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International Journal of Production Economics 191 (2017) 267–277

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International Journal of Production Economics

journal homepage: www.elsevier.com/locate/ijpe

Free or calculated shipping: Impact of delivery cost on supply chains moving to online retailing

Xiao-Feng Shao

Antai College of Economics and Management, Shanghai Jiao Tong University, 1954# Hua Shan Road, Shanghai, 200030, China

A R T I C L E I N F O

Keywords: Free shipping policy Calculated shipping policy Online retailing Geographical pricing Uniform pricing

E-mail address: [email protected].

http://dx.doi.org/10.1016/j.ijpe.2017.06.022 Received 28 April 2016; Received in revised form 12 Jun Available online 20 June 2017 0925-5273/© 2017 Elsevier B.V. All rights reserved.

A B S T R A C T

Shipping policy is an important element in the operations strategy for retailers moving online. In this study, we provide a framework to respectively investigate the impact of shipping policies in the multi-retailer and exclusive- retailer supply chains. We first develop an oligopoly model of price competition under calculated shipping policy and a Cournot model of quantity competition under free shipping policy in the supply chain with multiple competitive retailers. Our main finding is that while online retailing may result in a lower price for customers and a higher supply quantity for the supplier, none of the retailers can benefit from online move under calculated shipping policy, and only retailers with relatively small local market size may be better off under the free shipping policy. Both the supplier and customers prefer free shipping policy. For the supply chain consisting of an exclusive retailer, we develop a geographical pricing model and a uniform pricing model. We find that shipping policies with or without charges make no difference under geographical pricing approach. Under uniform pricing approach, customers in regions with shipping costs lower than the weighted average shipping cost prefer calculated shipping policy, while those in regions with shipping costs higher than the weighted average shipping cost prefer free shipping policy.

1. Introduction

In recent years, with the development of the Internet related infor- mation technology and the growth of third party logistics providers (Tsay and Agrawal, 2004), a growing number of traditional retailers have found it attractive to move online and employ the new means provided by the Internet to serve customers. Consequently, many retailers with brick-and-mortar stores are now entering the online marketplace to in- crease convenience for their local customers and sell to online shoppers from distant regions. Examples of companies making such a transition and having built online channels include Best Buy, Wal-Mart, Barnes & Noble, Tesco, Metro, Costco Wholesale etc. (NRF, 2014; Bernstein et al., 2008).

With increasing numbers of brick-and-mortar stores entering the online marketplace, e-commerce sales have climbed remarkably steadily for years, with continuous further growth expected. According to For- rester's latest five-year e-commerce forecast, US online retail sales will grow to $480 billion by 2019, up from 298.26 billion US dollars in 2014. Online retail sales in Canada are predicted to reach $39.9 billion, or 9.1% of total sales, in 2019, up from $22.3 billion in 2014, or 6.1% of total sales (Wray, 2014). And China is expected to become the first market to reach $1 trillion in online retail sales in 2019 (Meena, 2016).

e 2017; Accepted 15 June 2017

One of the most salient characteristics that differentiates online and offline shopping behavior is the low “transportation costs” required to visit an online store (Moe and Fader, 2004; Zhang et al., 2017; Xu et al., 2017). In the offline world, where the shopper incurs high “trans- portation costs” by taking the time and effort to visit stores located in other regions, it is more likely that he/she will choose to visit a local store. The low cost of visiting an online store site and the ease of acquiring online price information make the shopper more likely to search and compare the total prices before making a purchasing decision. Most consumers are sensitive to shipping charges, which are considered a main reason why online shoppers abandon their shopping carts. In a survey conducted by Kawamoto (2008), 72% of those surveyed respon- ded that if an online retailer starts charging a shipping cost, they would turn to another one that offered free shipping. Meanwhile shipping costs are anticipated to significantly impact online retailers' profitability, particularly when they offer free deliveries.

Although it has become a general trend for a brick-and-mortar retailer to open an online outlet, some fundamental questions remain to be answered. It is unclear as to whether the retail move from offline to online would benefit all parties in the supply chains. The online channels would cannibalize sales from the existing stores. According to Forrester's forecast, e-retail sales accounted for 7.4 percent of all retail sales

X.-F. Shao International Journal of Production Economics 191 (2017) 267–277

worldwide in 2015. This figure is expected to reach 12.8 percent in 2019. Thus, it is necessary to recognize the impact of retail move and the online shipping policy on the supply chains.

Furthermore, it is unclear how traditional retailers moving online should compete in terms of price. Most studies that compared price and service differences between online retailers have assumed that the de- livery costs to customers are equal. Since shipping costs vary depending on the distance between the retailers' original location and the customers' destination location, it is necessary to recognize this critical factor.

In this paper, we attempt to fill this gap by explicitly modeling the strategic interaction between firms operating online retail business and showing how the retail online move and the shipping policy affect the related parties in the supply chain. In particular, we try to answer the following questions.

● How does competition among retailers operating online channels affect their pricing strategies? What is the resulting effect on the supplier and customers?

● Who can benefit when the retailers move online? What is the impact of shipping policy on the related supply chain parties?

● Does channel structure affect the results? What if the products are distributed by one exclusive online retailer?

We first consider a supply chain in which a supplier selling an iden- tical product through its retailers located at different regions who are moving online. We model the price competition and Cournot competition among the retailers under the calculated shipping policy (i.e., the cus- tomers pay the shipping costs for products delivered from the retailers) and the free shipping policy (i.e., the retailers absorb the shipping costs), and study the retailers' optimal decisions on pricing and order quantity. To investigate the effect of channel structure, we consider a supply chain consisting a supplier and an exclusive retailer, and analyze the retailer's optimal decisions under both the geographical pricing approach and the uniform pricing approach.

The rest of the paper is organized as follows. In Section 2, we review the related literature. Section 3 describes the model. Section 4 derives the equilibrium results and compares the two shipping policies in the supply chain with multiple competitive retailers. Section 5 examines two different forms of pricing approach in the supply chain with an exclusive retailer. Finally, we draw our conclusions in Section 6. All proofs are provided in the Appendix.

2. Literature review

Our work is related to the research stream on shipping strategy. For online retailing operations, shipping policy is an important decision. There is a growing body of literature examining shipping policy. As Becerril-Arreola et al. (2013) discussed, a variety of shipping-related policies implemented by online retailers can be divided into three cate- gories: unconditional free shipping policy (i.e., under which the retailer absorbs the shipping costs); contingent free shipping policy (i.e., under which a retailer pays for the shipping costs if the orders equal to or larger than a value or quantity threshold; and customers pays for the shipping costs. Lewis et al. (2006) investigate the impact of shipping charges on consumer purchasing behavior. They show that consumers are sensitive to shipping charges, and promotions such as free shipping and free shipping for orders that exceed some size threshold are effective in generating additional sales. In another research paper, Lewis (2006) re- ported that the contingent free shipping policy is the most effective policy in increasing the revenues of online retailers. Based on an analytical model and subsequent empirical analyses using data collected from the online retailers of digital cameras and video games, Yao and Zhang (2012) find that online retailers will increase base prices when they offer free shipping. The value-based or quantity-based free shipping policy is widely adopted by online retailers as a common marketing promotion. Huang and Cheng (2015) examine the two forms of threshold

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free shipping policy from the customers' perspective. Boone and Gane- shan (2013) investigate how to structure value-based free shipping strategies and design inventory policies to maximize profits. Based on an optimization model that encompasses costs of procurement, ordering and holding inventory, and shipping to customers, they provide suggestions for the retailer to optimally determine the threshold value and replenish the inventory simultaneously.

Becerril-Arreola et al. (2013) consider a two-stage decision process in which the retailer first makes optimal decisions on the profit margin and the contingent free shipping threshold, and then determines the optimal inventory value. They show that variations in a positive finite free- shipping threshold affect both the average value and the standard devi- ation of the order sizes. To tackle the shipping-fee dilemma, Jiang et al. (2013) develop nonlinear mixed-integer programming models to concurrently determine the optimal shipping-fee schedules and product selling prices for single and multiple product transactions. To explore the free shipping policy in the context of newsvendor setting, Kwon and Cheong (2014) extend the base model developed by Zhou et al. (2009) to consider inventory issues when the exact distribution function of demand is not available. They present the optimal policies for the extended model and conduct numerical experiments to analyze the impacts of minimum free shipping quantity and the fixed shipping fee on the performance of the extended model.

Another stream of literature related to our work studies competition between online retailers (Abhishek et al., 2015; Wu et al., 2015) and the interactions between online firms and traditional firms (Yao and Liu, 2005, Hsiao and Chen, 2014; Chen et al., 2012; Wang et al., 2016). To capture the complexity that are associated with online retail competition, Dou and Ghose (2006) propose a Catastrophe Theory model and analyze the competitive influence that one online retailer can exert on its rival competing in the same online industry. They identify conditions under which catastrophes can occur in the customer base of a less-established online retailer competing with its more established competitor. Huang et al. (2013) formulate a Stackelberg game to investigate the dynamics between price and lead time for an e-retailing system with two duopo- listic suppliers and a retailer in a competitive environment. They suggest that when a supplier chooses a shorter lead time as the competitive strategy, the other supplier should choose a lower price for counteraction.

Forman et al. (2009) shows that the parameters in existing theoretical models of channel substitution such as offline transportation cost, online disutility cost, and the prices of online and offline retailers interact to determine consumer choice of channels. Mokhtarian (2004) analyzes the transportation and spatial impacts of online retailing and compares the advantages of brick-and-mortar stores and e-tailing. The author con- cludes that neither type uniformly dominates the other. Viswanathan (2005) develops a stylized spatial differentiation model to examine the impact of differences in network externalities and switching costs on competition between online, traditional, and hybrid firm. The results indicate that with network effects an increased market share does not translate into higher profits, and consumers rather than firms, benefit from increasing network externalities, with competitive effects out- weighing the surplus-extraction abilities of firms. To study the effect of the browse-and-switch behavior on the brick-and-mortar retailer and the online retailer, Balakrishnan et al. (2014) analyze a stylized economic model that incorporates uncertainty in consumers' valuation of the product and captures the heterogeneity among consumers in their inclination to purchase online. They shows that the browse-and-switch behavior intensifies competition, reducing the profits for both firms. Bernstein et al. (2008) show that clicks-and-mortar arises as the equi- librium channel structure, which does not necessarily imply higher profits for the competing firms.

There is scant literature, however, addressing the impact of shipping policy on the entire supply chain. Our work contributes to the literature in two main aspects. First, our work is the first to provide a compre- hensive comparison of the impact of calculated shipping policy and free

X.-F. Shao International Journal of Production Economics 191 (2017) 267–277

shipping policy on the related supply chain parties, including the sup- plier, retailers and customers. Second, we extends the setting to include multiple competing online retailers with different shipping costs to different regions.

3. Model setting

Consider a supplier selling an identical product at a given wholesale price w through n retailers located at different regions. The unit cost for retailer j is cj; j 2 N, which includes local operations expenses (hence, may be different among different retailers). In the offline scenario, we suppose the retailers independently make decisions in their respective regions due to the fact that customers in one region do not overflow to the retailers with lower prices located in other regions because of high search costs or transportation costs. To simplify the exposition, we model demand with a deterministic demand function. We assume customers in each region are heterogeneous and are uniformly distributed over [0,1]. A θ-type customer is willing to pay θv for a unit of product, where v is the product value. Each customer makes the decision after trading off his/her reservation value with the product price. If buying the product yields positive surplus, the customer purchases one unit of the product. The surplus of aθ-type customer in region j is given by

μ � θ; pj

� ¼ θv � pj:

where pj is the price of retailer j. Thus customers whose types are

distributed over � pj v; 1

� will choose to buy the product. Concretely, we

model demand in region j with the following linear demand function:

qj ¼ � 1 � pj

v

� dj;

where dj is the market size of region j, and 1� pjv is the conversion rate (i.e., probability that an arriving customer places an order with the retailer).

The jth retailer chooses its product price pj to maximize its profit:

max pj

πrj ¼ � pj � w � cj

�� 1 � pj

v

� dj:

We define the aggregate supply function as the total quantity distrib- uted by the supplier through n retailers for a given wholesale price w. The following result follows from the fact that the retailers independently chooses the prices to maximize their own profits.

Lemma 1. For the case with n offline exclusive retailers, the retailers' prices and the supplier's aggregate supply function are

pj ¼ v þ w þ cj

2 ;

Qs ¼ P

j2N � v � w � cj

� dj

2v :

The proof of this lemma and the proofs of all subsequent propositions and lemmas in our main paper are given in the Appendix. From Lemma 1, we find that the total quantity distributed in the supply chain is decreasing in the operating costs of the retailers, and increasing in the market size of the respective regions.

In the online scenario, we assume that there is no transportation cost for the customers. To a rational customer, the purchase decision ought to be based on the total price. The retailers can sell and ship the products to customers located in other regions. In our paper, shipping cost is the amount incurred in shipping the products from the retailer's location to the consumer's location, which is charged by a third party logistics pro- vider. Each unit of product shipped to region j from region i ði≠jÞ incurs a

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shipping cost cij. We assume that shipping costs are symmetric for a given location pair, i.e., cij ¼ cji. Shipping costs to local customers are assumed negligible (effectively zero), i.e., cjj ¼ 0. Assumption 1. cij � cilþ clj; ∀l 2 N:

This assumption is justified in the application context of our study, since the linear distance between two locations is usually the short- est path.

For simplicity, we ignore the setup costs of online business, and also ignore the changes of operations costs since the retailers moving online still operate the brick-and-mortar stores. These costs, however, can be readily incorporated into our model without affecting the main results.

Assumption 2. We assume throughout the paper that vþwþcj2 > wþ ciþ cij; ∀i; j 2 N:

This assumption ensures that the retail price in the jth region exceeds the marginal cost of the product shipped by a retailer from the ith region, so that there is a potential for profits and nonzero quantities in on- line retailing.

4. A supply chain with multiple retailers

In this section, we study the optimal decisions on pricing and order quantity under two different shipping policies in a supply chain with multiple retailers, and analyze the impact of shipping policy on the supply chain moving to online retailing. For this we focus on the com- parison of online supply chain performance with the offline counterpart. One of the main differences between online retailing and offline retailing is that, with the former, products are delivered to customers by the retailer, while in the latter, customers usually visit the local retailing outlet to buy the product (Gümüş et al., 2013). This means that shipping policy is an important consideration for online retailing. Another important difference is that online retailers are subject to fierce hori- zontal competition, while offline retailers are usually independent in their local markets. This means that retailers selling an identical product online engage in price competition or Cournot competition for the po- tential customers among themselves.

4.1. Calculated shipping policy

We now consider the case of n retailers in different regions moving to online retailing with calculated shipping policy. In the base case of offline retailing, each retailer acts as a monopolist in its local market. As dis- cussed before, they independently decide the prices to maximize the profits from their respective local markets. However, in the online case with calculated shipping policy, customers from one region can easily search the prices of the other retailers and make buy decisions based on the total prices. Therefore, retailers engage in a price competition game.

Under calculated shipping policy, each retailer simultaneously de- termines the product price. Customers then choose from which retailer to buy. The total price that customers need to pay is the sum of the product price plus the shipping cost from the retailer to the corresponding loca- tion. Given the other retailers' prices pi; i 2 N, the jth retailer chooses its price pj to maximize its profit. According to our assumption, local ship- ping cost is negligible. If pj > minfpiþ cijg, then pjþ cjk > minfpiþ cijgþ cjk ¼ plþ cliþ cik > plþ clk; k 2 N. Therefore, customers in the jth retailer's local market will switch to the lth retailer, and no customers from other locations would choose the jth retailer as well. If maxfpi� cjig � pj � minfpiþ cijg, customers in the jth retailer's local market will choose to buy from their local retailer, and no customers from other regions would switch to retailer j. If pj < maxfpi� cjig, customers from region i where pi > pjþ cji and pjþ cji ¼ minfpkþ ckig, k 2 N, will switch to the jth retailer. Define J ¼ fi

���∀pi > pjþ cji and pjþ cji ¼ minfpkþ ckig; k 2 Ng. Conse- quently, retailer j's profit function can be written as:

max pj

πrj ¼

8>>>>>< >>>>>:

0; if pj > min pi þ cij

� pj � w � cj

�� 1 � pj

v

� dj; ifmax

pi � cji

� pj � min

pi þ cij

� pj � w � cj

��� 1 � pj

v

� dj þ

X i2J

� 1 � pj þ cji

v

� di

# ; if pj < max

pi � cji

:

X.-F. Shao International Journal of Production Economics 191 (2017) 267–277

Proposition 1. In the online market with calculated shipping policy, there exists the Nash equilibrium among multiple competing retailers. At equilib- rium, retailer j's optimal price is

p*j ¼

8>>< >>:

v þ w þ cj 2

; if cj � min ck þ 2ckj

v þ w þ min

ck þ 2ckj

2

; if cj > min ck þ 2ckj

:

The above proposition shows the corresponding equilibrium prices for online retailers. Moreover, we find that the equilibrium price for a certain retailer depends on factors such as its own operating cost, the competitive retailers' operating cost, and the shipping costs from competitive retailers to its local region. We now analyze how changes in these parameters affect the equilibrium prices. Straightforward analysis using expressions obtained in Proposition 1 leads to the following observations:

1. At equilibrium, the lowest product price is quoted by the most effi- cient retailers with the lowest operating cost.

2. There exists a threshold of minfckþ 2ckjg. When cj is lower than the threshold, p*j increases in cj. When cj is above the threshold, p

* j keeps

constant. It implies that the jth retailer's optimal price p*j non- decreases in its own operating cost cj.

3. If the retailers are symmetric in terms of their operating costs, that is, cj ¼ c; ∀j 2 N, the optimal prices at equilibrium are also equal, and shipping costs do not affect the retailers' pricing decisions.

4. When the shipping costs are negligible, the optimal prices at equi- librium are determined by the most efficient retailer. When the shipping costs are high (e.g., the product being sold is large and/or heavy), the retailers independently decide their prices as in the offline retailing.

As in the case of offline retailing, the supplier's aggregate supply function under calculated online shipping policy is the aggregated de- mand across the retailers. Lemma 2 shows that the product quantity that the supplier distributes through less efficient retailers with high oper- ating costs actually increases due to the fact that online competition re- sults in a reduction in these retailers' prices, which in turn leads to an increase in demand.

Lemma 2. For the case with n online competitive retailers employing calculated shipping policy, the supplier's aggregate supply function is

Qs ¼ P

j2J1 � v � w � cj

� dj þ

P j2J2 � v � w � min

ck þ 2ckj

� dj

2v ;

where J1 ¼ fj ��∀cj � minfckþ 2ckjgg; J2 ¼ fj��∀cj > minfckþ 2ckjgg; k 2 N:

Next, we identify the benefit implications of supply chain adopting calculated shipping policy when moving to online retailing. In this light, we compare the equilibrium prices, the aggregate supply function, and the retailers' profits in our online model (where customers can search prices and make buy decisions based on the total prices) to those under an offline model where customers are restricted to buy from their local retailers.

Proposition 2. In online retailing with calculated shipping policy,

270

(i) The unit product price of retailer j is no higher than that in the offline case, that is, ponline*j � p

offline* j :

(ii) The supplier's aggregate supply quantity is no lower than that in the offline case, that is, Qonlines � Qofflines :

(iii) The jth retailer's profit is no more than that in the offline case, that is

πonlinej � π offline j :

The above proposition means that the supplier and customers may benefit from online retailing with calculated shipping policy. According to the proof of proposition 2, retailer j; j 2 J2 ¼ fj

��∀cj > minfckþ 2ckjg; k 2 Ng, would reduce its product price when the supply chain moves to online retailing. It implies that customers from regions with less efficient local retailers benefit from the competition among online retailers. And if J2≠∅, the supplier can distribute more products online than offline. However, retailers benefit nothing from the transition of supply chain. In fact, retailers with relatively high operating costs, that is, retailers belong to J2, are worse off in online retailing.

4.2. Free shipping policy

Under free shipping policy, online shoppers search the prices of all retailers and choose to buy from the one with the lowest price. A retailer offering a higher price would sell zero. Therefore, in this case, the prices offered by the retailers for the same region are equal. According to the demand function, the demand from the jth region Qj is a strictly decreasing function of pj with the inverse function p�j ðQjÞ. The mono- tonicity of QjðpjÞ implies that there is a one-to-one correspondence be- tween demand and the selling price in region j. Therefore, under free shipping policy, the problem of selecting a price is replaced with one of selecting a demand. Online retailers compete horizontally in order quantities for each region. Hence, they engage in a Cournot competi- tion game.

The sequence of events under free shipping policy is as follows:

1. Each retailer simultaneously determines the order quantity for each region. Once the decisions are made, retailers' inventories prepared for a particular region are fixed.

2. The price for each region is then determined accordingly based on the order quantities.

3. Customers make buy decisions and randomly choose from which retailer to buy.

Retailer i chooses its order quantity qij for region j to maximize its profit, anticipating the other retailers' reaction. Retailer i's profit function may be written as:

πri ¼ Xn j¼1

�� 1 � qij þ Q�ij

dj

� v � w � ci � cij

� qij;

where Q�ij is the total quantity ordered for region j by the rest of the retailers.

Proposition 3. There exists a unique equilibrium for the supply chain with multiple online retailers under free shipping policy. At equilibrium, the optimal price and retailer i's optimal order quantity for region j are as follows:

Fig. 1. Aggregate Supply Quantity Depending on Number of Retailers. Note: This figure depicts the aggregate supply quantity for a number of online competitive retailers ranging between 2 and 22, and for the cases with a low average operating and shipping cost ðP i2N

ðciþ cijÞ=n ¼ 3:5Þ, and a high average operating and shipping cost ð P i2N

ðciþ cijÞ=n ¼ 4Þ. We assume v ¼ 12; w ¼ 6; P

j2N dj ¼ 1,

P i2N

ðciþ cijÞ ¼ P i2N

ðciþ cikÞ:

X.-F. Shao International Journal of Production Economics 191 (2017) 267–277

p*j ¼ v þ nw þ Pk2N�ck þ ckj�

n þ 1 ;

q*ij ¼ dj

ðn þ 1Þv

" v � w þ

X k2N

� ck þ ckj

� � ðn þ 1Þ

� ci þ cij

�# :

Proposition 3 provides the retailers' equilibrium order quantities and the equilibrium price for each region. We find that the operating and shipping costs affect the retailers' order decisions. For a given market size of the jth region, the equilibrium order quantity of the ith retailer in- creases as the competitive retailers' operating costs and/or shipping costs increase, and decreases as its own costs increase. It also implies that for a given region, the retailer with the lowest total cost would order the highest quantity among the competitive retailers. If the retailers are symmetric in terms of their operating costs, that is, cj ¼ c; ∀j 2 N, for a given location, the local retailer would order more quantity than the other retailers. Moreover, we find that the equilibrium price for each region is affected by the cost factors of all competitive retailers. The equilibrium price increases as the retailers' costs increase.

Lemma 3. For the case with n online competitive retailers employing free shipping policy, the supplier's aggregate supply function is

Qs ¼ X j2N

dj ðn þ 1Þv

" nv � nw �

X i2N

� ci þ cij

�# :

Lemma 3 allow us to analyze how the supplier's aggregate supply quantity depend on the total costs of retailers and the total number of retailers in the supply chain. It is easy to see that the product quantity that the supplier distributes under free shipping policy increases as the retailers' operating and shipping costs decrease. Moreover, the result shows that the aggregate supply quantity increases in the number of retailers. Fig. 1 depicts the aggregate supply quantity as a function of the number of retailers for the cases with a low average operating and shipping cost ðP

i2N ðciþ cijÞ=n ¼ 3:5Þ, and a high average operating and

shipping cost ðP i2N

ðciþ cijÞ=n ¼ 4Þ. The figure confirms that the aggregate supply quantity is increasing in the number of retailers. The figure also

271

shows that the supplier benefit from a group of retailers with relatively lower operating and shipping costs.

Proposition 4. In online retailing with free shipping policy,

(i) The price for region j is lower than that in the offline case, that is,

ponline*j < p offline* j :

(ii) The supplier's aggregate supply quantity is larger than that in the offline case, that is, Qonlines > Q

offline s :

(iii) Suppose the operating costs are equal among the retailers and the shipping costs are negligible, the jth retailer's profit is higher than that in the offline case, that is πonlinej > π

offline j , when dj < 4

Pn i¼1ðdiÞ=ðn þ 1Þ2:

We know from Proposition 4 that, compared with the offline retailing with brick-and-mortar stores, customers and the supplier benefit from online retailing with free shipping policy. It is usually recognized that the competition among online retailers should be higher than that among the brick-and-mortar stores (Leng and Becerril-Arreola, 2010), which may lead to lower prices for customers. Our results confirms that online retailing with free shipping policy may result in a lower price for the customers and a higher supply quantity for the supplier. Proposition 4 shows that online retailing with free shipping policy may also benefit retailers with relatively small local market size. It implies that no retailers can benefit from online retailing if they have equal local market sizes. Next, we provide a numerical example to illustrate our analysis.

Example 1. Consider a case in which there are four retailers located in regions A, B, C, and D. We assume that the product value is v ¼ 12, and the wholesale price is w ¼ 6. The market sizes are dA ¼ 40; dB ¼ 100; dC ¼ 120; dD ¼ 140. Moreover, we assume that the operating costs of the retailers are equal, i.e., c1 ¼ c2 ¼ c3 ¼ c4 ¼ 2, and the unit shipping costs between different regions are equal, i.e., cij ¼ 0:2:

We find the optimal price for offline retailing is $10. The supplier's maximal supply quantity is 67. And the retailers' profits are $13.33 for Retailer 1, $33.33 for Retailer 2, $40 for Retailer 3, and $46.67 for Retailer 4. We also find that the optimal price for online retailing is $8.92 and the supplier's maximal supply quantity is 103. The resulting profits are $18.37 for Retailer 1, $20.01 for Retailer 2, $20.56 for Retailer 3, and $21.11 for Retailer 4, respectively. The above example indicates that, when the retailers move to online retailing with free shipping, it would benefit the customers and the supplier, and some retailers with small local market sizes, as shown in Proposition 4.

To investigate the impact of shipping cost on the retailers' optimal decisions and maximum profits in online retailing, we again consider Example 1 but change the market sizes to dA ¼ dB ¼ dC ¼ dD ¼ 100. In this sensitivity analysis, we increase the value of shipping cost from 0 to 2 in increments of 0.2, and compute optimal solutions and maximum profit for each retailer. We find from Fig. 2 that, as the shipping cost increases, the retailers should accordingly raise order quantity for their local mar- kets and decrease order quantity for the other regions to reduce the shipping-related expenses. High shipping costs may prevent online re- tailers from selling to other regions and protect their local markets.

In order to keep customers, the retailers have to order more and thus decrease the price when the shipping cost is low and online competition is fierce. As Fig. 2 indicates, as shipping cost increases, the retailers' profits first decreases but then increases. This interesting result reflects the following fact: when the shipping costs are relatively small, the re- tailers have to pay more shipping costs thus reduce their profits as the shipping costs increase. But when the shipping costs are relatively large, the retailers tend to order less for other regions which make the online competition less fierce. This implies that high shipping cost not always inevitably harms the performance of the retailers who can change order decisions to eliminate the negative impacts. Especially, when the ship- ping cost is very large (e.g., cij ¼ 2), the retailers' order quantity for other regions are close to zero. Referring to Fig. 2, we find that increasing shipping cost nonetheless deteriorates the supplier's performance.

Fig. 2. The Impact of the Shipping Cost on the Retailers' Optimal decisions, Profit and the Supplier's Aggregate Supply Quantity.

X.-F. Shao International Journal of Production Economics 191 (2017) 267–277

4.3. The impact of shipping policy

Unlike the calculated shipping policy (i.e., where customers pay for the shipment), the shipping costs are covered by the retailers under free shipping policy. This section shows the impact of shipping policy on the benefits of the supply chain parties.

Proposition 5. In online market with multiple competitive retailers, pCj > p F j ;

j 2 N; the customers always prefer free shipping policy to calculated ship- ping policy.

Proposition 5 shows that customers in each region pay less under free shipping policy. This may be explained as follows: Under calculated shipping policy, shipping costs actually prevent customers from choosing retailers in distant locations, which reduces competition among online retailers. While under free shipping policy, the retailers compete for customers in each region, which leads to a more fierce competition to reduce prices. This result can be partially validated by the empirical data collected by Gümüş et al. (2013). Through conducting an empirical analysis of data collecting from a large number of online retailers for

Fig. 3. The Impacts of the Shipping Cost an

272

digital cameras and printers, they find that the average total price per unit of a retailer charging a separate shipping fee is higher than the average total price per unit of a retailer offering free shipment.

Proposition 6. In the supply chain with multiple competitive online re- tailers, QFs > Q

C s ; the supplier always prefers free shipping policy to calculated

shipping policy.

Proposition 6 shows that the aggregate supply quantity is larger for the supplier when free shipping policy is adopted by the retailers. It offers a reason why the supplier may encourage horizontal competition among retailers in e-commerce scenario. Our result implies that the traditional channel policy in which exclusive retailers are assigned to different markets may be not optimal for the supplier engaging in moving its channels to online.

We now characterize the retailers' choice between calculated ship- ping policy and free shipping policy in the competitive online retailing.

Proposition 7. In a supply chain consisting of n identical online retailers with the same operating cost,

d Market Sizes on the Retailers' Profits.

X.-F. Shao International Journal of Production Economics 191 (2017) 267–277

(i) All retailers prefer calculated shipping policy to free shipping policy if each region has equal market size and the shipping costs between any two regions are equal.

(ii) When the shipping costs are negligible, retailers whose local market size is smaller than 4

Pn i¼1ðdiÞ=ðn þ 1Þ2 prefer free shipping policy to

calculated shipping policy, while those with local market sizes larger than 4

Pn i¼1ðdiÞ=ðn þ 1Þ2 prefer calculated shipping policy to free

shipping policy.

Our model is the first, to our knowledge, to consider the impact of shipping costs on the retailers' preference of shipping policy. Proposition 7 (i) is unique in showing conditions under which a calculated shipping policy will be preferred by all retailers. This result agrees with those of Shulman and Geng (2013) who identify how add-on pricing can lead to improved profitability for firms. It shows that in the online market con- sisting of multiple symmetric retailers with equal local market sizes, cross- selling to customers in other locations actually deteriorates retailers' profits as retailers have to cover additional shipping costs incurred during cross-selling. The second part of Proposition 7 implies that while it may benefit the smaller retailers, free shipment harms the larger retailers.

To investigate the impact of shipping cost and market size on the retailers' profits, we provide the following numerical example.

Example 2. Consider a case in which there are two retailers located in re- gions A and B, respectively. We assume that the product value is v ¼ 12, and the wholesale price is w ¼ 6. The aggregate market size is dAþ dB ¼ 200. Moreover, we assume that the operating costs of the retailers are equal, i.e., c1 ¼ c2 ¼ 2:

We first analyze the impact of shipping cost on the retailers' profits. In the analysis, we increase the value of shipping cost from 0 to 1 in in- crements of 0.1, and compute the profit for each retailer. As shown in Fig. 3, in the symmetric case, the retailers get less profits under the free shipping policy, and the profit difference increases as the shipping cost increases. We further investigate the asymmetric case in which the re- tailers have different local market sizes. Fig. 3 illustrates that when dA is small and dB is large, the smaller retailer (retailer 1) gets more profit and the larger retailer (retailer 2) get less profit under free shipping policy than under calculated shipping policy.

5. The case with one exclusive online retailer

In Section 4, we assumed that n online retailers compete in the market. To gain insight into what impact market competition might have on shipping policy preference of the supply chain parties, we consider a special case in this section. We remove market competition by reducing the number of retailers to one. In the supply chain with an exclusive retailer, there are usually two types of pricing strategies for online retailing in practice. In geographical pricing, the list price is based on the location of the customers. Hence, customers from different regions may pay different prices reflecting the costs of shipping to the corresponding locations. On the contrary, the same price is offered for customers from different geographical locations in uniform pricing.

5.1. Geographical pricing approach

Under free shipping policy, the profit for the exclusive retailer is as follows,

πFr ¼ Xn j¼1

� pj � w � cr � crj

�� 1 � pj

v

� dj;

where pj� w� cr� crj is the retailer's unit profit from region j. Given the retail price pj in region j, the total sales quantity for the supplier is,

273

QFs ¼ Xn j¼1

� 1 � pj

v

� dj:

Under calculated shipping policy, the profit for the exclusive retailer and the sales quantity for the supplier are as follows,

πCr ¼ Xn j¼1

� pj � w � cr

�� 1 � pj þ crj

v

� dj;

QCs ¼ Xn j¼1

� 1 � pj þ crj

v

� dj:

Lemma 4. Under geographical pricing approach, the exclusive retailer's optimal price for region j increases (respectively, decreases) in the corre- sponding shipping cost if free shipping policy (respectively, calculated shipping policy)is adopted.

The above lemma characterizes the effect of shipping cost on the retailer's pricing decisions for different regions under different shipping policies. It implies that the unit product price is higher for long-distance customers than for local customers under free shipping policy, which is rather intuitive as we know that in geographical pricing, the price for a certain location reflects its shipping cost. However, under calculated shipping policy, we find that the unit product price is lower for long- distance customers than for local customers. For long-distance cus- tomers, they are indeed paying higher shipping costs. The retailer may lower the product prices to attract customers to maximize its profits from those regions. So does it matter for the supply chain parties when different policies are applied? Hence, it is worthwhile to compare the optimal prices, the retailer's maximal profits, and the supplier's maximal sales qualities under the two policies.

Proposition 8. Under geographical pricing approach,

(i) Let pF*j and p C* j be the optimal product prices for region j under free and

calculated shipping policies, respectively. The customers pay the same total price under free and calculated shipping policies, that is, pF*j ¼ pC*j þ c1j:

(ii) The retailer's maximal profit under free shipping policy is equal to that under calculated shipping policy, that is, πF*r ¼ πC*r :

(iii) The supplier's maximal sales quantity under free shipping policy is equal to that under calculated shipping policy, that is, QF*s ¼ QC*s :

The above proposition implies that shipping policies with free or calculated cost make no difference to the supplier, the exclusive retailer, and the customers under geographical pricing approach. While cus- tomers generally assume that they are saving on delivery charges under free shipping policy, it shows that they are indeed paying the same total price as under calculated shipping policy. It implies that the benefit comes after all from a price one has paid.

5.2. Uniform pricing approach

Under uniform pricing approach, the retailer make a uniform price for all regions. If free shipping policy is adopted, long-distance customers pay as much as local customers. For each product delivered to region j, the retailer gets a unit profit of p� w� cr� crj: Thus the retailer makes a lower unit profit from long-distance regions with higher shipping costs. As the total price keeps equal, the retailer gets the same market share in all regions. The profit for the exclusive retailer and the sales quantity for the supplier with free shipping policy are as follows,

X.-F. Shao International Journal of Production Economics 191 (2017) 267–277

πFr ¼ Xn j¼1

� p � w � cr � crj

�� 1 � p

v

� dj;

QFs ¼ � 1 � p

v

�Xn j¼1

dj:

Under calculated shipping policy, long-distance customers pay higher total prices than local customers as they pay higher shipping costs. For each product delivered, the retailer gets an equal unit profit of p� w� cr: The market share in region j is equal to 1� pþcrjv : Thus the retailer gets a lower market share in long-distance regions with higher shipping costs. The profit for the exclusive retailer and the sales quantity for the supplier are as follows,

πCr ¼ ðp � w � crÞ Xn j¼1

� 1 � p þ crj

v

� dj;

QCs ¼ Xn j¼1

� 1 � p þ crj

v

� dj:

Lemma 5. Under uniform pricing approach, the retailer's optimal price in- creases (respectively, decreases) in the average shipping cost weighted by the

market size of n locations

Pn j¼1crjdjPn j¼1dj

! if free shipping policy (respectively,

calculated shipping policy)is adopted.

The above lemma shows that, under uniform pricing approach, the optimal price depends on the weighted average shipping cost. If the cost of shipping to a certain region increases, then the optimal price under free shipping policy increases and customers from other regions have to pay a higher price. Similarly, a decrease in shipping cost results in a price decrease for all regions. It implies that there exists a free-riding effect under free shipping policy. Under calculated shipping policy, we find that an increase in shipping cost to one region leads to a total price decrease for customers from other regions. And a decrease of shipping cost to one region results in a higher total price for other regions. It implies that there is a negative externality under calculated shipping policy. We next study how changes in shipping costs affect policy preference of the supply chain parties.

Proposition 9. Under uniform pricing approach,

(i) Customers in region i prefer calculated shipping policy to free shipping

policy if cri < Pn

j¼1crjdjPn j¼1dj

:

(ii) The retailer's maximal profit under free shipping policy is equal to that under calculated shipping policy, that is, πF*r ¼ πC*r :

(iii) The supplier's maximal sales quantity under free shipping policy is equal to that under calculated shipping policy, that is, QF*s ¼ QC*s :

The above proposition shows that, under uniform pricing approach, it does not matter to both the supplier and the exclusive retailer whether free shipping policy or calculated shipping policy is applied. But it does matter to the customers. Customers in regions where the shipping costs are lower than the weighted average shipping cost prefer calculated shipping policy, while those in regions with shipping costs higher than the weighted average shipping cost prefer free shipping policy.

274

6. Conclusion

Shipping policy is an important element in the operations strategy for online retailers. Motivated by both recent trends that brick-and-mortar retailers move online and challenges faced by online retailers, this research studies an important question: who can benefit from the retail move from offline to online when different shipping policies are employed in the supply chains? We follow an analytical framework to investigate the impact of calculated shipping and free shipping policies in the multi-retailer and exclusive-retailer supply chains. More specifically, we developed an oligopoly model of price competition under calculated shipping policy and a Cournot model of quantity competition under free shipping policy in the supply chain with multiple competitive retailers. For the supply chain consisting of one exclusive retailer, we developed a geographical pricing model and a uniform pricing model to investigate the impact of shipping policies.

Our results show that under both calculated shipping policy and free shipping policy, online retailing may result in a lower price for the cus- tomers and a higher supply quantity for the supplier. However, none of the retailers can benefit from online move under the calculated shipping policy, and only retailers with relatively small local market size may benefit from online move under the free shipping policy. In online market with multiple competitive retailers, both the supplier and customers al- ways prefer free shipping policy to calculated shipping policy. Our results indicate that while free shipping may benefit the smaller retailers, it does hurt the larger retailers. We demonstrate that shipping policies with free or calculated charges make no difference in the supply chain with one exclusive retailer under geographical pricing approach. Under uniform pricing approach, customers in locations where the shipping costs are lower than the weighted average shipping cost prefer calculated shipping policy, while those in locations with shipping costs higher than the weighted average shipping cost prefer free shipping policy.

Our study is limited by our assumption that the market size of each region keeps constant. Such an assumption is common in online competition literature, but the paper's findings are constrained if it is not true in practice. This research can be extended by allowing the market size of each region in online markets to be larger than the corresponding size in offline markets. It should be an interesting extension in examining other shipping polices beyond the calculated shipping policy and the free shipping policy because there are other shipping policies in online retail practice, such as value-based and quality-based shipping policies. Note that one of the assumptions in our theoretical modeling framework is that the retailers consider the same shipping policy while making pricing and ordering quantity decisions. A more comprehensive model should cap- ture how retailers interact with each other in price and delivery service competition. It could also be extended by examining supply chain governance structures to allow the supplier to game on the wholesale prices. Such extensions suggest complex interactions among supply chain parties, but it could potentially offer some additional insights for online retail business.

Acknowledgement

The work presented in this paper has been supported by grants from National Natural Science Foundation of China (71572106, 71372106 and 71632008) and the Program for New Century Excellent Talents in University (NCET-13-0369). The authors thank the referees for valuable suggestions and comments.

Appendix

Proof of Lemma 1. According to the profit function, it is clear that for a given wholesale price w, the optimal price for the jth retailer is p*j ¼ vþwþcj

2 .

X.-F. Shao International Journal of Production Economics 191 (2017) 267–277

Therefore, the demand in region j is

� � *! � �

qj p

* j ¼ 1 �

pj v

dj ¼ v � w � cj dj

2v :

Aggregating demand data across the retailers, we obtain the supplier's aggregate supply function. Proof of Proposition 1. We rearrange the retailers in orders according the operating cost ci. Without loss of generality, assume c1 � c2 � … � cn: Now

consider the pricing decision of retailer 1. If p1þ c1i < pi, potential customers in region i would all switch to retailer 1. Retailer i has zero profit. Thus, retailer i will decrease its retail price to avoid customer overflowing. Retailer 1 with the lowest operating cost makes the optimal price to maximize the profit from its local region. According to the profit function, we have p*1 ¼ vþwþc12 . Next we consider the pricing decision of retailer 2. To avoid customer overflowing, the price for retailer 2 must satisfy the condition: p2 � p*1þ c12: According to the profit function, if p*2 ¼ vþwþc22 and p2 � p*1þ c12; retailer 2 could get the maximal profit from its local region. Hence, we have the condition: c2 � c1þ 2c12: If c2 > c1þ 2c12; the optimal price for retailer 2 is equal to p*1þ c12; i.e., vþwþc1þ2c12

2 : Similarly, the price for retailer 3 must satisfy the condition: p3 � minfp*1þ c13; p*2þ c23g: According to the profit function, if p*3 ¼ vþwþc3

2 and p3 � minfp*1þ c13; p*2þ c23g; retailer 2 could get the maximal profit. If c2 � c1þ 2c12; we haveminfp*1þ c13; p*2þ c23g ¼ min

vþwþc1 2 þ c13;

vþwþc2 2 þ c23

� . Ifc2 > c1þ 2c12, we have minfp*1þ c13; p*2þ c23g ¼ min

vþwþc1

2 þ c13; vþwþc12 þ c12 þ c23 �

¼ vþwþc12 þ c13 ¼ min

vþwþc1 2 þ c13; vþwþc22 þ c23

� .

Thus, we have the condition: c3 � minfc1þ 2c13; c2þ 2c23g. If c3 > minfc1þ 2c13; c2þ 2c23g, the optimal price is equal to vþwþminfc1þ2c13;c2þ2c23g2 . Similarly, we can prove the optimal prices for the rest of the retailers.

Proof of Lemma 2. The result follows from straightforward calculations using the expressions obtained in Proposition 1.

Proof of Proposition 2. (i) From Proposition 1, if cj > minfckþ 2ckjg, the optimal price for retailer j is equal to vþwþminfckþ2ckjg2 , which is lower than the optimal offline price vþwþcj2 . (ii) Ifj 2 J2, then cj > minfckþ 2ckjg. As is evident from Lemma 1 and 2, the supplier can distribute more products online than offline. (iii) For retailer j; j 2 J1, the profit keeps constant as the online price is equal to its offline price. For retailer j; j 2 J2, the online profit is ðv�wþminfckþ2ckjg�2cjÞðv�w�minfckþ2ckjgÞdj

4v and the offline profit is ðv�w�cjÞ2dj

4v . We have ðv�wþminfckþ2ckjg�2cjÞðv�w�minfckþ2ckjgÞdj

4v � ðv�w�cjÞ2dj

4v ¼ �ðminfckþ2ckjg�cjÞ2dj

4v < 0. Proof of Proposition 3. We first show that the retailer equilibrium exists, and is unique. The ith retailer decision for location j is

�� � �

max qij

πrij ¼ 1 � qij þ Q�ij

dj v � w � ci � cij qij:

It is easy to see that the retailer problem is a strictly concave problem. The first-order optimality conditions for the retailer can be written as:

� �

1 � qij þ Q�ij

dj v � w � ci � cij �

v dj qij ¼ 0; i 2 N:

The above equation characterizes the retailer equilibrium best response. Solving the equations, we have that the optimal retailer order quantity for region j is

" #

q*ij ¼

dj ðn þ 1Þv v � w þ

X k2N

� ck þ ckj

� � ðn þ 1Þ

� ci þ cij

� :

Therefore, the total supply quality in the jth region is

" #

Q*j ¼

X i2N

q*ij ¼ dj

ðn þ 1Þv nv � nw � X i2N

� ci þ cij

� :

It is clear that for a given quantity selected by the retailers at equilibrium, the price selected by the retailers at equilibrium for the jth location is

� � *! P � �

pj Q

* j ¼ 1 �

Qj dj

v ¼ v þ nw þ k2N ck þ ckj n þ 1 :

Proof of Proposition 4. (i) Under the offline scenario, the optimal price in location j is poff*j ¼ vþwþcj

2 : Under the free shipping policy, the price at

equilibrium in location j is pon*j ¼ vþnwþ

Pn i¼1ðciþcijÞ

nþ1 : We have

pon*j � poff *j ¼ �ðn � 1Þv þ ðn � 1Þw þ 2Pni¼1�ci þ cij� � ðn þ 1Þcj

2ðn þ 1Þ ¼ �ðn � 1Þ

h v � w þ cj � 2

Pn i¼1;j≠i

� ci þ cij

�� ðn � 1Þ

i 2ðn þ 1Þ :

According to Assumption 1, Pn

i¼1;j≠iðciþcijÞ n�1 <

v�wþcj 2 . Therefore, p

on* j < p

off* j .

(ii) Under the offline scenario, the aggregate supply quantity is Qoffs ¼ Pn

j¼1

� v�w�cj

2v dj

� : In online retailing with free shipping policy, the total sales

quantity is Qons ¼ Pn

j¼1 dj

ðnþ1Þv ½nv� nw� Pn

i¼1ðciþ cijÞ�: We have Qons � Q off s ¼ 12ðnþ1Þv ½ðn� 1Þ

Pn j¼1djðv� wþ cjÞ� 2

Pn j¼1djð

Pn i¼1;j≠iðciþ cijÞÞ�.

According to Assumption 1, Pn

i¼1;j≠iðciþcijÞ n�1 <

v�wþcj 2 . Therefore, Q

on s > Q

off s .

(iii) Under the offline scenario, the profit of retailer j is πoffj ¼ ðv�w�cjÞ2

4v dj: Under the online scenario with free shipping policy, the profit of retailer j is

275

X.-F. Shao International Journal of Production Economics 191 (2017) 267–277

πonj ¼ Pn

i¼1 di

ðnþ1Þ2v½v � w þ Pn

k¼1ðck þ ckiÞ � ðn þ 1Þðcj þ cjiÞ�2: Suppose c1 ¼ c2 ¼ ⋯ ¼ cn and cji ¼ 0; ∀i; j 2 N. We have

� �2 " P #

πonj � πoffj ¼

v � w � cj v

n i¼1di

ðn þ 1Þ2 � 1 4 dj :

Thus, ifdj < 4 Pn

i¼1ðdiÞ=ðn þ 1Þ2, πonj > πoffj . Proof of Lemma 3. As we have proved in Proposition 3, the total supply quality in the jth region is

" #

Q*j ¼

X i2N

q*ij ¼ dj

ðn þ 1Þv nv � nw � X i2N

� ci þ cij

� :

Therefore, it is clear that the supplier's aggregate supply quantity is

" #

Qs ¼

X j2N

dj ðn þ 1Þv nv � nw �

X i2N

� ci þ cij

� :

Proof of Proposition 5. Under the calculated shipping policy, the optimal price for region j is pCj ¼ min

vþwþcj 2 ;

vþwþminfckþ2ckjg 2

� : Under the free

shipping policy, the price is pFj ¼ vþnwþ

Pn k¼1ðckþ2ckjÞ

nþ1 : We have p C j � pFj ¼

ðn�1Þv�ðn�1Þwþðnþ1Þminfcj;minfckþ2ckjgg�2 Pn

k¼1ðckþ2ckjÞ 2ðnþ1Þ .

If minfcj; minfckþ 2ckjgg ¼ cj, pCj � pFj ¼ ðn�1Þ

� v�wþcj�

2 Pn

k¼1;k≠j ðckþ2ckjÞ

ðn�1Þ

� 2ðnþ1Þ . According to Assumption 1, v� wþ cj >

2 Pn

k¼1;k≠jðckþ2ckjÞ ðn�1Þ , we have p

C j > p

F j . Ifminfcj;

minfckþ 2ckjgg ¼ minfckþ 2ckjg ¼ clþ 2clj, we have pCj � pFj ¼ ðn�1Þðv�wÞþðnþ1Þðclþ2cljÞ�2

Pn j¼1ðcjþcjiÞ

2ðnþ1Þ ¼ ðn�1Þðv�wÞþðn�1Þðclþ2cljÞ�2

Pn k¼1;k≠lðckþ2ckjÞ

2ðnþ1Þ � ðn�1Þ½v�wþcl�2

Pn k¼1;k≠lðckþ2ckjÞ=ðn�1Þ�

2ðnþ1Þ > 0.

Proof of Proposition 6. Under free shipping policy, the total sales quantity is QFs ¼ Pn

j¼1 dj

ðnþ1Þv ½nv� nw� Pn

k¼1ðckþ 2ckjÞ�: Under calculated shipping policy, the total sales quantity isQCs ¼

Pn j¼1

dj 2v ½v� w� minfcj; minfckþ 2ckjgg�. DefineJ1 ¼ fj

��∀cj � minfckþ 2ckjgg; J2 ¼ fj��∀cj > minfckþ 2ckjgg; and minfckþ 2ckjg ¼ clj þ 2cljj. We have

"

n #

QFs � QCs ¼ n � 1

2ðn þ 1Þv X j2J1

dj v � w þ cj � 2 X

k¼1;k≠j

� ck þ 2ckj

�� ðn � 1Þ þ n � 1

2ðn þ 1Þv X j2J2

dj

" v � w þ clj þ 2cljj � 2

Xn k¼1;k≠lj

� ck þ 2ckj

�� ðn � 1Þ

#

> n � 1

2ðn þ 1Þv X j2J1

dj

" v � w þ cj � 2

Xn k¼1;k≠j

� ck þ 2ckj

�� ðn � 1Þ

# þ n � 1 2ðn þ 1Þv

X j2J2

dj

" v � w þ clj � 2

Xn k¼1;k≠lj

� ck þ 2ckj

�� ðn � 1Þ

# > 0:

Thus, we have QFs > Q C s .

Proof of Proposition 7. (i) Under calculated shipping policy, the profit of retailer i is

πCi ¼ ðv � w þ minfci; minfck þ 2ckigg � 2ciÞðv � w � minfci; minfck þ 2ckiggÞ

4v di:

Under free shipping policy, the profit of retailer i is

" #2

πFi ¼

Xn j¼1

dj ðn þ 1Þ2v

v � w þ Xn k¼1

� ck þ ckj

� � ðn þ 1Þ

� ci þ cij

� :

Under the scenario with identical retailers and equal local market sizes, we define ci ¼ c, cij ¼ cs; and di ¼ d; ∀i; j ¼ 1; 2; ⋯; n; i≠j. We have

"

2 2

πFi � πCi ¼ d v

ðn � 1Þðv � w � c � 2csÞ þ ðv � w � c þ ðn � 1ÞcsÞ ðn þ 1Þ2

� ðv � w � cÞ 2

4

#

¼ �dðn � 1Þðv � w � c � 2csÞ½2ðn þ 3Þcs þ ðn � 1Þðv � w � cÞ� 4vðn þ 1Þ2

< 0:

(ii) Suppose c1 ¼ c2 ¼ ⋯ ¼ cn and cji ¼ 0; ∀i; j 2 N. We have

� �2 " Pn #

πFj � πCj ¼ v � w � cj

v i¼1di

ðn þ 1Þ2 � 1 4 dj :

Thus, ifdj < 4 Pn

i¼1ðdiÞ=ðn þ 1Þ2, πFj > πCj . Proof of Lemma 4. Maximizing the profit function of the retailer leads to the following optimal pricing solutions: pF*j ¼

vþwþcrþcrj 2 and p

C* j ¼

vþwþcr�crj 2 .

Hence, an increase in crj implies an increase in pF*j and a decrease in p C* j .

Proof of Proposition 8. Under free shipping policy, the unit product price for region j is pF*j . Under calculated shipping policy, customers in region j

276

X.-F. Shao International Journal of Production Economics 191 (2017) 267–277

pay a total price of pC*j þ crj for each unit of product. According to the proof of Lemma 4, we get pC*j þ crj ¼ vþwþcrþcrj

2 ¼ pF*j . Further, we get QF�s ¼ QC�s ¼Pn j¼1

� v�w�cr�crj

2v

� dj and πF*r ¼ πC*r ¼

Pn j¼1

� v�w�cr�crj

4v

�2 dj.

Proof of Lemma 5. Maximizing the profit function of the retailer leads to the following optimal pricing solutions: pF* ¼ vþwþcr2 þ Pn

j¼1crjdj

2 Pn

j¼1dj and pC* ¼

vþwþcr 2 �

Pn j¼1crjdj

2 Pn

j¼1dj . Hence, an increase in

Pn j¼1crjdjPn j¼1dj

implies an increase in pF* and a decrease in pC*.

Proof of Proposition 9. According to the proof of Lemma 5, we have pF* ¼ vþwþcr2 þ Pn

j¼1crjdj

2 Pn

j¼1dj and pC* ¼ vþwþcr2 �

Pn j¼1crjdj

2 Pn

j¼1dj . Hence, we get QF*s ¼ QC*s ¼

ðv�w�crÞ Pn

j¼1dj� Pn

j¼1crjdj 2v and π

F* r ¼ πC*r ¼

Pn j¼1dj 4v

v � w � cr �

Pn j¼1crjdjPn j¼1dj

!2 . Further, we get pF*� ðpC*þ criÞ ¼

Pn j¼1crjdjPn j¼1dj

� cri.

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