (33) and q=qc∗(αΔ,cΔ) of (34) into Πcp∣αΔ,cΔ of (32), we can get the entire supply chain’s profit:(38)Πc∗αΔ,cΔ=a+θαm+αΔ-bcs-bcm-cΔ24b-fαΔ-gcΔ.This leads to the following result.Proposition 7.
In a centralized model, the optimal total capital invested in the quality improvement activity and cost reduction activity is(39)k¯c∗=kc∗,ifkc∗<K,K,ifkc∗≥K,where(40)kc∗=a+θαm-bcs-bcmA-1/2Bθ2+A1/2b2θ2B2+ABb21/24bθ2B2+ABb2-A-1/2Bθ2+A1/2b222.Moreover, the optimal capital invested in quality improvement activity and cost reduction activity, respectively, is(41)AαΔ∗k¯c∗2=B2θ2θ2B2+ABb2·k¯c∗,(42)BcΔ∗k¯c∗2=ABb2θ2B2+ABb2·k¯c∗.Proof of Proposition7.
This proof is analog to the proof of Proposition4; we omit it here.By comparing Propositions7 and 4, we obtain the following corollary.Corollary 8.
IfK≤k∗, then the optimal investment strategies are identical in the decentralized and centralized models; that is, A(αΔ∗(k¯∗))2=A(αΔ∗(k¯c∗))2 and B(cΔ∗(k¯∗))2=B(cΔ∗(k¯c∗))2; if K>k∗, then the optimal quality improvement investment level and cost reduction investment level in the decentralized model are both lower than those in the decentralized model; that is, AαΔ∗k¯∗2<AαΔ∗k¯c∗2 and BcΔ∗k¯∗2<BcΔ∗k¯c∗2.Proof of Corollary8.
First, ifK≤k∗, then AαΔ∗k¯∗2=B2θ2/θ2B2+ABb2·K=AαΔ∗k¯c∗2 and BcΔ∗k¯∗2=ABb2/θ2B2+ABb2·K=BcΔ∗k¯c∗2. Second, if k∗<K<kc∗, then AαΔ∗k¯∗2=B2θ2/θ2B2+ABb2·k∗<B2θ2/(θ2B2+ABb2)·K=AαΔ∗k¯c∗2 and BcΔ∗k¯∗2=ABb2/(θ2B2+ABb2)·k∗<ABb2/(θ2B2+ABb2)·K=BcΔ∗k¯c∗2. Third, if K≥kc∗, then AαΔ∗k¯∗2=B2θ2/(θ2B2+ABb2)·k∗<B2θ2/(θ2B2+ABb2)·kc∗=AαΔ∗k¯c∗2 and BcΔ∗k¯∗2=ABb2/(θ2B2+ABb2)·k∗<ABb2/(θ2B2+ABb2)·kc∗=BcΔ∗k¯c∗2. We thus complete the proof of Corollary 8.Corollary8 shows that the optimal investment strategy in the decentralized model may deviate from that in the centralized model. Specifically, when K≤k∗, the investment strategies are identical in the two models. This is because when K≤k∗, the whole capital will be invested in each of the two models. However, when k∗<K<kc∗, the capital constraint has effect in the centralized model but has no effect in the decentralized model, and then the whole capital will be invested in the centralized model but will not be invested fully in the decentralized model. When k>kc∗, the capital constraint has no effect in each of the two models and the investment level in the decentralized model is lower than that in the centralized model. Additionally, substituting pc∗(αΔ,cΔ) of (33), qc∗(αΔ,cΔ) of (34), AαΔ∗k¯c∗2 of (41), and BcΔ∗k¯c∗2 of (42) into (32), we can get the optimal profit of the centralized supply chain, denoted by Πc∗.


## 4.2. Coordination Mechanism

In Section4.1, we have shown that the optimal investment strategy and retail price decision in the decentralized model may deviate from that in the centralized model. Thus, the supply chain profit under the decentralized decision making is suboptimal. In this subsection, we develop a contract mechanism to coordinate the decentralized supply chain. Our mechanism suggests that, in addition to the wholesale price policy in the decentralized model, the mechanism has three other policies: a profit sharing policy (see, e.g., [17]), a production cost sharing policy, and an investment cost sharing policy. For convenience, we denote the developed mechanism as {w,x,y,z}, specifically as follows.(a)
The wholesale price policy suggests that the supplier charges the manufacturer a wholesale pricew per unit for its component, and then the manufacturer sets a markup m above wholesale price w and sells the product to the market at the price of p=w+m.(b)
The profit sharing policy suggests that the supplier sharesx percentage of its net profit to the manufacturer.(c)
The production cost sharing policy suggests that the supplier sharesy percentage of the manufacturer’s production cost.(d)
The investment cost sharing policy suggests that the supplier sharesz percentage of the manufacturer’s R&D investment cost.Then, under the above proposed mechanism{w,x,y,z}, the total transfer payment from the manufacturer to the supplier is given by(43)Tw,x,y,z=wa-bw+m+θαm+αΔ-xw-csa-bw+m+θαm+αΔ-ycm-cΔa-bw+m+θαm+αΔ-zfαΔ+gcΔ.The first term in the above equation corresponds to the payment caused by the wholesale price policy, the second term corresponds to the payment caused by the profit sharing policy, the third term corresponds to the payment caused by the production cost sharing policy, and the last term is the payment caused by the investment cost sharing policy.Consequently, the profit functions of the two players in the supply chain under the proposed mechanism{w,x,y,z} are as follows:(44)ΠT,sw,x,y,z=Tw,x,y,z-csa-bw+m+θαm+αΔ,(45)ΠT,mw,x,y,z=p-cm-cΔa-bw+m+θαm+αΔ-fαΔ+gcΔ-Tw,x,y,z.From (44) and (45), we have the following result.Proposition 9.
Under mechanism{w,x,y,z}, if(46)y=1-x,z=1-x,m=0,then, for any given 0<x<1, the optimal pricing decisions and investment strategies are identical in the decentralized and centralized supply chains; that is, pT∗(αΔ,cΔ)=pc∗αΔ,cΔ, AαΔ∗k¯T∗2=AαΔ∗k¯c∗2, and BcΔ∗k¯T∗2=BcΔ∗k¯c∗2, where pT∗(αΔ,cΔ), AαΔ∗k¯T∗2, and BcΔ∗k¯T∗2 represent the optimal retail price, capital invested in quality improvement activity, and capital invested in cost reduction activity under the proposed mechanism, respectively. That is, the decentralized supply chain is coordinated.Proof of Proposition9.
Substituting (43) and (46) into ΠT,s(w,x,y,z) of (44) and ΠT,m(w,x,y,z) of (45), respectively, we have, after some algebra, that (47)ΠT,sx=1-xp-cm-cs+cΔa-bp+θαm+αΔ-fαΔ+gcΔ=1-xΠcp∣αΔ,cΔ,ΠT,mx=xp-cm-cs+cΔa-bp+θαm+αΔ-fαΔ+gcΔ=xΠcp∣αΔ,cΔ.Thus, for any given x, ΠT,s(x), ΠT,m(x), and Πcp∣αΔ,cΔ achieve their maxima at the same point. Accordingly, under the proposed mechanism, the optimal retail price decision and investment strategy in the centralized model will be chosen. We thus complete the proof of Proposition 9.Proposition9 shows that the decentralized supply chain can be coordinated by the proposed mechanism {w,x,y,z} with appropriate parameters. Here we should point out that zero markup chosen by the downstream manufacturer is used to solve the double marginalization problem, while the profit sharing policy is to ensure that the manufacturer can get a positive profit, and thus the manufacturer has incentive to participate in the coordination. As a matter of fact, many downstream firms, such as Amazon (https://www.amazon.com/) and Joybuy (http://www.joybuy.com/), usually set a zero markup but share upstream suppliers’ revenue. This is quite similar to our coordination mechanism, which requires that the downstream manufacturer sets a zero markup above the upstream supplier’s wholesale price w but shares the supplier’s profit.In reality, an effective coordination mechanism should achieve Pareto improvement; otherwise, there exists at least one player having no incentive to participate in the coordination. At the end of this subsection, we will show that both the supplier and the manufacturer can benefit from participating in the proposed coordination mechanism, that is,ΠT,s(x)>Πs∗ and ΠT,m(x)>Πm∗, where Πs∗ and Πm∗ represent the optimal profits of the supplier and the manufacturer in the decentralized supply chain without coordination mechanism. Note from the proof of Proposition 9 that, under the proposed mechanism {w,x,y,z}, the profits earned by the supplier and the manufacturer are ΠT,s(x)=(1-x)Πc∗ and ΠT,m(x)=xΠc∗, respectively, where Πc∗ represents the optimal profit of the centralized supply chain. This together with the fact Πc∗>Πs∗+Πm∗, that there must exist a set of values for parameter x, such that (1-x)Πc∗>Πs∗ and xΠc∗>Πm∗ holds.


## 5. Concluding Remarks and Future Research

In practice, it is quite common that the manufacturer conducts R&D activity, which can achieve quality improvement and/or cost reduction. However, The existing studies on R&D activity in supply chains assume that R&D activity can either only lead to quality improvement or only result in cost reduction. This paper considers a manufacturer Stackelberg supply chain with a price and quality sensitive demand and investigates the problem of the manufacturer’s R&D activity, which can achieve not only quality improvement but also cost reduction. By establishing a three-stage game model, the paper derives the optimal R&D investment strategy for the manufacturer and corresponding pricing decisions for both the supplier and the manufacturer in the decentralized supply chain. Moreover, a contract mechanism is developed to coordinate the decentralized supply chain. The key findings obtained in this paper are as follows.(1)
In the decentralized supply chain, there exists a unique solution to the three-stage game model, in the sense that the optimal investment strategy and pricing decisions exist and are unique.(2)
The optimal investment strategy for the manufacturer requires that the manufacturer must simultaneously invest in both quality improvement activity and cost reduction activity.(3)
The optimal investment strategy and pricing decisions in the decentralized supply chain may deviate from those in the centralized supply chain. The mechanism consisting of a wholesale price policy, a profit sharing policy, a production cost sharing policy, and an investment cost sharing policy can be used to coordinate the decentralized supply chain and achieve Pareto improvement.This paper can be extended in several directions. In this paper, we focus on the problem of the downstream manufacturer’s R&D investment in both quality improvement activity and cost reduction activity. It would be interesting to study the upstream supplier’s R&D investment in both quality improvement activity and cost reduction activity. Additionally, in this study, we assume that there is only one component supplier in the supply chain. A more general setting may involve multiple component suppliers.



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*Source: 9879874-2017-01-18.xml*