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Financing online retailers: Bank vs. electronic business platform, equilibrium, and coordinating strategy
Accepted Manuscript
Financing Online Retailers: Bank vs. Electronic Business Platform, Equilibrium, and Coordinating Strategy Chengfu Wang, Xiaojun Fan, Zhe Yin PII: DOI: Reference:
S0377-2217(19)30011-6 https://doi.org/10.1016/j.ejor.2019.01.009 EOR 15581
To appear in:
European Journal of Operational Research
Received date: Revised date: Accepted date:
29 September 2017 30 June 2018 4 January 2019
Please cite this article as: Chengfu Wang, Xiaojun Fan, Zhe Yin, Financing Online Retailers: Bank vs. Electronic Business Platform, Equilibrium, and Coordinating Strategy, European Journal of Operational Research (2019), doi: https://doi.org/10.1016/j.ejor.2019.01.009
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Highlights • Online retailers obtain financing from a bank or an electronic business platform. • Electronic business platform is divided into an active and a conservative form.
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• We perform a Stackelberg game analysis of electronic business platform financing. • Active electronic business platform financing can achieve coordination.
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• The coordinated active electronic business platform financing is preferable.
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Financing Online Retailers: Bank vs. Electronic Business Platform, Equilibrium, and Coordinating Strategy Chengfu Wanga,b , Xiaojun Fana ,1
a The School of Management, Shanghai University, Shanghai, China Dongfang College, Zhejiang University of Finance & Economics, Hangzhou, China
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b
Zhe Yina
Abstract
We consider a new financing source for online retailers, namely an electronic business (EB) platform,
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which is divided into an active and a conservative form according to the level of integration of lending and leasing businesses. The newsvendor-like online retailer is capital-constrained and can choose between EB platform financing and bank credit financing (BCF). We model their interaction as a Stackelberg game, where the online retailer functions as the follower who decides the order quantity and the lender (bank or EB platform) functions as the leader who declares the loan interest rate or
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the EB platform’s usage fee rate. From the equilibrium and coordinating analyses, we conclude that the active EB platform financing can achieve supply chain financing (SCF) coordination, yielding
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a larger order quantity and larger participants’ profits than those yielded through BCF, while the conservative may not. When the coordination condition on order quantity is satisfied, if the initial
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working capital is a constant, there exists a linear decreasing trend between the EB platform’s loan interest rate and the usage fee rate. When the weak coordination conditions are satisfied, the active
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EB platform can offer many coordinated financing contracts, which are applicable to different initial working capital conditions. For achieving SCF coordination, the EB platform should be active to adjust his loan interest rate and usage fee rate together. Moreover, even if the online retailer has
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sufficient working capital, she always prefers using the coordinated active EB platform financing to refusing any outside financing. Keywords: Finance; Online retailer; Trade credit; Stackelberg game; Supply chain coordination.
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Corresponding author: E-mail: [email protected]
Preprint submitted to Elsevier
January 11, 2019
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1. Introduction Limited working capital is a common constraint for online retailers (usually small and mediumsized enterprises or SMEs) in procurement decisions. Conversely, they tend to find it difficult to obtain financing from banks, owing to their lack of collateral, lack of credit history, and the tenuous
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nature of their business establishment (Vandenberg, 2003). One of the most effective methods of accessing funds is supply chain financing (SCF), in which SMEs can use the core enterprise’s credit and real transaction background to improve their own accountability. In general, SCF can be divided into two widely used financing modes: bank credit financing (BCF) and trade credit financing (TCF). Besides BCF, supplier financing of retailer inventory, as a form of TCF, has
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become popular in many countries, such as the United States, China, and some less-developed countries (Petersen and Rajan, 1997; Fisman and Love, 2003; Kouvelis and Zhao, 2012). In recent years, the rapid development of electronic business (EB) has provided online retailers with a new financing source, an EB platform.
Under TCF, a corporation extends its credit to an upstream/downstream supply chain partner
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by using short-term loans (Yan et al., 2016). The EB platform, considered as an important supply chain partner of the online retailer, plays a significant role in the online retailer’s achievement of its
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sales target, and can also extend trade credit to the online retailer by running a lending business. Therefore, EB platform financing of online retailer inventory can be defined as a new form of TCF. In China, EB platforms, such as Alibaba and Jingdong Mall, have introduced a lending business
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based on their original EB platform’s leasing business. By using the big data collected from dayto-day trading on these EB platforms, such financing form obtains many competitive advantages,
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including the ability to borrow without collateral or guarantee, a lower default risk, and the quicker granting of loans.
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The EB platform, such as Alibaba or Jingdong Mall, carries out its lending operation on the
premise of obtaining the original leasing profit, and consider its leasing business to be independent of its lending business. Therefore, the only decision variable is the EB platform’s loan interest rate. The EB platform’s usage fee rate for the leasing business remains constant, regardless of whether or not it lends to an online retailer. This is a form of conservative EB platform financing. We introduce a new financing form of active EB platform financing, in which the EB platform integrates the leasing business and lending business perfectly. Here, the EB platform simultaneously sets suitable decision variables, the EB platform’s usage fee rate and the EB platform’s loan interest rate, in 3
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order to optimize its profit or to achieve SCF coordination. Is the active option better than the conservative option? Or in other words, should the conservative EB platform change its lending strategy? SCF not only diversifies funding sources or financing forms, but also improves the financial
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efficiency of the entire supply chain system by aligning the finance and operation functions. The prevalence of EB platform financing practices brings to the forefront several research questions about financial efficiency: What are the optimal decision variables (the EB platform’s loan interest rate, the EB platform’s usage fee rate, and the online retailer’s order quantity) under EB platform financing? Why and how should EB platforms provide financing for online retailers instead of letting them use bank financing? How does the online retailer choose a suitable financing mode or
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design an appropriate loan portfolio?
To answer these questions, we introduce a stylized model of a supply chain, with a newsvendorlike online retailer selling goods on an EB platform. The market demand follows a certain distribution when the retail price is specified exogenously. The online retailer is capital-constrained and
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can choose EB platform financing or BCF. For the sake of describing the relationship among all participants and the sequence of events properly, we consider the financing process as a Stackelberg game between the online retailer (as the follower) and her lender (as the leader). Initially, the EB
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platform is either conservative or active, and designs his financing contract based on two variables: the EB platform’s loan interest rate and the EB platform’s usage fee rate. The risk-neutral bank
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runs the BCF business in a perfectly competitive capital market, with risk-free interest income being its expected revenue. Then, the online retailer selects a financing form (conservative EB
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platform financing, active EB platform financing, BCF, or refusing any outside financing), and decides the order quantity in response to the leader’s decision.
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Our work contributes to the literature in four ways. First, we establish the functions of the participants’ profits by using the final actual cash flow and the opportunity costs of the initial investments, which differs from the previous studies (Lee and Rhee , 2011; Kouvelis and Zhao, 2012). The online retailer’s initial working capital and the EB platform’s loan-granting quota are the initial investments of the online retailer and the EB platform, respectively. When they deposit these initial investments in a bank, rather than investing in an EB platform financing business, they can obtain the interest incomes at a risk-free interest rate. We refer to those interest incomes as the opportunity costs of these initial investments, which should be deducted when we calculate
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the corresponding participant’s profit. Second, we propose a new TCF contract offered by an EB platform, and formulate a Stackelberg game including an EB platform and an online retailer. From the Stackelberg equilibrium analysis, we find that the EB platform’s loan interest rate and the EB platform’s usage fee rate both have impact on the online retailer’s order quantity decision, and further affect the participants’ profits. Third, we analyze the efficiency of EB platform financing
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from the perspective of an SCF coordination contract. The active EB platform financing can achieve SCF coordination, yielding a larger order quantity and larger participants’ profits than the case of BCF. However, the conservative EB platform financing may not be able to achieve SCF coordination. Fourth, we have analyzed the relations among parameters, which are useful for the active EB platform to run a coordinated SCF business: (i) When the coordination condition on
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order quantity is satisfied, if the online retailer’s initial working capital remains constant, there exists a linear decreasing trend between the EB platform’s loan interest rate and the EB platform’s usage fee rate. (ii) When the weak coordination conditions are satisfied, the active EB platform can offer many coordinated financing contracts, which are applicable to different initial working capital conditions, while the conservative EB platform could design very few coordinated financing
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contracts. (iii) The online retailer always prefers using the coordinated active EB platform financing to refusing any outside financing, even if she has sufficient initial working capital.
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The rest of the paper is organized as follows. Section 2 reviews the relevant literature. Section 3 describes our basic model of active EB platform financing and bank credit financing, including
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the sequence of events, notations, assumptions, and profit functions. In Section 4, we present the equilibrium and coordination analysis under BCF. In Section 5, we analyze the SCF equilibrium and
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coordination condition under active EB platform financing. When the active EB platform financing achieves SCF coordination, Section 6 demonstrates the relations among parameters, which offers insights on how to run a coordinated active EB platform financing business. The final section
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concludes the paper and presents future research directions. The appendix includes proofs of some propositions.
2. Literature Review The interface of operations and financial decisions has attracted researchers’ interests for a long time. Modigliani and Miller (1958) are the first scholars to express the irrelevance between operations and financial decisions in a perfect capital market. However, when market imperfections 5
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are introduced, the irrelevance no longer holds. According to this theoretical framework, financing can improve or create firm value by reducing taxes and contract costs, by affecting real investments in operations, or by financial hedging of operation risks (Smith and Stulz, 1985; Gaur and Seshadri, 2005; Ni et al., 2017; Wang et al., 2018). Several researchers have examined the risk management in SCF processes (Ding et al., 2007; Zhao and Huchzermeier, 2015, Yang and Birge, 2017). Other
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researchers have focused on a capital-constrained firm’s operations and financing decisions for profit maximization, such as production and financing decisions (Lee and Tang, 1997; Buzacott and Zhang, 2004; Xu and Birge, 2004) and inventory and financing decisions. We pay attention to a special form of TCF, namely EB platform financing of online retailer inventory. This could be described as a type of financing the newsvendor problem and belongs to the research area of
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inventory and financing decisions, but differs from the TCF studies with EOQ or EPQ models under a determined market demand environment (Teng et al., 2012; Ouyang et al., 2013; Wu et al., 2014; Shah and C´ ardenas-Barr´ on, 2015; Chen and Teng, 2015; Wu et al., 2016; Tiwari et al., 2016; Jaggi et al.,2017). We analyze two research categories on financing the newsvendor problem: participants’ equilibrium in a decentralized financing system and the coordination contract in the
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financing system.
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2.1. Participants’ Equilibrium in a Decentralized Financing System The classic newsvendor model helps to find the optimal order quantity, which can minimize
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the operating cost or maximize the profit of the purchaser in a single period. Several researchers have extended this model to product purchasing with multiple categories and budget constraints
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(Lau and Lau, 1996; Mostard and Teunter, 2006; Chen and Chen, 2010). These studies use the nonlinear programming method to maximize the capital-constrained purchaser’s profit, but ignore other participants’ profits as well as the possibility of outside financing. Raghavan and Mishra
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(2011) study a two-level supply chain financing problem with a single retailer and a manufacturer, who are both capital-constrained and are linked in the activities of ordering, producing, and buyback. They could obtain the lender’s optimal loan amount, but ignore the borrower’s decision on the order quantity and the lender’s decision on the interest rate. Considering the external capital injection, each participant makes decision successively in order to maximize his/her own profit, and these decisions are interrelated in a decentralized financing system. The Stackelberg game model has been used to analyze SCF problems. Kouvelis and Zhao (2011) are the first to incorporate both the bankruptcy cost and the bankruptcy risk into 6
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BCF operation, and find the equilibrium solutions of the order quantity and the wholesale price. Then, Kouvelis and Zhao (2012) consider the bankruptcy problem under SCF. Both the retailer and the supplier are capital-constrained, and the retailer can choose between BCF and TCF. Their equilibrium analysis results show that under certain conditions (relate to parameters of the interest rate, the wholesale price, the working capital, and the collateral), TCF is more attractive than
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BCF. However, these articles only analyze the equilibrium problem within the financing process, but have not taken the SCF coordination problem into account. Additionally, Kouvelis and Zhao (2012) handle the problem of optimal parameter setting by maximizing participants’ terminal cash flows, rather than maximizing participants’ profits. In our research, we will use the terminal cash flows and the opportunity costs of the initial capital investments to establish participants’ profit
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functions, which is in line with Raghavan and Mishra (2012).
Following the above research methods, Jing et al. (2012) analyze the effect of the production cost, the capital level, and the demand variability on the efficiency of BCF and TCF. Chen (2015) compares BCF and TCF under a wholesale price contract and a revenue-sharing contract to find how the product cost, the internal capital, and the demand variability affect the participants’
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equilibrium. Gao et al. (2018) analyze the optimal Stackelberg strategies for financing a supply chain through an online peer-to-peer lending platform. Similarly, we discuss the participants’
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equilibrium and compare different financing forms. However, in our case, the EB platform financing of an online retailer is a new form of TCF, where the lender’s decision variables are the EB platform’s
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loan interest rate and the EB platform’s usage fee rate, as opposed to the wholesale price under supplier financing. Furthermore, there is no collateral or guarantee in our model.
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2.2. Coordination Contract in the Financing System In a supply chain context, owing to the independent decisions of the participants, the total profit
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in a decentralized supply chain is less than that in a centralized supply chain, where the same firm produces and sells directly to its end customers. This phenomenon is commonly called double marginalization (Spengler, 1950). For the sake of mitigating double marginalization and improving the integrated efficiency of SCF, we derive a coordination scheme in which the Stackelberg leader (generally the lender or supplier) induces the follower (generally the borrower or the retailer) to replicate the decision outcome of the centralized supply chain. Dada and Hu (2008) are the first to consider the SCF coordination problem under the well-
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known two-part tariff contract. Lee and Rhee (2010) investigate four coordination mechanisms (all-unit quantity discount, buybacks, two-part tariff, and revenue-sharing), and find that, under positive inventory financing costs, these mechanisms fail to achieve coordination in direct financing from a bank, whereas under TCF they can fully coordinate the supply chain. Then, Lee and Rhee (2011) show that, when the supplier grants a markdown allowance to the retailer in SCF system,
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he can fully coordinate the retailer’s decision for the largest joint profit under TCF. However, the coordination cannot be realized under BCF. Zhang et al. (2014) propose a modified quantity discount contract based on both order quantity and advance payment, and find that this contract can entice the retailer to make a decision to achieve TCF coordination. Yan et al. (2016) propose a new SCF form, which mixes TCF and BCF, and find that supply chain coordination can be
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achieved when we set the suitable guarantee coefficient and wholesale price.
Unlike these researches, we consider the decision variables of the EB platform’s loan interest rate and the EB platform’s usage fee rate in our coordination analysis, and compare the optimal order quantities and participants’ profits under BCF and under EB platform financing. Then, we further characterize the EB platform financing coordination conditions from various perspectives,
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including the EB platform’s loan interest rate, the EB platform’s usage fee rate, and the online retailer’s initial working capital. Our results illustrate that, in aspects of the optimal order quantity
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and participants’ profits, EB platform financing is superior to BCF, which is consistent with the fact that “70-80% of trade between firms is still conducted in open account (trade credit)” (Zhou
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and Groenevelt, 2008).
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3. Model Formulation
3.1. Assumptions, Notations and Sequence of Events
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Following the standard approach in the classic newsvendor model of prior SCF studies, we
consider the online retailer as a newsvendor who has an opportunity to purchase products from a supplier before the sales season in order to satisfy the market’s uncertain demand. However, in contrast to previous studies, the online retailer (referred to as “she”) is capital-constrained and can borrow from a lender (referred to as “he”), the EB platform or the bank. The leftover value of unsold products and the goodwill loss for unmet demand do not change the nature of the problem, and thus we ignore them without loss of generality. We also assume that, as long as the capital-constrained online retailer borrows from the EB platform or the bank, 8
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she can be granted a sufficient loan-granting quota (i.e., the loan-granting quota is equal to the loan amount that the online retailer applies for). Notations are summarized in Table 1. Table 1: List of notations
market demand quantity
f (x)
the probability density function (PDF) of x
F (x)
the cumulative distribution function (CDF) of x
F (x)
the complementary CDF of x
w
supplier’s wholesale price
p
retail price of the product
k
demand threshold with no bankruptcy
m
online retailer’s initial working capital, m ≥ 0
r
loan interest rate for the time of order up to
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x
the end of the sales season, r ∈ [0, 1]
EB platform’s usage fee rate, λ ∈ ([0, 1]
q
online retailer’s order quantity, q ≥ 0
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λ
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We assume that the retail price p is a constant and the market’s demand distribution in the sales season can be confirmed. The uncertain demand is denoted by a nonnegative stochastic variable x.
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The probability density function (PDF) of x is f (·), the cumulative distribution function (CDF) of x is F (·), and the complementary CDF of x is F (·) = 1 − F (·). We assume that F (·) is differentiable and increasing and F (x = 0) = 0. Consequently, F (x) is continuous and f (x) > 0. We define f (x) F (x)
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h(x) =
as the failure rate and H(x) = xh(x) as the generalized failure rate. Following the
previous studies (Kouvelis and Zhao, 2011; Jing et al., 2012), we assume that h(x) and H(x) are
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both monotonically increasing in x. In other words, the demand distribution has a characteristic of increasing failure rate (IFR). Since the conservative EB platform’s decision variable, the EB platform’s loan interest rate, is
only one of the decision variables of the active EB platform, we regard the conservative EB platform financing as a special case of the more general active EB platform financing. After analyzing the equilibrium and coordination condition under active EB platform financing, we can easily obtain the equilibrium and coordination condition under conservative EB platform financing only by replacing the variable of the EB platform’s usage fee rate with a constant risk-free EB platform’s usage fee 9
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rate. Therefore, we set the following tie-breaking rule: unless stated otherwise, our study is mainly focused on the active EB platform financing form. The online retailer can choose between active EB platform financing and BCF. The active EB platform’s decision variables are λa and ra , and the bank’s decision variable under BCF is rb . Let
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λf and rf denote the EB platform’s risk-free usage fee rate and the risk-free loan interest rate, respectively. Specifically, the EB platform’s usage fee rate is λf under BCF. We assume that the capital market is perfectly competitive under BCF, which means that the bank’s revenue from issuing a loan to the online retailer is equal to the risk-free interest income. The order quantity in the decentralized SCF system is denoted by qi (i = a, b), which is the online retailer’s decision variable when she uses active EB platform financing (i = a) or BCF (i = b). Furthermore, the
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superscript ∗ denotes the corresponding cases when the participants choose the optimal parameters in the Stackelberg equilibrium.
The length of the credit period is equal to that of the sales period. Then, the sequence of events and the decision protocol are both shown in Figure 1. Before the sales season, the active EB
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platform declares his loan interest rate (ra ) and usage fee rate (λa ) under an EB platform financing contract, and the bank determines the loan interest rate (rb ) under a BCF contract. Then, the online retailer calculates her preliminary optimal order quantity under the market’s demand distribution
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and estimates whether her own initial working capital is sufficient for procurement. When it is not sufficient, the online retailer could choose an outside financing source, such as BCF or active
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EB platform financing, or refuse any outside financing. Thereafter, if the online retailer chooses an outside financing source, she decides the updated optimal order quantity (qa or qb ) and the
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loan amount (wqa − m or wqb − m) in response to the chosen lender’s financing contract. Then, she applies for the loan amount from the chosen lender, and after receiving the loan amount the online retailer purchases products from a supplier and pays in full. All of the above procedures
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occur before the sales season. During the sales season t ∈ (0, T ), the online retailer sells on the EB platform. With every unit being sold, the online retailer should pay a certain percentage (λa or λf ) of the sales revenue as the usage fee to the EB platform. Lastly, at the end of the sales season (t = T ), demand is realized. The online retailer should repay the loan debt. Then, if her sales revenue after subtracting the EB platform’s usage fee is sufficiently large, the loan debt is repayed successfully. Otherwise, the loan debt cannot be covered and the online retailer goes bankrupt. In this case, she must transfer all remaining sales revenue to the chosen lender.
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Online retailer chooses Online retailer decides the an outside financing order quantity, q a or q b , source or refuses any obtains the loan amount, outside financing. wqa - m or wqb - m , and purchases products.
With every unit being sold, online retailer pays a certain percentage, l a or l f , of the sales revenue as the usage fee to EB platform.
t =0
Before the sales season
During the sales season
Figure 1: Sequence of events
Online retailer can repay the loan debt successfully, or is forced into bankruptcy.
t =T
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Active EB platform and bank offer a TCF contract ( l a , ra ) and a BCF contract ( rb ), respectively.
As previously discussed, we formulate a two-stage Stackelberg game model to analyze the interaction between the SCF participants. Both the bank and the active EB platform are the Stackelberg leaders. The bank determines his loan interest rate, and the active EB platform sets his loan inter-
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est rate and usage fee rate simultaneously. Afterwards, the online retailer, as the follower, chooses a financing source (or refuses any outside financing) and decides the optimal order quantity. All participants are rational and select a best-reply policy to pursue the Stackelberg equilibrium, and the leaders can coordinate the total supply chain benefits by setting suitable decision variables.
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3.2. Profit Functions Under Bank Credit Financing and EB Platform Financing For notational convention, let a+ = max{a, 0}. πij (i = a, b and j = p, r, b, s) denotes the
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expected profit of the EB platform (j = p), the online retailer (j = r), the bank (j = b), and the centralized SCF system (j = s) at the end of the selling season under active EB platform financing
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(i = a) and BCF (i = b). Given the online retailer’s opportunity cost, the online retailer’s expected profit can be derived as
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πir = {(1 − EB platform’s usage fee rate)E[sales revenue] − E[online retailer’s actual repayment]}+ −opportunity cost of initial working capital,
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where i = a or b, and the online retailer’s actual repayment is the smaller one of the agreed repayment and the sales revenue after subtracting the EB platform’s usage fee. With every unit being sold, the online retailer should pay a certain percentage of the sales revenue as the usage fee to the EB platform simultaneously. However, she repays the loan debt at the end of the sales season. This process also obeys the PMSA rules (Lee and Rhee, 2011), because the EB platform’s usage fee is a kind of enterprise operation cost rather than a debt. Under BCF, other participants’ expected profits can be derived as πbp = (EB platform’s usage fee rate)E[sales revenue] 11
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and πbb = E[online retailer’s actual repayment] − opportunity cost of loan-granting quota. Under active EB platform financing, the EB platform’s expected profit is given by
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πap = E[online retailer’s actual repayment] + (EB platform’s usage fee rate)E[sales revenue] −opportunity cost of loan-granting quota.
The expected profit of the centralized BCF system is πbs = πbr + πbp + πbb , and the expected profit of
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the centralized active EB platform financing system is πas = πar + πap .
4. Equilibrium and Coordination Analysis Under Bank Credit Financing Before analyzing the BCF and the active EB platform financing models, we first consider the
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relation among parameters in a general model. Given the order quantity q, the online retailer’s Rq R∞ Rq expected sales quantity is s(q) = E{min(x, q)} = 0 xf (x)dx + q qf (x)dx = 0 F (x)dx.
Lemma 1. When a capital-constrained online retailer borrows from a lender, the demand threshold (wq−m)(1+r) , (1−λ)p
and the online retailer’s expected actual repayment to the
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with no bankruptcy is k =
lender is (1 − λ)ps(k) = (1 − λ)pE{min(x, k)}. (See Appendix A.1 for the proof ).
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When the online retailer borrows from a lender, her agreed repayment is (wq − m)(1 + r). Then, only if the market demand is sufficient to reach or exceed the demand threshold k =
(wq−m)(1+r) , (1−λ)p
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can the online retailer’s agreed repayment be covered at the end of the sales season. Otherwise, the online retailer goes bankrupt and transfers all remaining sales revenue to the EB platform. The purchase cost per unit is w(1 + r), which includes the loan interest, and (1 − λ)p is the sales revenue
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per unit after subtracting the EB platform’s usage fee. Then, we obtain w(1 + r) ≤ (1 − λ)p, and thus k ≤ q. The demand threshold with no bankruptcy is not bigger than the order quantity. Furthermore, it is obvious that s(k) < k, and thus, (1 − λ)ps(k) < (1 − λ)pk = (wq − m)(1 + r). It means that the online retailer’s expected actual repayment is less than the online retailer’s agreed repayment. In other words, the lender shares the market volatility risk with the online retailer by providing credit financing. Here, k (or s(k)) has different representations under BCF and EB platform financing, which is meaningful for establishing the participants’ profits functions and simplifying the analysis procedure. 12
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4.1. SCF Equilibrium Under Bank Credit Financing When the capital-constrained online retailer borrows from a bank in a perfectly competitive capital market, the bank can only get the market risk-free interest income by setting a suitable loan interest rate rb . Specifically, under BCF the EB platform does not provide financing services
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and his usage fee rate is λf . With every unit being sold, the online retailer should pay the usage fee λf p to the EB platform at the same time. Thus, the bank’s revenue equilibrium expression is given as follows:
E{min[(wqb − m)(1 + rb ), (1 − λf )pmin(x, qb )]} = (wqb − m)(1 + rf ).
(1)
This means that the bank’s expected revenues, either from issuing a loan amount (wqb − m) to the
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online retailer with rb or from investing this capital amount (wqb − m) with rf , are the same. We can also derive that rb > rf .
Proposition 1. In the decentralized BCF system, when the online retailer borrows from a bank in a perfectly competitive capital market, the market demand threshold with no bankruptcy is kb = and the bank’s revenue equilibrium expression is (1 − λf )ps(kb ) = (wqb − m)(1 + rf ).
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(wqb −m)(1+rb ) , (1−λf )p
Moreover, the expected profits of the online retailer, the EB platform, and the bank are πbr = qb∗ satisfies F (qb∗ ) =
w(1+rf ) (1−λf )p .
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(1 − λf )ps(qb ) − wqb (1 + rf ), πbp = λf ps(qb ), and πbb = 0, respectively. The optimal order quantity (See Appendix A.2 for the proof ).
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Proposition 1 shows that, when the online retailer borrows from a bank in a perfectly competitive capital market, her optimal order quantity qb∗ is fixed and independent from the bank’s loan interest
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rate rb . Then, we can find the expected participants’ profits, πbr (qb∗ ) and πbp (qb∗ ), which are both fixed. When the online retailer has sufficient initial working capital and does not borrow from outside, her optimal order quantity also satisfies F (q ∗ ) =
w(1+rf ) (1−λf )p .
Therefore, the capital-constrained online
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retailer’s optimal order quantity under BCF is identical to that of the capital-unconstrained online retailer, which is similar to the finding of Kouvelis and Zhao (2012). By substituting qb∗ into the bank’s revenue equilibrium expression Eq.(1), we could further obtain the bank’s loan interest rate rb . 4.2. SCF Coordination Under Bank Credit Financing Proposition 2. When the online retailer borrows from a bank in a perfectly competitive capital market, the expected profit of the centralized BCF system is πbs = πbr +πbp +πbb = ps(qbs )−wqbs (1+rf ), 13
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the optimal order quantity satisfies F (qbs∗ ) =
w(1+rf ) , p
and the SCF coordination cannot be achieved.
(See Appendix A.3 for the proof ). qbs denotes the order quantity in the centralized BCF system. Proposition 2 demonstrates that qbs∗ , πbs (qbs∗ ) and kb (qbs∗ ) in the centralized BCF system are all fixed, since rf , λf , w, m and p are
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all constant. By comparing F (qb∗ ) with F (qbs∗ ), we find that, only if λf = 0 is satisfied, can qb∗ be equal to qbs∗ . However, the case of λf = 0 is almost impossible in the real economy. In other words, the optimal order quantity in the decentralized BCF system is always smaller than that in the centralized BCF system. We could further find that the sum of participants’ profits under the decentralized BCF system is always smaller than the joint profit of the centralized BCF system
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(i.e., πbr (qb∗ ) + πbp (qb∗ ) + πbb (qb∗ ) < πbs (qbs∗ )). Hence, under BCF, SCF coordination cannot be achieved and the supply chain efficiency cannot be improved to the optimal level of the centralized BCF system. If the optimal order quantity under active EB platform financing can be consistent with that in the centralized BCF system, i.e., qa∗ could be equal to qbs∗ , EB platform financing performs
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better than BCF does in terms of supply chain efficiency improvement.
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5. Equilibrium and Coordination Analysis Under EB Platform Financing 5.1. SCF Equilibrium Under EB Platform Financing
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In the Stackelberg game, the active EB platform is the leader, with the decision variables of the EB platform’s loan interest rate ra and the EB platform’s usage fee rate λa . The online retailer decides her optimal order quantity qa following the EB platform’s decision. Specifically,
λa = λf .
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the conservative EB platform financing is a special case of the active EB platform financing with
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Proposition 3. In the decentralized active EB platform financing system, the market demand threshold with no bankruptcy is ka =
(wqa −m)(1+ra ) . (1−λa )p
Let Ω =
w(1+ra ) (1−λa )p
for simplicity.
(3.1) As the Stackelberg game follower, the online retailer’s expected profit is πar = (1−λa )p[s(qa )− s(ka )] − m(1 + rf ). Given the IFR assumption, the optimal order quantity satisfies F (qa∗ ) = ∗ ∂qa∗ ∂ka (qa∗ ) ∂ka (qa∗ ) a F ka (qa∗ )) Ω, ∂λ ≤ ∂q < 0. ∂ra < 0, and ∂λa ≤ ∂ra a (3.2) As the Stackelberg game leader, the EB platform’s expected profit is πap = (1 − λa )ps(ka ) +
λa ps(qa )−(wqa −m)(1+rf ). Then, the EB platform’s optimal decision variables satisfy w(1+ra∗ ) =
(1 − λ∗a )p and m(1 + ra∗ ) = 0. Furthermore, from the above Stackelberg equilibrium results, it could 14
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be further deduced that πar (ra∗ , λ∗a , qa∗ ) = −m(1 + rf ) and πap (ra∗ , λ∗a , qa∗ ) = ps(qa∗ ) − wqa∗ (1 + rf ). (See Appendix A.4 for the proof ). In Proposition 3.1 the online retailer’s expected profit is πar = (1 − λa )p[
R +∞ qa
R qa ka
(x − ka )f (x)dx +
(qa − ka )f (x)dx] − m(1 + rf ), which is similar to the finding of Yan et al. (2016). Since the
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partial derivatives of qa∗ with respect to ra and λa are both less than 0, there exists a one-to-one mapping between qa∗ and (ra , λa ), and qa∗ decreases monotonously as (ra , λa ) increases. This implies that, by setting a bigger usage fee rate or a bigger loan interest rate, the EB platform may not be able to obtain a larger profit, because of this monotonicity property of qa∗ . Additionally, the absolute value of
∂qa∗ ∂λa
is bigger than that of
∂qa∗ ∂ra .
Hence, by adjusting the usage fee rate λa , the EB
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platform can more easily induce the online retailer to enlarge the optimal order quantity qa∗ .
In Proposition 3.2, even though the EB platform obtains his maximum expected profit πap (ra∗ , λ∗a , qa∗ ), the online retailer’s maximum expected profit πar (ra∗ , λ∗a , qa∗ ) is not larger than 0. Then, the online retailer will refuse to borrow from the EB platform. Therefore, the solutions of the EB platform’s optimal decision variables do not exist. The EB platform, as the Stackelberg game leader, must
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consider the participation constraint that the online retailer’s expected profit should be larger than 0. Furthermore, in order to entice the online retailer to select EB platform financing rather than
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BCF, the online retailer’s expected profit under EB platform financing should also be larger than that under BCF, which will be discussed in subsection 5.2. Note that under conservative EB platform financing we can substitute λa = λf into Proposition 3 to obtain the corresponding optimal
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order quantity and optimal loan interest rate.
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5.2. Coordinating Strategy Under EB Platform Financing EB platform financing is a new form of TCF, and numerous previous studies ( Lee and Rhee,
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2011; Kouvelis and Zhao, 2012) have proven that supplier financing, as a traditional form of TCF, could be better than BCF in terms of the optimal order quantity and participants’ profits. In this section, we will analyze whether TCF offered by the EB platform could also be better than BCF, and under what condition this occurs. According to the previous research (Yan et al., 2016), if a TCF contract can achieve SCF coordination, it must ensure that each member’s profit is not lower than the profit level under BCF. On one hand, only when the EB platform’s profit from running a lending business is bigger than his profit under BCF, can he grant the online retailer a loan. On the other hand, if the 15
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online retailer’s profit under EB platform financing is bigger than her profit under BCF, she will prefer EB platform financing to BCF. Hence, if the two above-mentioned necessary conditions are satisfied simultaneously, SCF coordination is possible to be achieved. As we all know, internet companies follow Metcalf’s law, and satisfying a customer’s purchasing demand is an important way to entice the customer to visit the EB platform website. Therefore, even though the members’
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profits are the key to determining whether the online retailer and the EB platform participate in EB platform financing, the order quantity must also be sufficiently large in order to satisfy customers’ purchasing demand. Next, we will analyze the coordination conditions from the perspectives of the order quantity and the participants’ profits.
Corollary 1. Given p, w, m, rf , λf , and the IFR assumption, after comparing the optimal order
we conclude that:
qa∗ > qbs∗ , if F (ka (qa∗ )) <
qa∗ = qbs∗ , if F (ka (qa∗ )) =
(1−λa )(1+rf ) . 1+ra
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quantities in the centralized BCF system and the decentralized active EB platform financing system, (1−λa )(1+rf ) ; 1+ra
qa∗ < qbs∗ , if F (ka (qa∗ )) >
(1−λa )(1+rf ) ; 1+ra
(See Appendix A.5 for the proof ).
Note that both qa∗ and F (ka (qa∗ )) are functions of ra and λa . From Corollary 1, the optimal
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order quantity in the decentralized active EB platform financing system can be bigger than that in the centralized BCF system by setting suitable ra and λa , which is similar to the conclusion of Yan
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et al. (2016). This will help internet companies attract more registered users at a lower shortage rate. Obviously, these suitable ra and λa to realize qa∗ ≥ qbs∗ in Corollary 1 are different from these
ra∗ and λ∗a to maximize πap in Proposition 3. This suggests that the EB platform can set suitable
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ra and λa to obtain a bigger order quantity, but his profit under active EB platform financing may not be larger than that under BCF. Let qas denote the order quantity in the centralized active
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EB platform financing system, we could get the following proposition to characterize the SCF coordination conditions.
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Proposition 4. Given p, w, m, rf , λf , and the IFR assumption, the expected profit of the centralized active EB platform financing system is πas = πar + πap = ps(qas ) − wqas (1 + rf ). Then, the maximum profits and the optimal order quantities in the centralized active EB platform financing system and the centralized BCF system satisfy πas (qas∗ ) = πbs (qbs∗ ) and F (qas∗ ) = F (qbs∗ ) = When F ka (qa∗ )) =
(1−λa )(1+rf ) , 1+ra
w(1+rf ) . p
πar (qa∗ ) ≥ πbr (qb∗ ), and πap (qa∗ ) ≥ πbp (qb∗ ) are concurrently satisfied,
the active EB platform financing can achieve SCF coordination. However, the conservative EB platform financing may not be able to achieve SCF coordination. (See Appendix A.6 for the proof ). From Propositions 1 and 2, the optimal order quantity in the centralized BCF system is big16
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ger than that in the decentralized BCF system, i.e., qbs∗ > qb∗ , and thus the participants’ profits satisfy πbs (qbs∗ ) > πbr (qb∗ ) + πbp (qb∗ ). Specifically, qb∗ , qbs∗ , πbr (qb∗ ), πbp (qb∗ ) and πbs (qbs∗ ) are all fixed. Then Corollary 1 and Proposition 4 show that, when F ka (qa∗ )) =
(1+rf )(1−λa ) 1+ra
(which
is the coordination condition on order quantity) is satisfied by setting suitable ra and λa , the active EB platform can entice the online retailer to replicate the optimal order quantity deci-
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sion of the centralized BCF system or the centralized active EB platform financing system, i.e., qa∗ = qas∗ = qbs∗ > qb∗ , where qa∗ , qas∗ , qbs∗ and qb∗ are all fixed. Moreover, when the coordination condition on order quantity F ka (qa∗ )) =
(1+rf )(1−λa ) 1+ra
is satisfied, the total profit of the decentral-
ized active EB platform financing system is equal to that of the centralized active EB platform financing system or the centralized BCF system, but bigger than that of the decentralized BCF
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system, i.e., πar (qa∗ ) + πap (qa∗ ) = πas (qas∗ ) = πbs (qbs∗ ) > πbr (qb∗ ) + πbp (qb∗ ), where πas (qas∗ ), πbs (qbs∗ ), πbr (qb∗ ) and πbp (qb∗ ) are all fixed, and πar (qa∗ ) and πap (qa∗ ) are both functions of ra and λa . Therefore, when F ka (qa∗ )) =
(1+rf )(1−λa ) 1+ra
is satisfied, πar (qa∗ ) ≥ πbr (qb∗ ) and πap (qa∗ ) ≥ πbp (qb∗ ) (which are the coordina-
tion conditions on participants’ profits) are easy to be satisfied under active EB platform financing by setting suitable λa and ra . It implies that the active EB platform financing can achieve SCF
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coordination.
Under conservative EB platform financing, for the constant value of m and λa = λf , we can find (1+rf )(1−λa ) is satisfied. However, this unique solution 1+ra πap (qa∗ ) ≥ πbp (qb∗ ) simultaneously. This suggests that the
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the unique solution of ra when F ka (qa∗ )) = of ra may not satisfy πar (qa∗ ) ≥ πbr (qb∗ ) and
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conservative EB platform financing may not be able to achieve SCF coordination. Proposition 4 implies that the active EB platform financing performs better than the conser-
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vative case in terms of SCF coordination. When the active EB platform financing achieves SCF coordination, not only the optimal order quantity under active EB platform financing can be equal to that in the centralized SCF system, but also the participants’ profits under active EB platform fi-
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nancing can be larger than (or at least equal to) those under BCF. Therefore, the coordinated active EB platform financing can improve the supply chain efficiency and is attractive for all participants.
6. Numerical Studies and Extensions We now present numerical studies and extensions in order to clarify our findings and to illustrate the impacts of different parametric variables on SCF coordination. We maintain the following assumptions over the numerical studies: (i) demand x follows an exponential distribution with a 17
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mean of 100, which is similar to the parameter set by Buzacott and Zhang (2004), and (ii) the base values of the parameters are p = 1.6, w = 0.4, λf = 0.25, rf = 0.1, and m = 7. 6.1. Coordination Analysis Based on Different EB Platform’s Loan Interest Rates Proposition 5. Given p, w, m, rf , λf , and the IFR assumption, when the coordination condition dπar (qa∗ ) dra
> 0, and
dπap (qa∗ ) dra
(1+rf )(1−λa ) , 1+ra
is satisfied,
1−λa 1+ra
is a fixed value,
< 0. (See Appendix A.7 for the proof ).
a = − 1−λ 1+ra < 0,
λa
0.45
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0.4
0.35
0.3
0.25
0.2
0.15
ra(λa=0.25)=0.376
0
0.1
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EB platform’s usage fee rate λa when the
coordination condition on order quantity is satisfied
0.5
dλa dra
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on order quantity, F ka (qa∗ )) =
0.2 0.3 EB platform’s loan interest rate ra
0.4
0.5
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Figure 2: EB platform’s usage fee rate under different ra
40
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Participants’ profits when the coordination condition on order quantity is satisfied
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πpa(q*a)
38
πra(q*a)
πpb(q*b)
πrb(q*b)
36 34 32 30
coordination zone
28 26 24 22 20
ra(λa=0.25)=0.376 0
0.1
0.2 0.3 EB platform’s loan interest rate ra
0.4
0.5
Figure 3: Participants’ profits under different ra
If the coordination condition on order quantity F ka (qa∗ )) =
(1+rf )(1−λa ) 1+ra
is satisfied and m is a
constant, we obtain qa∗ = qas∗ = qbs∗ , which are all fixed. Then, there exists a one-to-one mapping 18
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between ra and λa from the equation F ka (qa∗ )) =
(1+rf )(1−λa ) . 1+ra
Provided m = 7 and ra changes
from 0 to 0.5 with an increment of 0.02, Figure 2 shows a linear decreasing trend between λa and ra , which suggests that
dλa dra
is a fixed negative number. In Figure 3, the respective profits of
the online retailer and the EB platform in the decentralized BCF system, πbr (qb∗ ) and πbp (qb∗ ), are independent of the EB platform’s loan interest ra . However, the respective profits of the online
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retailer and the EB platform in the decentralized active EB platform financing system, πar (qa∗ ) and πap (qa∗ ), are tightly tied to the EB platform’s loan interest ra . Here, πar (qa∗ ) increases in ra , whereas πap (qa∗ ) decreases in ra . Moreover, for any ra ∈ [0.4, 0.46], πar (qa∗ ) ≥ πbr (qb∗ ) and πap (qa∗ ) ≥ πbp (qb∗ ) are satisfied simultaneously, and thus SCF coordination can be achieved. If ra < 0.4, we obtain πar (qa∗ ) < πbr (qb∗ ), which implies that the online retailer would prefer BCF to EB platform financing.
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Similarly, if ra > 0.46, we obtain πap (qa∗ ) < πbp (qb∗ ), which implies that the EB platform will not choose EB platform financing. Specifically, when the active EB platform sets his loan interest rate at ra (λa = λf = 0.25) = 0.376, he cannot achieve SCF coordination. This suggests that the conservative EB platform may not be able to achieve SCF coordination.
From Proposition 4, when the coordination condition on order quantity is satisfied, we get
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πar (qa∗ ) + πap (qa∗ ) > πbr (qb∗ ) + πbp (qb∗ ), in which πbr (qb∗ ) and πbp (qb∗ ) are both fixed. Considering that with ra increasing the monotonicity of πar (qa∗ ) is different from that of πap (qa∗ ) in Proposition 5, we can
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further deduce that πar (qa∗ ) ≥ πbr (qb∗ ) and πap (qa∗ ) ≥ πbp (qb∗ ) can be satisfied simultaneously at some
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reasonable points of ra . This has been confirmed by the numerical study in Figure 3. 6.2. Coordination Analysis Based on Different EB Platform’s Usage Fee Rates
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Proposition 6. Given p, w, m, rf , λf , and the IFR assumption, when the coordination condition on order quantity, F ka (qa∗ )) = < 0, and
is satisfied,
1+ra 1−λa
> 0. (See Appendix A.8 for the proof ).
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dπar (qa∗ ) dλa
dπap (qa∗ ) dλa
(1+rf )(1−λa ) , 1+ra
19
is a fixed value,
dra dλa
1+ra = − 1−λ < 0, a
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coordination condition on order quantity is satisfied
0.9 ra
EB platform¡¯s loan interest rate ra when the
0.8 0.7 0.6 0.5
0.3 0.2 0.1 0
λa=λf=0.25 0
0.05
0.1
0.15 0.2 0.25 0.3 0.35 EB platform’s usage fee rate λa
0.4
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0.4
0.45
πpa(q*a) 40
35
30
πpb(q*b)
πrb(q*b)
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25
20
15
0
0.05
0.1
λa=λf=0.25
0.15 0.2 0.25 0.3 0.35 EB platform’s usage fee rate λa
0.4
0.45
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10
πra(q*a)
coordination zone
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Participants’ profits when the coordination condition on order quantity is satisfied
45
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Figure 4: EB platform’s loan interest under different λa
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Figure 5: Participants’ profits under different λa
Provided the online retailer’s initial working capital m = 7, if F ka (qa∗ )) =
(1+rf )(1−λa ) 1+ra
is
satisfied, we demonstrate the relations among the participants’ profits, EB platform’s loan interest
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rate and the EB platform’s usage fee rate. A linear decreasing trend between λa and ra suggests dλa dra
is a fixed negative number in Figure 4. Figure 5 shows that the online retailer’s profit in
the decentralized EB platform financing system πar (qa∗ ) decreases in the usage fee rate λa , and has only one intersection point with the πbr (qb∗ ) curve. The EB platform’s profit in the decentralized EB platform financing system πap (qa∗ ) increases in the usage fee rate λa , and has an intersection point with the πbp (qb∗ ) curve. For any λa ∈ [0.20, 0.22], both πar (qa∗ ) ≥ πbr (qb∗ ) and πap (qa∗ ) ≥ πbp (qb∗ ) are satisfied simultaneously, and SCF coordination can be achieved. Note that if λa < 0.20 or λa > 0.22, either πar (qa∗ ) < πbr (qb∗ ) or πap (qa∗ ) < πbp (qb∗ ) occurs, and the online retailer or the EB 20
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platform will not select EB platform financing. Moreover, the fluctuation of πar and πap in Figure 5 is bigger than that in Figure 3, with either λa or ra ranging from 0 to 0.46. The reason may be that
∂qa∗ ∂λa
≤
∂qa∗ ∂ra
< 0 and λa has a greater impact on qa∗ from Proposition 2. Specifically, if
λa = λf = 0.25, we have πar (qa∗ ) < πbr (qb∗ ), which means the conservative EB platform may not be
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able to achieve SCF coordination. When the coordination condition on order quantity is satisfied, with λa increasing the monotonicity of πar (qa∗ ) is different from that of πap (qa∗ ) in Proposition 6. We have known that the participants’ profits satisfy πar (qa∗ ) + πap (qa∗ ) > πbr (qb∗ ) + πbp (qb∗ ) in Proposition 4. Additionally, πbr (qb∗ ) and πbp (qb∗ ) are both fixed. Thus, πar (qa∗ ) > πbr (qb∗ ) and πap (qa∗ ) > πbp (qb∗ ) can be satisfied simultaneously at some reasonable points of λa , which has been confirmed by the numerical study in Figure
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5.
6.3. EB Platform Financing Coordination Under a Two Part-tariff Contract In the case of a two-part tariff, after the EB platform financing is granted, the capital-constrained
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online retailer should not only pay the usage fee during the sales period and repay the loan debt at the end of sales period, but also pay a lump sum payment at the beginning of the sales period, as it is frequently asked as an entry fee to transactions in the real marketplace (Lee and Rhee, 2010).
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The entry fee is independent of the order quantity and is a decision variable of the EB platform. Let A denote the entry fee. Then, the participants’ profits are πar = (1 − λa )p[s(qa ) − s(ka )] − m(1 +
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rf ) − A(1 + rf ) and πap = (1 − λa )ps(ka ) + λa ps(qa ) − (wqa − m)(1 + rf ) + A(1 + rf ).
Because A is independent of the order quantity qa , the SCF coordination conditions are still (1−λa )(1+rf ) , 1+ra
πar (qa∗ ) ≥ πbr (qb∗ ), and πap (qa∗ ) ≥ πbp (qb∗ ). It can be further concluded that,
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F ka (qa∗ )) =
after we add the entry fee as an EB platform’s decision variable, both the active EB platform and
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the conservative EB platform can achieve SCF coordination. Previous numerical studies have demonstrated the existence of coordination zone when the ini-
tial working capital m is a constant. However, when m is within a certain range, we could find many groups of (ra , λa ) that can achieve SCF coordination. Are there some significant characteristics of these coordination points? Is there a threshold of m that determines whether the online retailer applies for an outside financing or refuses any outside financing? We will answer these questions next. In the following numerical studies, the parameters values of no special instructions are consistent with those of previous sections.
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6.4. The Relations among the Usage Fee Rate, the Loan Interest Rate, and the Initial Working Capital When the Weak Coordination Conditions and the BCF Equilibrium are Satisfied Proposition 7. Given p, w, rf , λf , and the IFR assumption, for m within a certain range, then, (1+rb )2 h(kb (qb∗ )) drb dm = − (1−λf )p[1−H(kb (qb∗ ))] < 0. (1+rf )(1−λa ) and πar (qa∗ ) = πbr (qb∗ ) (7.2) when the weak coordination conditions, F ka (qa∗ )) = 1+ra (1+rf )[H(ka (qa∗ ))−λa ] (1+ra ) dra ∗ a are satisfied, dλ dm = p[1−H(ka (qa∗ ))][s(qa∗ )−s(ka (qa∗ ))] , dm = − (1−λa )p[1−H(ka (qa∗ ))] [h(ka (qa ))(1 + ra ) + H(ka (qa∗ ))−λa dka (qa∗ ) (1+ra ) . = − (1−λa )p[1−H(k ∗ dm s(qa∗ )−s(ka (qa∗ )) (1 + rf )], and a (qa ))] dka (qa∗ ) dH(ka (qa∗ )) (1) We have dm < 0 and < 0; dm
(2) If λa < H(ka (qa∗ )) < 1, we have
dλa dm
w(1+rf ) (1−λf )p ,
> 0 and
(3) If 0 < H(ka (qa∗ )) < λa , we have (i)
dλa dm
dra dm
are satisfied,
< 0;
< 0, (ii) dra dm
< 0 for h(ka (qa∗ ))(1 + ra )[s(qa∗ ) −
> 0 for h(ka (qa∗ ))(1 + ra )[s(qa∗ ) − s(ka (qa∗ ))] −
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s(ka (qa∗ ))] − [λa − H(ka (qa∗ ))](1 + rf ) > 0, and (iii)
dra dm
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(7.1) when the BCF equilibrium, F (qb∗ ) =
[λa − H(ka (qa∗ ))](1 + rf ) < 0. (See Appendix A.9 for the proof ).
1.8
r
a
r
b
M
1.4
λf
1.2 1
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Loan interest rate when the weak coordination conditions and the BCF equilibrium are satisfied
1.6
λa
1.6 1.4 1.2 1
0.8
0.6
0.6
0.4
0.4
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0.8
0.2
0.2
0
−0.12 0
2
4
0 6
−0.12 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Online retailer’s initial working capital m
Figure 6: ra ,λa , rb , and rf under different m
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EB platform’s usage fee rate when the weak coordination conditions and the BCF equilibrium are satisfied
1.8
22
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0.76
0.76 0.7
0.64
0.64
0.58
0.58
0.52
0.52
0.46
0.46
0.4
0.4
0.34
0.34
0.28
0.28
0.22
0.22
0.16
0.16
0.1
0.1
0.04 −0.02
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Generalized failure rate when the weak coordination conditions are satisfied
λa
EB platform’s usage fee rate when the weak coordination conditions are satisfied
H(ka(q*a))
0.7
0.04
0
2
4
6
8
10
12 14 16 18 20 22 24 26 28 Online retailer’s initial working capital m
30
32
34
36
38
−0.02 40
We refer to F ka (qa∗ )) =
(1+rf )(1−λa ) 1+ra
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Figure 7: H(ka (qa∗ )) and λa under different m
and πar (qa∗ ) = πbr (qb∗ ) as the weak coordination conditions.
Accordingly, when m is within a certain range from 0 to 40, we obtain many groups of (λa , ra ) that satisfy the weak coordination conditions. From Proposition 1, after we find the optimal order
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quantity qb∗ under BCF, we can find rb from the equation (1 − λf )ps(kb (qb∗ )) = (wqb∗ − m)(1 + rf ) for any given m. Figure 6 has depicted the impacts of the initial working capital on the loan interest
equilibrium are satisfied.
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and on the EB platform’s usage fee rate, when the weak coordination conditions and the BCF
We could find that rb decreases with m in Proposition 7.1, which has been confirmed by the
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numerical study in Figure 6. That ka (qa∗ ) and H(ka (qa∗ )) both decrease with m in Proposition 7.2 has also been confirmed by the numerical study in Figure 7. In Figure 6 and Figure 7, we could
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find that, when m < 20 (i.e., H(ka (qa∗ )) > λa ), λa increases with m. However, when m > 20 (i.e., H(ka (qa∗ )) < λa ), λa decreases with m. This monotonicity of λa is consistent with Proposition 7.2.
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In Figure 6, we also find that ra always decreases with m. For m ∈ [20, 40] (i.e., H(ka (qa∗ )) < λa ) in
our numerical study, h(ka (qa∗ ))(1 + ra )[s(qa∗ ) − s(ka (qa∗ ))] − [λa − H(ka (qa∗ ))](1 + rf ) is always bigger than 0. Thus, this monotonicity of ra is also consistent with Proposition 7.2. In Proposition 7.2, if λa < H(ka (qa∗ )) < 1,
dλa dm
> 0; otherwise,
dλa dm
< 0. It suggests that, with
the initial working capital m increasing within a certain range, the EB platform’s usage fee rate λa does not remain constant. From the monotonicity analysis of ra in Proposition 7.2, we also find that, the EB platform’s loan interest rate also changes with the initial working capital. It could be concluded that the EB platform should adjust the variables of λa and ra together in order to 23
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achieve SCF coordination. If λa < H(ka (qa∗ )) < 1, λa monotonically increases with m in Proposition 7.2. Thus, under the condition of λa < H(ka (qa∗ )) < 1 there exists at most one intersection point of λa and λf . When m = 8, we could find this intersection point in Figure 6. Only in this intersection point, can the
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conservative EB platform financing achieve SCF coordination. However, in Figure 6, within a large range of the online retailer’s initial working capital m ∈ (2, 24), the reasonable active EB platform’s coordinated financing contracts, where ra and λa are both in the range of [0,1], could be designed. This again demonstrates that, in order to achieve SCF coordination, the EB platform should be active to adjust variables of λa and ra together.
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From the Proposition 7.2 and Figure 6, there exist very few intersection points of λa and λf where the conservative EB platform financing could achieve SCF coordination. It will stunt the development of the financing business. However, the active EB platform can offer many groups of (ra , λa ) to design his coordinated financing contracts, which can be applicable to different conditions of the online retailer’s initial working capital. In other words, the online retailer will have a variety
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of coordinated financing options. It will facilitate the operation of the active EB platform financing. The active EB platform is more flexible to achieve SCF coordination than the conservative one.
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6.5. Online Retailer’s Selection: Using an Outside Financing, or Refusing Any Outside Financing When the online retailer refuses any outside financing, she uses her own initial working capital
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to purchase and the EB platform’s usage fee rate is λf . Let qnl denote the online retailer’s order r . quantity and the expected profit of the online retailer is denoted by πnl
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r = Proposition 8. When the online retailer refuses any outside financing, her expected profit is πnl −1 w(1+rf ) ( (1−λf )p ), the optimal −1 w(1+rf ) F ( (1−λf )p ). Moreover,
(1 − λf )ps(qnl ) − wqnl (1 + rf ) with a constraint of wqnl ≤ m. If m < wF
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∗ = order quantity is qnl
m w.
∗ = Otherwise, the optimal order quantity is qnl
r (q ∗ ) ≤ π r (q ∗ ). when the active EB platform financing can achieve SCF coordination, we have πnl a a nl
(See Appendix A.10 for the proof ). ∗ and corresponding maximum For any given m, we can find the optimal order quantity qnl r (q ∗ ) from Proposition 8. With m increasing from 0 to infinity, the online expected profit πnl nl r (q ∗ = F retailer’s maximum expected profit is πnl nl
−1 w(1+rf ) ( (1−λf )p )),
which is identical to her maximum
expected profit under BCF in Proposition 1. Moreover, from Proposition 4, when the coordination
24
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r (q ∗ = F of the active EB platform financing is achieved, we could deduce that πnl nl
−1 w(1+rf ) ( (1−λf )p ))
=
πbr (qb∗ ) ≤ πar (qa∗ ). Therefore, we could conclude that, in order to obtain a bigger profit, the online retailer will prefer using the coordinated active EB platform financing to refusing any outside
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financing, even if she has sufficient initial working capital.
7. Conclusion
In this paper, we present two financing options (EB platform financing and BCF) for an online retailer in order to study the interaction of the financing and operating decisions. Our model offers some analyses of the equilibrium and coordination of EB platform financing from a supply
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chain perspective. In the decentralized EB platform financing system, we obtain the Stackelberg equilibrium and local optimal parameters. Furthermore, we analyze the SCF coordination problem by comparing the decentralized EB platform financing system, the decentralized BCF system, and the centralized BCF system (or the centralized EB platform financing system) to find the global optimal parameters and to design a properly structured financing contract.
M
The Stackelberg equilibrium analysis of the EB platform financing reveals that the EB platform’s loan interest rate and the EB platform’s usage fee rate both have an impact on the online retailer’s
ED
order quantity decision, as well as the participants’ profits. From the coordination analysis, we conclude that EB platform financing performs better than BCF does in terms of SCF efficiency
PT
improvement, and that the active EB platform financing can achieve SCF coordination, but the conservative EB platform financing may not. When SCF coordination is achieved, as shown in Proposition 4, the optimal order quantity in the decentralized EB platform financing system is
CE
equal to that in the centralized BCF system, and the participants’ profits in the decentralized EB platform financing system are not less than those in the decentralized BCF system. Furthermore,
AC
in order to achieve SCF coordination, the EB platform should adjust variables of the EB platform’s loan interest rate and the EB platform’s usage fee rate together. When the coordination condition on order quantity is satisfied, if the online retailer’s initial working capital is a constant, the EB platform’s loan interest rate is linearly decreasing in the EB platform’s usage fee rate. When the weak coordination conditions are satisfied, the active EB platform can offer many coordinated financing contracts, which are applicable to different initial working capital conditions, while the conservative EB platform could only design very few coordinated financing contracts. Compared with the decentralized BCF system, the EB platform’s profit under the coordinated 25
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active EB platform financing could be improved by inducing a larger online retailer’s order quantity. Thus, the EB platform is willing to finance the online retailer, even though he needs to invest a lot of capital as the loan-granting quota. The online retailer also prefers the coordinated active EB platform financing to BCF, because her profit under the coordinated active EB platform financing is larger than (or at least equal to) that in the decentralized BCF system. Moreover, the optimal
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order quantity under the coordinated active EB platform financing is bigger than that in the decentralized BCF system, which is conducive to improving the customer service level.
By formulating a model, in which the online retailer refuses any outside financing, and comparing her maximum profits in different cases, we analyze the impact of the initial working capital on the online retailer’s financing selection. We conclude that, in order to obtain a bigger profit,
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the online retailer will always prefer using the coordinated active EB platform financing to refusing any outside financing, even though she has sufficient initial working capital. We have various future research directions to explore. First, this study establishes that the active EB platform financing can achieve SCF coordination and is better than BCF in terms of
M
the participants’ profits and the order quantity. However, the conservative EB platform may not achieve SCF coordination. This raises the question that why not all EB platforms set their usage fee rate and loan interest rate at the same time, or why not all EB platforms merge their lending
ED
business and leasing business. We might conjecture that isolating risks from different businesses, online retailer’s operating information asymmetry, and market imperfections, such as bankruptcy
PT
costs and taxes, have a relatively large impact on the EB platform’s decision. We will study these issues in future works. Second, we also intend to analyze the issue of lending competition among
CE
multiple EB platforms, which could be studied by a special Stackelberg game with multiple leaders and one follower. Third, we can also apply our research method to analyze the equilibrium and coordination of a new SCF form, which is operated on a peer-to-peer lending platform. Fourth,
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the capital-constrained online retailer’s dynamic inventory equilibrium problem in the case of a multi-period setting is also worth studying. Acknowledgements The authors thank the editor and two anonymous reviewers of this paper for their constructive suggestions and comments. This research was funded by the National Natural Science Foundation of China (No. 71701182, No. 71372187 and No. 71772115). Appendix:
26
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A.1. Proof of Lemma 1 The online retailer’s actual repayment at the end of selling season is the smaller one of the agreed repayment and the sales revenue after subtracting the EB platform’s usage fee, i.e., min[(wq−m)(1+ r), (1 − λ) p min(x, q)] = (1 − λ) p min[ (wq−m)(1+r) , min(x, q)], where (wq − m)(1 + r) is the agreed (1−λ) p
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repayment, and (1 − λ) p is the sales revenue per unit after subtracting the EB platform’s usage fee. Additionally, w(1 + r) is the purchasing cost per unit, which includes the loan interest. We obtain that w(1+r) ≤ (1−λ) p and
(wq−m)(1+r) (1−λ) p
≤ q. Thus, we have (1−λ) p min[ (wq−m)(1+r) , min(x, q)] = (1−λ) p
(1 − λ) p min[ (wq−m)(1+r) , x]. This implies that if x ≥ (1−λ) p
(wq−m)(1+r) , (1−λ)p
the online retailer’s actual
repayment can be (wq − m)(1 + r), and she can repay the full loan debt; If x <
(wq−m)(1+r) , (1−λ)p
the
online retailer cannot repay the full loan debt and must transfer all sales revenue (1 − λ)px to the
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. The online retailer’s lender. Hence, the demand threshold with no bankruptcy is k = (wq−m)(1+r) (1−λ)p Rk expected actual repayment is (1 − λ)pE[min(x, k)] = (1 − λ)p 0 F (x)dx = (1 − λ)ps(k). A.2. Proof of Proposition 1 Substituting kb =
(wqb −m)(1+rb ) (1−λf )p
into Eq.(1), we have (1 − λf ) p E{min[kb , min(x, qb )]} = (wqb −
M
m)(1+rf ). Because of kb ≤ qb , it follows (1−λf ) p E{min[kb , min(x, qb )]} = (1−λf ) p E{min[kb , x]}. Hence, Eq.(1) can be further converted to (1−λf )ps(kb ) = (wqb −m)(1+rf ). From the participants’
ED
profits expressions in subsection 3.2, under BCF the online retailer’s expected profit is πbr = E{(1 − λf )pmin(x, qb ) − min[(wqb − m)(1 + rb ), (1 − λf )pmin(x, qb )]}+ − m(1 + rf )
PT
= (1 − λf )pE{min(x, qb ) − min[kb , x]} − m(1 + rf ) = (1 − λf )p{s(qb ) − s(kb )} − m(1 + rf )
CE
= (1 − λf )ps(qb ) − wqb (1 + rf ).
(A.1)
AC
The EB platform can only obtain the usage fee and thus his expected profit is πbp = λf ps(qb ).
(A.2)
πbb = (1 − λf )ps(kb ) − (wqb − m)(1 + rf ) = 0.
(A.3)
The bank’s expected profit is
Taking the first-order and second-order derivative of πbr with respect to qb , we obtain that the optimal order quantity qb∗ satisfies F (qb∗ ) = F (kb (qb∗ ))
w(1 + rf ) w(1 + rb ) = . (1 − λf )p (1 − λf )p 27
(A.4)
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A.3. Proof of Proposition 2 From Proposition 1, πbs = πbr + πbp + πbb = ps(qbs ) − wqbs (1 + rf ). After taking the first-order and
second-order derivative of πbs with respect to qbs , we deduce that the second-order derivative is not w(1+rf ) . p
great than 0, and thus F (qbs∗ ) =
Since F (qb∗ ) =
w(1+rf ) (1−λf )p
in Proposition 1, it can be deduced
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that, only if λf = 0, can we obtain qb∗ = qbs∗ . However, the case of λf = 0 is almost impossible in the real economy. Therefore, the SCF coordination cannot be achieved under BCF. A.4. Proof of Proposition 3
Under active EB platform financing, the market demand threshold with no bankruptcy is ka = (wqa −m)(1+ra ) . (1−λa )p
According to Lemma 1, we obtain ka ≤ qa .
is
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From the participants’ profits expressions in subsection 3.2, the online retailer’s expected profit
πar = E{(1 − λa )pmin(x, qa ) − min[(wqa − m)(1 + ra ), (1 − λa )pmin(x, qa )]}+ − m(1 + rf ) = (1 − λa )p[s(qa ) − s(ka )] − m(1 + rf ),
M
and the EB platform’s expected profit is
(A.5)
ED
πap = (1 − λa )ps(ka ) + λa ps(qa ) − (wqa − m)(1 + rf ). Proof of Proposition 3.1: Let Ω =
w(1+ra ) (1−λa )p
(A.6)
for simplicity. According to Lemma 1, we know
PT
that w(1 + ra ) ≤ (1 − λa )p and Ω ≤ 1. After we drive the first and second derivative of πar dπar dqa
CE
f (qa )]. When
dπar dqa
2 r
= (1 − λa )p[F (qa ) − ΩF (ka )] and ddqπ2a = (1 − λa )p[Ω2 f (ka ) − a 2 r = 0, we get F (qa∗ ) = F ka (qa∗ ) Ω. Thus, we further deduce that ddqπ2a |qa =qa∗ =
with respect to qa , we get that
a
(1 − λa )pF (qa∗ )[Ωh(ka (qa∗ )) − h(qa∗ )]. Since Ω ≤ 1, qa∗ ≥ ka (qa∗ ) and the IFR assumption, we get d2 πar dqa2
AC
Ωh(ka (qa∗ )) − h(qa∗ ) ≤ 0 and
≤ 0. Then, the optimal order quantity satisfies F (qa∗ ) = F ka (qa∗ ) Ω.
(A.7)
After taking partial derivative of Eq.(A.7) with respect to ra on both sides, we obtain
1−H(ka (qa∗ )) (1+ra )(Ωh(ka (qa∗ ))−h(qa∗ )) .
to ra on both sides, We further obtain 1− Ω=
w(1+ra ) (1−λa )p
Then, we take the partial derivative of ka (qa∗ ) =
∂ka (qa∗ ) ∂ra (wqa∗ −m)
=Ω
w
∂qa∗ ∂ra
+
wqa∗ −m (1−λa )p
=
Ω−ka (qa∗ )h(qa∗ ) (1+ra )(Ωh(ka (qa∗ ))−h(qa∗ ))
(wqa∗ −m)(1+ra ) (1−λa )p
≤ 1, ka ≤ qa and the IFR assumption. 28
=
with respect
∗ −m) (wqa 1− h(qa∗ ) w (1+ra )(Ωh(ka (qa∗ ))−h(qa∗ )) (wqa∗ −m)(1+ra ) ∗
=Ω
h(qa∗ ) ≤ 1−H(ka (qa∗ )), because of H(ka (qa∗ )) =
∂qa∗ ∂ra
(1−λa )p
.
h(ka (qa )),
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Similar to Chen and Wang (2012), Yan and Sun et al. (2016) and Gao and Fan et al.(2018), we also use proof by contradiction to imply deduce that
∂ka (qa∗ ) ∂ra
≥ 0, owing
and the IFR assumption. Let
∂qa∗ ∂ra
< 0. Assume that the inequality of
∂qa∗ ∂ra
∗ to Ωh(ka (qa∗ )) − h(qa∗ ) < 0 , 1 − (wqaw−m) h(qa∗ ) ≤ 1 ∗ ∂ka (qa∗ ) |ra =r0 = 0, it follows (wqa (rw0 )−m) h(qa∗ (r0 )) ∂ra
≥ 0 holds, we
− H(ka (qa∗ )) ≤ 0 = 1. Note that
0 ≤ ra ≤ 1 , we have three cases to discuss for any ra .
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∗ (wqa∗ (ra )−m) h(qa∗ (ra )) < (wqa (rw0 )−m) h(qa∗ (r0 )) = 1, because w ∗ ∗ a (qa ) Therefore, 1 − (wqa (rwa )−m) h(qa∗ (ra )) > 0, and thus ∂k∂r < 0, a
(1)If r0 ≥ 1, for any ra < 1, we have
of
∂qa∗ ∂ra
≥ 0 and the IFR assumption.
which is inconsistent with the assumption. (2)If 0 < r0 < 1, we have
∂ka (qa∗ ) ∂ra
< 0 for any ra ∈ (0, r0 ). It is also a contradiction to the
(3)If r0 = 0, we get ∂qa∗ ∂ra |ra =r0 ≤
0 and it is a contradiction to the assumption.
Therefore, the inequalities of wqa∗ −m h(qa∗ ) w
∗
0, and then, 0 = 1 − wqa (rw0 )−m h(qa∗ (r0 )) ≤ 1 − H(ka (qa∗ , r0 )).
∂qa∗ ∂ra
< 0 and
≤ 1.
Similarly, we derive
∂qa∗ ∂λa
=
Because of 1 + ra ≥ 1 − λa , it
∂ka (qa∗ ) ∂ra
< 0 hold, which further lead to H(ka (qa∗ )) <
∗ ∗ 1−H(ka (qa∗ )) ∂ka (qa∗ ) a (qa )h(qa ) = (1−λaΩ−k ∂λa (1−λa )(Ωh(ka (qa∗ ))−h(qa∗ )) < 0 and )(Ωh(ka (qa∗ ))−h(qa∗ )) ∗ ∂qa∗ ∂ka (qa∗ ) ∂ka (qa∗ ) a is obvious that ∂λ ≤ ∂q < 0. ∂ra < 0 and ∂λa ≤ ∂ra a
M
Hence,
∂ka (qa∗ ) |ra =r0 = ∂ra
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assumption.
< 0.
ED
Proof of Proposition 3.2: When F (qa∗ ) = F ka (qa∗ ) Ω, after taking partial derivative of
Eq.(A.6) with respect to ra , we obtain
PT
∂ka (qa∗ ) ∂πap (qa∗ ) ∂q ∗ = (1 − λa )pF ka (qa∗ ) + [λa pF (qa∗ ) − w(1 + rf )] a = 0, ∂ra ∂ra ∂ra
(A.8)
CE
which can be converted to
(A.9)
AC
(1 − λa )pF ka (qa∗ ) [Ω − ka (qa∗ )h(qa∗ )] + [λa pF (qa∗ ) − w(1 + rf )][1 − H(ka (qa∗ ))] = 0.
When F (qa∗ ) = F ka (qa∗ ) Ω, after taking partial derivative of Eq.(A.6) with respect to λa , we
obtain
∂ka (qa∗ ) ∂πap (qa∗ ) ∂q ∗ = ps(qa∗ ) − ps(ka (qa∗ )) + (1 − λa )pF ka (qa∗ ) + [λa pF (qa∗ ) − w(1 + rf )] a = 0. ∂λa ∂λa ∂λa
(A.10)
29
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From Eq.(A.10)-Eq.(A.8)=0, we get 1 1 Ω − ka (qa∗ )h(qa∗ ) − ) ( ∗ ∗ Ωh(ka (qa )) − h(qa ) 1 − λa 1 + ra 1 − H(ka (qa∗ )) 1 1 + [λa pF (qa∗ ) − w(1 + rf )] − ) = 0. ( Ωh(ka (qa∗ )) − h(qa∗ ) 1 − λa 1 + ra
ps(qa∗ ) − ps(ka (qa∗ )) + (1 − λa )pF ka (qa∗ ))
(A.11)
Substituting Eq.(A.9) into Eq.(A.11), we obtain ps(qa∗ ) − ps(ka (qa∗ )) = 0, i.e., qa∗ = ka (qa∗ ). This p
p
∗
∗
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a (qa ) a (qa ) = 0 and ∂π∂λ = 0 be satisfied simultaneously. means that, only when qa∗ = ka (qa∗ ), can ∂π∂r a a ∗ −m)(1+r ) w(1+r ) (wq a a a Then, owing to F (qa∗ ) = F ka (qa∗ ) (1−λa )p and ka (qa∗ ) = , if qa∗ = ka (qa∗ ), we can obtain (1−λa )p
that the EB platform’s optimal decision variables satisfy w(1 + ra∗ ) = (1 − λ∗a )p and m(1 + ra∗ ) = 0. Therefore, after we get the above Stackelberg equilibrium, the EB platform’s expected profit
expression (i.e., Eq.(A.6)) can be convert to πap (ra∗ , λ∗a , qa∗ ) = ps(qa∗ )−(wqa∗ −m)(1+rf ), and the online
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retailer’s expected profit expression (i.e., Eq.(A.5)) can be convert to πar (ra∗ , λ∗a , qa∗ ) = −m(1 + rf ) which is not larger than 0. A.5. Proof of Corollary 1
ED
A.6. Proof of Proposition 4
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a) s∗ From Proposition 2 and Proposition 3, we know that F (qa∗ ) = F ka (qa∗ ) w(1+r (1−λa )p and F (qb ) = (1+ra ) w(1+rf ) . If F ka (qa∗ ) (1−λ < 1 + rf , it follows F (qa∗ ) < F (qbs∗ ), i.e., qa∗ > qbs∗ . Similarly, if p a) (1+ra ) (1+ra ) F ka (qa∗ ) (1−λ > 1 + rf , we get qa∗ < qbs∗ , and, if F ka (qa∗ ) (1−λ = 1 + rf , we get qa∗ = qbs∗ . a) a) From Proposition 3, we obtain πas = πar + πap = ps(qas ) − wqas (1 + rf ), which is similar to the
PT
expression of πbs in Proposition 2. After taking derivative, we get qas∗ = qbs∗ and F (qbs∗ ) = F (qas∗ ) =
w(1 + rf ) . p
(A.12)
CE
From Propositions 1 and 2, we obtain qbs∗ > qb∗ and πbs (qbs∗ ) > πbr (qb∗ ) + πbp (qb∗ ). Then, from Corollary 1 and Eq.(A.12), if F (ka (qa∗ )) =
(1−λa )(1+rf ) , 1+ra
which is the coordination condition on
AC
order quantity, is satisfied, we yield qa∗ = qas∗ = qbs∗ > qb∗ and πar (qa∗ ) + πap (qa∗ ) = πas (qas∗ ) = πbs (qbs∗ ) > πbr (qb∗ ) + πbp (qb∗ ), where qa∗ , qas∗ , qbs∗ , qb∗ , πas (qas∗ ), πbs (qbs∗ ), πbr (qb∗ ), and πbp (qb∗ ) are all fixed, and πar (qa∗ ) and πap (qa∗ ) are both functions of ra and λa . According to Proposition 1, qb∗ is the optimal order quantity in the decentralized BCF system. In order to entice the participants to support this coordination strategy, we can get the coordination conditions on participants’ profits, πar (qa∗ ) ≥ πbr (qb∗ ) and πap (qa∗ ) ≥ πbp (qb∗ ). If F (ka (qa∗ )) =
(1−λa )(1+rf ) 1+ra
is satisfied, we have obtained that πar (qa∗ ) + πap (qa∗ ) > πbr (qb∗ ) + πbp (qb∗ ),
where the parameters of qa∗ , qb∗ , πbr (qb∗ ), and πbp (qb∗ ) are all fixed, and πar (qa∗ ) and πap (qa∗ ) are both 30
ACCEPTED MANUSCRIPT
functions of ra and λa . Hence, under active EB platform financing, we can figure out the feasible region of (λa , ra ), for any constant value of m, from the coordination conditions F (ka (qa∗ )) = (1−λa )(1+rf ) , 1+ra
πar (qa∗ ) ≥ πbr (qb∗ ) and πap (qa∗ ) ≥ πbp (qb∗ ). However, under conservative EB platform
financing, for any constant value of m and λa = λf , we can find an unique solution of ra from and πap (qa∗ )
unique solution of ra may not be able to satisfy πar (qa∗ ) ≥ πbr (qb∗ ) time. Therefore, the active EB platform financing can achieve
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(1−λa )(1+rf ) , but this 1+ra p ∗ ≥ πb (qb ) at the same
F (ka (qa∗ )) =
SCF coordination, while the conservative EB platform financing may not be able to. A.7. Proof of Proposition 5
From Proposition 4, if the coordination condition on order quantity, F
wqa∗ −m 1+ra p 1−λa )
= (1 +
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a) ∗ s∗ ∗ rf ) (1−λ (1+ra ) , is satisfied, we can deduce that qa = qa and qa is fixed. Then, if m is a constant, this
1−λa 1−λa 1+ra . Thus, 1+ra (1−λa )(1+rf ) with = 1+ra
coordination condition on order quantity is an equation with only one variable, is fixed. Furthermore, the first derivative of the right-hand side of F (ka (qa∗ )) dλa
respect to ra is −(1 + rf ) dra
(1+ra )+(1−λa ) (1+ra )2
If m is a constant and F (ka (qa∗ )) = is fixed. Then,
dπar (qa∗ ) dra
=
ED
A.8. Proof of Proposition 6 From Appendix A.7, we yield
dra dλa
> 0 and
dπap (qa∗ ) dra
=
1−λa ∗ ∗ 1+ra p[s(ka (qa )) − s(qa )]
1+ra = − 1−λ < 0. Then, if qa∗ = qas∗ and a
= −p[s(qa∗ ) − s(ka (qa∗ ))] < 0 and
PT
a = − 1−λ 1+ra < 0.
is satisfied, we have obtained that qa∗ = qas∗ and
1−λa ∗ ∗ 1+ra p[s(qa ) − s(ka (qa ))]
0.
dπar (qa∗ ) dλa
(1−λa )(1+rf ) 1+ra
dλa dra
M
1−λa 1+ra
= 0, so that
dπap (qa∗ ) dλa
1−λa 1+ra
<
is fixed, it follows
= p[s(qa∗ ) − s(ka (qa∗ ))] > 0.
A.9. Proof of Proposition 7
CE
Proof of Proposition 7.1: From Eq.(A.4), we obtain that qb∗ is fixed and F (kb (qb∗ )) = Appendix A.2. Then, we derive the first derivative of F (kb (qb∗ )) =
AC
to m,
1+rf 1+rb
1+rf 1+rb
in
on both sides with respect
1 + rf drb f (kb (qb∗ )) drb [−(1 + rb ) + (wqb∗ − m) ]= , (1 − λf )p dm (1 + rb )2 dm 1 + rf drb (wqb∗ − m)f (kb (qb∗ )) f (kb (qb∗ ))(1 + rb ) [ − ] = , (1 − λf )p (1 + rb )2 dm (1 − λf )p [kb (qb∗ )f (kb (qb∗ )) − F (kb (qb∗ ))]
f (kb (qb∗ ))(1 + rb )2 drb = , dm (1 − λf )p
(1 + rb )2 h(kb (qb∗ )) drb =− . dm (1 − λf )p(1 − H(kb (qb∗ )))
According to Appendix A.4, H(kb (qb∗ )) is less than 1, so that 31
(A.13) drb dm
< 0.
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Proof of Proposition 7.2: From Proposition 4, if F ka (qa∗ )) =
(1+rf )(1−λa ) , 1+ra
we get qa∗ = qas∗ =
qbs∗ , which are all fixed. Then, πar (qa∗ ) = πbr (qb∗ ) can be convert to πar (qa∗ ) − πbr (qb∗ ) = 0, and the first derivative of the equation πar (qa∗ ) − πbr (qb∗ ) = 0 with respect to m is
Taking derivative of F ka (qa∗ )) =
1 + rf dλa dka (qa∗ ) + (1 − λa )F (ka (qa∗ )) =− . dm dm p
(1+rf )(1−λa ) 1+ra
and ka (qa∗ ) =
(wqa∗ −m)(1+ra ) (1−λa )p
to m, respectively, it follows that
(A.14)
on both sides with respect
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[s(qa∗ ) − s(ka (qa∗ ))]
1 + rf dka (qa∗ ) dλa dra = [(1 + ra ) + (1 − λa ) ]. 2 ∗ dm (1 + ra ) f (ka (qa )) dm dm
(A.15)
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dka (qa∗ ) 1 dλa dra = [(wqa∗ − m)(1 + ra ) + (wqa∗ − m)(1 − λa ) − (1 − λa )(1 + ra )]. 2 dm (1 − λa ) p dm dm
(A.16)
We can observe equations, Eq. (A.14), Eq.(A.15) and Eq.(A.16), have three unknown variables dra dλa dm , dm
and
dka∗ dm ,
and, by substituting F ka (qa∗ )) =
can deduce that
and Eq.(A.15) into Eq.(A.14), we
(1 + rf )2 (1 − λa )2 dλa (1 + rf )2 (1 − λa )3 dra 1 + rf ] + =− . ∗ 2 f (ka (qa ))(1 + ra ) dm f (ka (qa∗ ))(1 + ra )3 dm p
M
[s(qa∗ ) − s(ka (qa∗ )) +
(1+rf )(1−λa ) 1+ra
ED
Therefore,
PT
(1 + rf )2 (1 − λa )2 dλa dra f (ka (qa∗ ))(1 + ra )3 f (ka (qa∗ ))(1 + ra )3 ∗ ∗ =− [s(q ) − s(k (q )) + ] − . a a a dm (1 + rf )2 (1 − λa )3 f (ka (qa∗ ))(1 + ra )2 dm p(1 + rf )(1 − λa )3 By substituting Eq.(A.16) and F ka (qa∗ )) =
(1+rf )(1−λa ) 1+ra
into Eq.(A.14), we can obtain
(wqa∗ − m)(1 + rf ) dλa (1 + rf )(wqa∗ − m)(1 − λa ) dra λa (1 + rf ) ] + =− . p dm p(1 + ra ) dm p
CE
[s(qa∗ ) − s(ka (qa∗ )) +
(A.17)
(A.18)
AC
Substituting Eq.(A.17) into Eq.(A.18), we yield (1 + rf )[H(ka (qa∗ )) − λa ] dλa = . dm p[1 − H(ka (qa∗ ))][s(qa∗ ) − s(ka (qa∗ ))]
(A.19)
Because of H(ka (qa∗ )) < 1 and qa∗ > ka (qa∗ ) in Proposition 3 and Lemma 1, it can be deduced that if λa < H(ka (qa∗ )) < 1,
dλa dm
> 0; if H(ka (qa∗ )) < λa ,
dλa dm
< 0.
By substituting Eq.(A.19) into (A.17) and simplifying formulas, we have (H(ka (qa∗ )) − λa )(1 + rf ) dra (1 + ra ) ∗ =− [h(k (q ))(1 + r ) + ]. a a a dm (1 − λa )p[1 − H(ka (qa∗ ))]) s(qa∗ ) − s(ka (qa∗ )) 32
(A.20)
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If λa < H(ka (qa∗ )) < 1, we have
dra dm
< 0. If H(ka (qa∗ )) < λa ,
s(ka (qa∗ ))] − [λa − H(ka (qa∗ ))](1 + rf ) > 0, and
dra dm
[λa − H(ka (qa∗ ))](1 + rf ) < 0.
dra dm
< 0 for h(ka (qa∗ ))(1 + ra )[s(qa∗ ) −
> 0 for h(ka (qa∗ ))(1 + ra )[s(qa∗ ) − s(ka (qa∗ ))] −
∗
(1+ra ) a (qa ) = − (1−λa )p[1−H(k <0 We can further deduce that dkdm ∗ a (q ))] a
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A.10. Proof of Proposition 8 When the online retailer refuses any outside financing and uses her own initial working capital to purchase, her opportunity cost of the procurement is wqnl (1 + rf ). Therefore, after deducting the opportunity cost, we can deduce that her expected profit is
with a constraint of wqnl ≤ m.
r with respect to q , We take the derivative of πnl nl r dπnl dqnl
r dπnl dqnl
= (1 − λf )pF (qnl ) − w(1 + rf ). When
r −1 w(1+rf ) −1 w(1+r ) dπnl ( (1−λf )p ). Therefore, if qnl < F ( (1−λf f)p ), we have dqnl > 0 and r −1 w(1+rf ) dπnl r decreases in q . According ≤ 0 and πnl qnl ≥ F ( (1−λf )p ), we have dqnl nl
= 0, it follows qnl = F
r increases in q . If πnl nl
(A.21)
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r = (1 − λf )ps(qnl ) − wqnl (1 + rf ), πnl
m < wF
−1 w(1+rf ) ( (1−λf )p ),
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r and the constraint of wq ≤ m , we can further infer that if to this monotonicity analysis of πnl nl ∗ = the optimal order quantity is qnl
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∗ = optimal order quantity is qnl
−1 w(1+r ) F ( (1−λf f)p ).
m w,
and that if m ≥ wF
−1 w(1+rf ) ( (1−λf )p ),
the
Therefore, with m increasing from 0 to infinity, the
r (q ∗ = F online retailer’s maximum expected profit is πnl nl
−1 w(1+rf ) ( (1−λf )p )).
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If the online retailer refuses any outside financing, her expected profit expression Eq.(A.21) r (q ∗ = F and maximum expected profit πnl nl
−1 w(1+rf ) ( (1−λf )p ))
are both identical to those under BCF
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in proposition 1 and proposition 4. Hence, if the active EB platform financing can achieve SCF −1 w(1+rf ) ( (1−λf )p ))
= πbr (qb∗ ) ≤ πar (qa∗ ).
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r (q ∗ = F coordination, we could conclude that πnl nl
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Financing Online Retailers: Bank vs. Electronic Business Platform, Equilibrium, and Coordinating Strategy Chengfu Wang, Xiaojun Fan, Zhe Yin PII: DOI: Reference:
S0377-2217(19)30011-6 https://doi.org/10.1016/j.ejor.2019.01.009 EOR 15581
To appear in:
European Journal of Operational Research
Received date: Revised date: Accepted date:
29 September 2017 30 June 2018 4 January 2019
Please cite this article as: Chengfu Wang, Xiaojun Fan, Zhe Yin, Financing Online Retailers: Bank vs. Electronic Business Platform, Equilibrium, and Coordinating Strategy, European Journal of Operational Research (2019), doi: https://doi.org/10.1016/j.ejor.2019.01.009
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights • Online retailers obtain financing from a bank or an electronic business platform. • Electronic business platform is divided into an active and a conservative form.
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• We perform a Stackelberg game analysis of electronic business platform financing. • Active electronic business platform financing can achieve coordination.
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• The coordinated active electronic business platform financing is preferable.
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Financing Online Retailers: Bank vs. Electronic Business Platform, Equilibrium, and Coordinating Strategy Chengfu Wanga,b , Xiaojun Fana ,1
a The School of Management, Shanghai University, Shanghai, China Dongfang College, Zhejiang University of Finance & Economics, Hangzhou, China
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b
Zhe Yina
Abstract
We consider a new financing source for online retailers, namely an electronic business (EB) platform,
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which is divided into an active and a conservative form according to the level of integration of lending and leasing businesses. The newsvendor-like online retailer is capital-constrained and can choose between EB platform financing and bank credit financing (BCF). We model their interaction as a Stackelberg game, where the online retailer functions as the follower who decides the order quantity and the lender (bank or EB platform) functions as the leader who declares the loan interest rate or
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the EB platform’s usage fee rate. From the equilibrium and coordinating analyses, we conclude that the active EB platform financing can achieve supply chain financing (SCF) coordination, yielding
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a larger order quantity and larger participants’ profits than those yielded through BCF, while the conservative may not. When the coordination condition on order quantity is satisfied, if the initial
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working capital is a constant, there exists a linear decreasing trend between the EB platform’s loan interest rate and the usage fee rate. When the weak coordination conditions are satisfied, the active
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EB platform can offer many coordinated financing contracts, which are applicable to different initial working capital conditions. For achieving SCF coordination, the EB platform should be active to adjust his loan interest rate and usage fee rate together. Moreover, even if the online retailer has
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sufficient working capital, she always prefers using the coordinated active EB platform financing to refusing any outside financing. Keywords: Finance; Online retailer; Trade credit; Stackelberg game; Supply chain coordination.
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Corresponding author: E-mail: [email protected]
Preprint submitted to Elsevier
January 11, 2019
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1. Introduction Limited working capital is a common constraint for online retailers (usually small and mediumsized enterprises or SMEs) in procurement decisions. Conversely, they tend to find it difficult to obtain financing from banks, owing to their lack of collateral, lack of credit history, and the tenuous
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nature of their business establishment (Vandenberg, 2003). One of the most effective methods of accessing funds is supply chain financing (SCF), in which SMEs can use the core enterprise’s credit and real transaction background to improve their own accountability. In general, SCF can be divided into two widely used financing modes: bank credit financing (BCF) and trade credit financing (TCF). Besides BCF, supplier financing of retailer inventory, as a form of TCF, has
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become popular in many countries, such as the United States, China, and some less-developed countries (Petersen and Rajan, 1997; Fisman and Love, 2003; Kouvelis and Zhao, 2012). In recent years, the rapid development of electronic business (EB) has provided online retailers with a new financing source, an EB platform.
Under TCF, a corporation extends its credit to an upstream/downstream supply chain partner
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by using short-term loans (Yan et al., 2016). The EB platform, considered as an important supply chain partner of the online retailer, plays a significant role in the online retailer’s achievement of its
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sales target, and can also extend trade credit to the online retailer by running a lending business. Therefore, EB platform financing of online retailer inventory can be defined as a new form of TCF. In China, EB platforms, such as Alibaba and Jingdong Mall, have introduced a lending business
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based on their original EB platform’s leasing business. By using the big data collected from dayto-day trading on these EB platforms, such financing form obtains many competitive advantages,
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including the ability to borrow without collateral or guarantee, a lower default risk, and the quicker granting of loans.
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The EB platform, such as Alibaba or Jingdong Mall, carries out its lending operation on the
premise of obtaining the original leasing profit, and consider its leasing business to be independent of its lending business. Therefore, the only decision variable is the EB platform’s loan interest rate. The EB platform’s usage fee rate for the leasing business remains constant, regardless of whether or not it lends to an online retailer. This is a form of conservative EB platform financing. We introduce a new financing form of active EB platform financing, in which the EB platform integrates the leasing business and lending business perfectly. Here, the EB platform simultaneously sets suitable decision variables, the EB platform’s usage fee rate and the EB platform’s loan interest rate, in 3
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order to optimize its profit or to achieve SCF coordination. Is the active option better than the conservative option? Or in other words, should the conservative EB platform change its lending strategy? SCF not only diversifies funding sources or financing forms, but also improves the financial
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efficiency of the entire supply chain system by aligning the finance and operation functions. The prevalence of EB platform financing practices brings to the forefront several research questions about financial efficiency: What are the optimal decision variables (the EB platform’s loan interest rate, the EB platform’s usage fee rate, and the online retailer’s order quantity) under EB platform financing? Why and how should EB platforms provide financing for online retailers instead of letting them use bank financing? How does the online retailer choose a suitable financing mode or
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design an appropriate loan portfolio?
To answer these questions, we introduce a stylized model of a supply chain, with a newsvendorlike online retailer selling goods on an EB platform. The market demand follows a certain distribution when the retail price is specified exogenously. The online retailer is capital-constrained and
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can choose EB platform financing or BCF. For the sake of describing the relationship among all participants and the sequence of events properly, we consider the financing process as a Stackelberg game between the online retailer (as the follower) and her lender (as the leader). Initially, the EB
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platform is either conservative or active, and designs his financing contract based on two variables: the EB platform’s loan interest rate and the EB platform’s usage fee rate. The risk-neutral bank
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runs the BCF business in a perfectly competitive capital market, with risk-free interest income being its expected revenue. Then, the online retailer selects a financing form (conservative EB
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platform financing, active EB platform financing, BCF, or refusing any outside financing), and decides the order quantity in response to the leader’s decision.
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Our work contributes to the literature in four ways. First, we establish the functions of the participants’ profits by using the final actual cash flow and the opportunity costs of the initial investments, which differs from the previous studies (Lee and Rhee , 2011; Kouvelis and Zhao, 2012). The online retailer’s initial working capital and the EB platform’s loan-granting quota are the initial investments of the online retailer and the EB platform, respectively. When they deposit these initial investments in a bank, rather than investing in an EB platform financing business, they can obtain the interest incomes at a risk-free interest rate. We refer to those interest incomes as the opportunity costs of these initial investments, which should be deducted when we calculate
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the corresponding participant’s profit. Second, we propose a new TCF contract offered by an EB platform, and formulate a Stackelberg game including an EB platform and an online retailer. From the Stackelberg equilibrium analysis, we find that the EB platform’s loan interest rate and the EB platform’s usage fee rate both have impact on the online retailer’s order quantity decision, and further affect the participants’ profits. Third, we analyze the efficiency of EB platform financing
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from the perspective of an SCF coordination contract. The active EB platform financing can achieve SCF coordination, yielding a larger order quantity and larger participants’ profits than the case of BCF. However, the conservative EB platform financing may not be able to achieve SCF coordination. Fourth, we have analyzed the relations among parameters, which are useful for the active EB platform to run a coordinated SCF business: (i) When the coordination condition on
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order quantity is satisfied, if the online retailer’s initial working capital remains constant, there exists a linear decreasing trend between the EB platform’s loan interest rate and the EB platform’s usage fee rate. (ii) When the weak coordination conditions are satisfied, the active EB platform can offer many coordinated financing contracts, which are applicable to different initial working capital conditions, while the conservative EB platform could design very few coordinated financing
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contracts. (iii) The online retailer always prefers using the coordinated active EB platform financing to refusing any outside financing, even if she has sufficient initial working capital.
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The rest of the paper is organized as follows. Section 2 reviews the relevant literature. Section 3 describes our basic model of active EB platform financing and bank credit financing, including
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the sequence of events, notations, assumptions, and profit functions. In Section 4, we present the equilibrium and coordination analysis under BCF. In Section 5, we analyze the SCF equilibrium and
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coordination condition under active EB platform financing. When the active EB platform financing achieves SCF coordination, Section 6 demonstrates the relations among parameters, which offers insights on how to run a coordinated active EB platform financing business. The final section
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concludes the paper and presents future research directions. The appendix includes proofs of some propositions.
2. Literature Review The interface of operations and financial decisions has attracted researchers’ interests for a long time. Modigliani and Miller (1958) are the first scholars to express the irrelevance between operations and financial decisions in a perfect capital market. However, when market imperfections 5
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are introduced, the irrelevance no longer holds. According to this theoretical framework, financing can improve or create firm value by reducing taxes and contract costs, by affecting real investments in operations, or by financial hedging of operation risks (Smith and Stulz, 1985; Gaur and Seshadri, 2005; Ni et al., 2017; Wang et al., 2018). Several researchers have examined the risk management in SCF processes (Ding et al., 2007; Zhao and Huchzermeier, 2015, Yang and Birge, 2017). Other
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researchers have focused on a capital-constrained firm’s operations and financing decisions for profit maximization, such as production and financing decisions (Lee and Tang, 1997; Buzacott and Zhang, 2004; Xu and Birge, 2004) and inventory and financing decisions. We pay attention to a special form of TCF, namely EB platform financing of online retailer inventory. This could be described as a type of financing the newsvendor problem and belongs to the research area of
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inventory and financing decisions, but differs from the TCF studies with EOQ or EPQ models under a determined market demand environment (Teng et al., 2012; Ouyang et al., 2013; Wu et al., 2014; Shah and C´ ardenas-Barr´ on, 2015; Chen and Teng, 2015; Wu et al., 2016; Tiwari et al., 2016; Jaggi et al.,2017). We analyze two research categories on financing the newsvendor problem: participants’ equilibrium in a decentralized financing system and the coordination contract in the
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financing system.
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2.1. Participants’ Equilibrium in a Decentralized Financing System The classic newsvendor model helps to find the optimal order quantity, which can minimize
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the operating cost or maximize the profit of the purchaser in a single period. Several researchers have extended this model to product purchasing with multiple categories and budget constraints
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(Lau and Lau, 1996; Mostard and Teunter, 2006; Chen and Chen, 2010). These studies use the nonlinear programming method to maximize the capital-constrained purchaser’s profit, but ignore other participants’ profits as well as the possibility of outside financing. Raghavan and Mishra
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(2011) study a two-level supply chain financing problem with a single retailer and a manufacturer, who are both capital-constrained and are linked in the activities of ordering, producing, and buyback. They could obtain the lender’s optimal loan amount, but ignore the borrower’s decision on the order quantity and the lender’s decision on the interest rate. Considering the external capital injection, each participant makes decision successively in order to maximize his/her own profit, and these decisions are interrelated in a decentralized financing system. The Stackelberg game model has been used to analyze SCF problems. Kouvelis and Zhao (2011) are the first to incorporate both the bankruptcy cost and the bankruptcy risk into 6
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BCF operation, and find the equilibrium solutions of the order quantity and the wholesale price. Then, Kouvelis and Zhao (2012) consider the bankruptcy problem under SCF. Both the retailer and the supplier are capital-constrained, and the retailer can choose between BCF and TCF. Their equilibrium analysis results show that under certain conditions (relate to parameters of the interest rate, the wholesale price, the working capital, and the collateral), TCF is more attractive than
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BCF. However, these articles only analyze the equilibrium problem within the financing process, but have not taken the SCF coordination problem into account. Additionally, Kouvelis and Zhao (2012) handle the problem of optimal parameter setting by maximizing participants’ terminal cash flows, rather than maximizing participants’ profits. In our research, we will use the terminal cash flows and the opportunity costs of the initial capital investments to establish participants’ profit
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functions, which is in line with Raghavan and Mishra (2012).
Following the above research methods, Jing et al. (2012) analyze the effect of the production cost, the capital level, and the demand variability on the efficiency of BCF and TCF. Chen (2015) compares BCF and TCF under a wholesale price contract and a revenue-sharing contract to find how the product cost, the internal capital, and the demand variability affect the participants’
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equilibrium. Gao et al. (2018) analyze the optimal Stackelberg strategies for financing a supply chain through an online peer-to-peer lending platform. Similarly, we discuss the participants’
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equilibrium and compare different financing forms. However, in our case, the EB platform financing of an online retailer is a new form of TCF, where the lender’s decision variables are the EB platform’s
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loan interest rate and the EB platform’s usage fee rate, as opposed to the wholesale price under supplier financing. Furthermore, there is no collateral or guarantee in our model.
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2.2. Coordination Contract in the Financing System In a supply chain context, owing to the independent decisions of the participants, the total profit
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in a decentralized supply chain is less than that in a centralized supply chain, where the same firm produces and sells directly to its end customers. This phenomenon is commonly called double marginalization (Spengler, 1950). For the sake of mitigating double marginalization and improving the integrated efficiency of SCF, we derive a coordination scheme in which the Stackelberg leader (generally the lender or supplier) induces the follower (generally the borrower or the retailer) to replicate the decision outcome of the centralized supply chain. Dada and Hu (2008) are the first to consider the SCF coordination problem under the well-
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known two-part tariff contract. Lee and Rhee (2010) investigate four coordination mechanisms (all-unit quantity discount, buybacks, two-part tariff, and revenue-sharing), and find that, under positive inventory financing costs, these mechanisms fail to achieve coordination in direct financing from a bank, whereas under TCF they can fully coordinate the supply chain. Then, Lee and Rhee (2011) show that, when the supplier grants a markdown allowance to the retailer in SCF system,
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he can fully coordinate the retailer’s decision for the largest joint profit under TCF. However, the coordination cannot be realized under BCF. Zhang et al. (2014) propose a modified quantity discount contract based on both order quantity and advance payment, and find that this contract can entice the retailer to make a decision to achieve TCF coordination. Yan et al. (2016) propose a new SCF form, which mixes TCF and BCF, and find that supply chain coordination can be
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achieved when we set the suitable guarantee coefficient and wholesale price.
Unlike these researches, we consider the decision variables of the EB platform’s loan interest rate and the EB platform’s usage fee rate in our coordination analysis, and compare the optimal order quantities and participants’ profits under BCF and under EB platform financing. Then, we further characterize the EB platform financing coordination conditions from various perspectives,
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including the EB platform’s loan interest rate, the EB platform’s usage fee rate, and the online retailer’s initial working capital. Our results illustrate that, in aspects of the optimal order quantity
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and participants’ profits, EB platform financing is superior to BCF, which is consistent with the fact that “70-80% of trade between firms is still conducted in open account (trade credit)” (Zhou
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and Groenevelt, 2008).
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3. Model Formulation
3.1. Assumptions, Notations and Sequence of Events
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Following the standard approach in the classic newsvendor model of prior SCF studies, we
consider the online retailer as a newsvendor who has an opportunity to purchase products from a supplier before the sales season in order to satisfy the market’s uncertain demand. However, in contrast to previous studies, the online retailer (referred to as “she”) is capital-constrained and can borrow from a lender (referred to as “he”), the EB platform or the bank. The leftover value of unsold products and the goodwill loss for unmet demand do not change the nature of the problem, and thus we ignore them without loss of generality. We also assume that, as long as the capital-constrained online retailer borrows from the EB platform or the bank, 8
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she can be granted a sufficient loan-granting quota (i.e., the loan-granting quota is equal to the loan amount that the online retailer applies for). Notations are summarized in Table 1. Table 1: List of notations
market demand quantity
f (x)
the probability density function (PDF) of x
F (x)
the cumulative distribution function (CDF) of x
F (x)
the complementary CDF of x
w
supplier’s wholesale price
p
retail price of the product
k
demand threshold with no bankruptcy
m
online retailer’s initial working capital, m ≥ 0
r
loan interest rate for the time of order up to
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x
the end of the sales season, r ∈ [0, 1]
EB platform’s usage fee rate, λ ∈ ([0, 1]
q
online retailer’s order quantity, q ≥ 0
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λ
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We assume that the retail price p is a constant and the market’s demand distribution in the sales season can be confirmed. The uncertain demand is denoted by a nonnegative stochastic variable x.
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The probability density function (PDF) of x is f (·), the cumulative distribution function (CDF) of x is F (·), and the complementary CDF of x is F (·) = 1 − F (·). We assume that F (·) is differentiable and increasing and F (x = 0) = 0. Consequently, F (x) is continuous and f (x) > 0. We define f (x) F (x)
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h(x) =
as the failure rate and H(x) = xh(x) as the generalized failure rate. Following the
previous studies (Kouvelis and Zhao, 2011; Jing et al., 2012), we assume that h(x) and H(x) are
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both monotonically increasing in x. In other words, the demand distribution has a characteristic of increasing failure rate (IFR). Since the conservative EB platform’s decision variable, the EB platform’s loan interest rate, is
only one of the decision variables of the active EB platform, we regard the conservative EB platform financing as a special case of the more general active EB platform financing. After analyzing the equilibrium and coordination condition under active EB platform financing, we can easily obtain the equilibrium and coordination condition under conservative EB platform financing only by replacing the variable of the EB platform’s usage fee rate with a constant risk-free EB platform’s usage fee 9
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rate. Therefore, we set the following tie-breaking rule: unless stated otherwise, our study is mainly focused on the active EB platform financing form. The online retailer can choose between active EB platform financing and BCF. The active EB platform’s decision variables are λa and ra , and the bank’s decision variable under BCF is rb . Let
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λf and rf denote the EB platform’s risk-free usage fee rate and the risk-free loan interest rate, respectively. Specifically, the EB platform’s usage fee rate is λf under BCF. We assume that the capital market is perfectly competitive under BCF, which means that the bank’s revenue from issuing a loan to the online retailer is equal to the risk-free interest income. The order quantity in the decentralized SCF system is denoted by qi (i = a, b), which is the online retailer’s decision variable when she uses active EB platform financing (i = a) or BCF (i = b). Furthermore, the
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superscript ∗ denotes the corresponding cases when the participants choose the optimal parameters in the Stackelberg equilibrium.
The length of the credit period is equal to that of the sales period. Then, the sequence of events and the decision protocol are both shown in Figure 1. Before the sales season, the active EB
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platform declares his loan interest rate (ra ) and usage fee rate (λa ) under an EB platform financing contract, and the bank determines the loan interest rate (rb ) under a BCF contract. Then, the online retailer calculates her preliminary optimal order quantity under the market’s demand distribution
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and estimates whether her own initial working capital is sufficient for procurement. When it is not sufficient, the online retailer could choose an outside financing source, such as BCF or active
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EB platform financing, or refuse any outside financing. Thereafter, if the online retailer chooses an outside financing source, she decides the updated optimal order quantity (qa or qb ) and the
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loan amount (wqa − m or wqb − m) in response to the chosen lender’s financing contract. Then, she applies for the loan amount from the chosen lender, and after receiving the loan amount the online retailer purchases products from a supplier and pays in full. All of the above procedures
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occur before the sales season. During the sales season t ∈ (0, T ), the online retailer sells on the EB platform. With every unit being sold, the online retailer should pay a certain percentage (λa or λf ) of the sales revenue as the usage fee to the EB platform. Lastly, at the end of the sales season (t = T ), demand is realized. The online retailer should repay the loan debt. Then, if her sales revenue after subtracting the EB platform’s usage fee is sufficiently large, the loan debt is repayed successfully. Otherwise, the loan debt cannot be covered and the online retailer goes bankrupt. In this case, she must transfer all remaining sales revenue to the chosen lender.
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Online retailer chooses Online retailer decides the an outside financing order quantity, q a or q b , source or refuses any obtains the loan amount, outside financing. wqa - m or wqb - m , and purchases products.
With every unit being sold, online retailer pays a certain percentage, l a or l f , of the sales revenue as the usage fee to EB platform.
t =0
Before the sales season
During the sales season
Figure 1: Sequence of events
Online retailer can repay the loan debt successfully, or is forced into bankruptcy.
t =T
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Active EB platform and bank offer a TCF contract ( l a , ra ) and a BCF contract ( rb ), respectively.
As previously discussed, we formulate a two-stage Stackelberg game model to analyze the interaction between the SCF participants. Both the bank and the active EB platform are the Stackelberg leaders. The bank determines his loan interest rate, and the active EB platform sets his loan inter-
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est rate and usage fee rate simultaneously. Afterwards, the online retailer, as the follower, chooses a financing source (or refuses any outside financing) and decides the optimal order quantity. All participants are rational and select a best-reply policy to pursue the Stackelberg equilibrium, and the leaders can coordinate the total supply chain benefits by setting suitable decision variables.
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3.2. Profit Functions Under Bank Credit Financing and EB Platform Financing For notational convention, let a+ = max{a, 0}. πij (i = a, b and j = p, r, b, s) denotes the
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expected profit of the EB platform (j = p), the online retailer (j = r), the bank (j = b), and the centralized SCF system (j = s) at the end of the selling season under active EB platform financing
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(i = a) and BCF (i = b). Given the online retailer’s opportunity cost, the online retailer’s expected profit can be derived as
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πir = {(1 − EB platform’s usage fee rate)E[sales revenue] − E[online retailer’s actual repayment]}+ −opportunity cost of initial working capital,
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where i = a or b, and the online retailer’s actual repayment is the smaller one of the agreed repayment and the sales revenue after subtracting the EB platform’s usage fee. With every unit being sold, the online retailer should pay a certain percentage of the sales revenue as the usage fee to the EB platform simultaneously. However, she repays the loan debt at the end of the sales season. This process also obeys the PMSA rules (Lee and Rhee, 2011), because the EB platform’s usage fee is a kind of enterprise operation cost rather than a debt. Under BCF, other participants’ expected profits can be derived as πbp = (EB platform’s usage fee rate)E[sales revenue] 11
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and πbb = E[online retailer’s actual repayment] − opportunity cost of loan-granting quota. Under active EB platform financing, the EB platform’s expected profit is given by
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πap = E[online retailer’s actual repayment] + (EB platform’s usage fee rate)E[sales revenue] −opportunity cost of loan-granting quota.
The expected profit of the centralized BCF system is πbs = πbr + πbp + πbb , and the expected profit of
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the centralized active EB platform financing system is πas = πar + πap .
4. Equilibrium and Coordination Analysis Under Bank Credit Financing Before analyzing the BCF and the active EB platform financing models, we first consider the
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relation among parameters in a general model. Given the order quantity q, the online retailer’s Rq R∞ Rq expected sales quantity is s(q) = E{min(x, q)} = 0 xf (x)dx + q qf (x)dx = 0 F (x)dx.
Lemma 1. When a capital-constrained online retailer borrows from a lender, the demand threshold (wq−m)(1+r) , (1−λ)p
and the online retailer’s expected actual repayment to the
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with no bankruptcy is k =
lender is (1 − λ)ps(k) = (1 − λ)pE{min(x, k)}. (See Appendix A.1 for the proof ).
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When the online retailer borrows from a lender, her agreed repayment is (wq − m)(1 + r). Then, only if the market demand is sufficient to reach or exceed the demand threshold k =
(wq−m)(1+r) , (1−λ)p
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can the online retailer’s agreed repayment be covered at the end of the sales season. Otherwise, the online retailer goes bankrupt and transfers all remaining sales revenue to the EB platform. The purchase cost per unit is w(1 + r), which includes the loan interest, and (1 − λ)p is the sales revenue
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per unit after subtracting the EB platform’s usage fee. Then, we obtain w(1 + r) ≤ (1 − λ)p, and thus k ≤ q. The demand threshold with no bankruptcy is not bigger than the order quantity. Furthermore, it is obvious that s(k) < k, and thus, (1 − λ)ps(k) < (1 − λ)pk = (wq − m)(1 + r). It means that the online retailer’s expected actual repayment is less than the online retailer’s agreed repayment. In other words, the lender shares the market volatility risk with the online retailer by providing credit financing. Here, k (or s(k)) has different representations under BCF and EB platform financing, which is meaningful for establishing the participants’ profits functions and simplifying the analysis procedure. 12
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4.1. SCF Equilibrium Under Bank Credit Financing When the capital-constrained online retailer borrows from a bank in a perfectly competitive capital market, the bank can only get the market risk-free interest income by setting a suitable loan interest rate rb . Specifically, under BCF the EB platform does not provide financing services
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and his usage fee rate is λf . With every unit being sold, the online retailer should pay the usage fee λf p to the EB platform at the same time. Thus, the bank’s revenue equilibrium expression is given as follows:
E{min[(wqb − m)(1 + rb ), (1 − λf )pmin(x, qb )]} = (wqb − m)(1 + rf ).
(1)
This means that the bank’s expected revenues, either from issuing a loan amount (wqb − m) to the
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online retailer with rb or from investing this capital amount (wqb − m) with rf , are the same. We can also derive that rb > rf .
Proposition 1. In the decentralized BCF system, when the online retailer borrows from a bank in a perfectly competitive capital market, the market demand threshold with no bankruptcy is kb = and the bank’s revenue equilibrium expression is (1 − λf )ps(kb ) = (wqb − m)(1 + rf ).
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(wqb −m)(1+rb ) , (1−λf )p
Moreover, the expected profits of the online retailer, the EB platform, and the bank are πbr = qb∗ satisfies F (qb∗ ) =
w(1+rf ) (1−λf )p .
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(1 − λf )ps(qb ) − wqb (1 + rf ), πbp = λf ps(qb ), and πbb = 0, respectively. The optimal order quantity (See Appendix A.2 for the proof ).
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Proposition 1 shows that, when the online retailer borrows from a bank in a perfectly competitive capital market, her optimal order quantity qb∗ is fixed and independent from the bank’s loan interest
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rate rb . Then, we can find the expected participants’ profits, πbr (qb∗ ) and πbp (qb∗ ), which are both fixed. When the online retailer has sufficient initial working capital and does not borrow from outside, her optimal order quantity also satisfies F (q ∗ ) =
w(1+rf ) (1−λf )p .
Therefore, the capital-constrained online
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retailer’s optimal order quantity under BCF is identical to that of the capital-unconstrained online retailer, which is similar to the finding of Kouvelis and Zhao (2012). By substituting qb∗ into the bank’s revenue equilibrium expression Eq.(1), we could further obtain the bank’s loan interest rate rb . 4.2. SCF Coordination Under Bank Credit Financing Proposition 2. When the online retailer borrows from a bank in a perfectly competitive capital market, the expected profit of the centralized BCF system is πbs = πbr +πbp +πbb = ps(qbs )−wqbs (1+rf ), 13
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the optimal order quantity satisfies F (qbs∗ ) =
w(1+rf ) , p
and the SCF coordination cannot be achieved.
(See Appendix A.3 for the proof ). qbs denotes the order quantity in the centralized BCF system. Proposition 2 demonstrates that qbs∗ , πbs (qbs∗ ) and kb (qbs∗ ) in the centralized BCF system are all fixed, since rf , λf , w, m and p are
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all constant. By comparing F (qb∗ ) with F (qbs∗ ), we find that, only if λf = 0 is satisfied, can qb∗ be equal to qbs∗ . However, the case of λf = 0 is almost impossible in the real economy. In other words, the optimal order quantity in the decentralized BCF system is always smaller than that in the centralized BCF system. We could further find that the sum of participants’ profits under the decentralized BCF system is always smaller than the joint profit of the centralized BCF system
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(i.e., πbr (qb∗ ) + πbp (qb∗ ) + πbb (qb∗ ) < πbs (qbs∗ )). Hence, under BCF, SCF coordination cannot be achieved and the supply chain efficiency cannot be improved to the optimal level of the centralized BCF system. If the optimal order quantity under active EB platform financing can be consistent with that in the centralized BCF system, i.e., qa∗ could be equal to qbs∗ , EB platform financing performs
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better than BCF does in terms of supply chain efficiency improvement.
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5. Equilibrium and Coordination Analysis Under EB Platform Financing 5.1. SCF Equilibrium Under EB Platform Financing
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In the Stackelberg game, the active EB platform is the leader, with the decision variables of the EB platform’s loan interest rate ra and the EB platform’s usage fee rate λa . The online retailer decides her optimal order quantity qa following the EB platform’s decision. Specifically,
λa = λf .
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the conservative EB platform financing is a special case of the active EB platform financing with
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Proposition 3. In the decentralized active EB platform financing system, the market demand threshold with no bankruptcy is ka =
(wqa −m)(1+ra ) . (1−λa )p
Let Ω =
w(1+ra ) (1−λa )p
for simplicity.
(3.1) As the Stackelberg game follower, the online retailer’s expected profit is πar = (1−λa )p[s(qa )− s(ka )] − m(1 + rf ). Given the IFR assumption, the optimal order quantity satisfies F (qa∗ ) = ∗ ∂qa∗ ∂ka (qa∗ ) ∂ka (qa∗ ) a F ka (qa∗ )) Ω, ∂λ ≤ ∂q < 0. ∂ra < 0, and ∂λa ≤ ∂ra a (3.2) As the Stackelberg game leader, the EB platform’s expected profit is πap = (1 − λa )ps(ka ) +
λa ps(qa )−(wqa −m)(1+rf ). Then, the EB platform’s optimal decision variables satisfy w(1+ra∗ ) =
(1 − λ∗a )p and m(1 + ra∗ ) = 0. Furthermore, from the above Stackelberg equilibrium results, it could 14
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be further deduced that πar (ra∗ , λ∗a , qa∗ ) = −m(1 + rf ) and πap (ra∗ , λ∗a , qa∗ ) = ps(qa∗ ) − wqa∗ (1 + rf ). (See Appendix A.4 for the proof ). In Proposition 3.1 the online retailer’s expected profit is πar = (1 − λa )p[
R +∞ qa
R qa ka
(x − ka )f (x)dx +
(qa − ka )f (x)dx] − m(1 + rf ), which is similar to the finding of Yan et al. (2016). Since the
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partial derivatives of qa∗ with respect to ra and λa are both less than 0, there exists a one-to-one mapping between qa∗ and (ra , λa ), and qa∗ decreases monotonously as (ra , λa ) increases. This implies that, by setting a bigger usage fee rate or a bigger loan interest rate, the EB platform may not be able to obtain a larger profit, because of this monotonicity property of qa∗ . Additionally, the absolute value of
∂qa∗ ∂λa
is bigger than that of
∂qa∗ ∂ra .
Hence, by adjusting the usage fee rate λa , the EB
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platform can more easily induce the online retailer to enlarge the optimal order quantity qa∗ .
In Proposition 3.2, even though the EB platform obtains his maximum expected profit πap (ra∗ , λ∗a , qa∗ ), the online retailer’s maximum expected profit πar (ra∗ , λ∗a , qa∗ ) is not larger than 0. Then, the online retailer will refuse to borrow from the EB platform. Therefore, the solutions of the EB platform’s optimal decision variables do not exist. The EB platform, as the Stackelberg game leader, must
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consider the participation constraint that the online retailer’s expected profit should be larger than 0. Furthermore, in order to entice the online retailer to select EB platform financing rather than
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BCF, the online retailer’s expected profit under EB platform financing should also be larger than that under BCF, which will be discussed in subsection 5.2. Note that under conservative EB platform financing we can substitute λa = λf into Proposition 3 to obtain the corresponding optimal
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order quantity and optimal loan interest rate.
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5.2. Coordinating Strategy Under EB Platform Financing EB platform financing is a new form of TCF, and numerous previous studies ( Lee and Rhee,
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2011; Kouvelis and Zhao, 2012) have proven that supplier financing, as a traditional form of TCF, could be better than BCF in terms of the optimal order quantity and participants’ profits. In this section, we will analyze whether TCF offered by the EB platform could also be better than BCF, and under what condition this occurs. According to the previous research (Yan et al., 2016), if a TCF contract can achieve SCF coordination, it must ensure that each member’s profit is not lower than the profit level under BCF. On one hand, only when the EB platform’s profit from running a lending business is bigger than his profit under BCF, can he grant the online retailer a loan. On the other hand, if the 15
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online retailer’s profit under EB platform financing is bigger than her profit under BCF, she will prefer EB platform financing to BCF. Hence, if the two above-mentioned necessary conditions are satisfied simultaneously, SCF coordination is possible to be achieved. As we all know, internet companies follow Metcalf’s law, and satisfying a customer’s purchasing demand is an important way to entice the customer to visit the EB platform website. Therefore, even though the members’
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profits are the key to determining whether the online retailer and the EB platform participate in EB platform financing, the order quantity must also be sufficiently large in order to satisfy customers’ purchasing demand. Next, we will analyze the coordination conditions from the perspectives of the order quantity and the participants’ profits.
Corollary 1. Given p, w, m, rf , λf , and the IFR assumption, after comparing the optimal order
we conclude that:
qa∗ > qbs∗ , if F (ka (qa∗ )) <
qa∗ = qbs∗ , if F (ka (qa∗ )) =
(1−λa )(1+rf ) . 1+ra
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quantities in the centralized BCF system and the decentralized active EB platform financing system, (1−λa )(1+rf ) ; 1+ra
qa∗ < qbs∗ , if F (ka (qa∗ )) >
(1−λa )(1+rf ) ; 1+ra
(See Appendix A.5 for the proof ).
Note that both qa∗ and F (ka (qa∗ )) are functions of ra and λa . From Corollary 1, the optimal
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order quantity in the decentralized active EB platform financing system can be bigger than that in the centralized BCF system by setting suitable ra and λa , which is similar to the conclusion of Yan
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et al. (2016). This will help internet companies attract more registered users at a lower shortage rate. Obviously, these suitable ra and λa to realize qa∗ ≥ qbs∗ in Corollary 1 are different from these
ra∗ and λ∗a to maximize πap in Proposition 3. This suggests that the EB platform can set suitable
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ra and λa to obtain a bigger order quantity, but his profit under active EB platform financing may not be larger than that under BCF. Let qas denote the order quantity in the centralized active
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EB platform financing system, we could get the following proposition to characterize the SCF coordination conditions.
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Proposition 4. Given p, w, m, rf , λf , and the IFR assumption, the expected profit of the centralized active EB platform financing system is πas = πar + πap = ps(qas ) − wqas (1 + rf ). Then, the maximum profits and the optimal order quantities in the centralized active EB platform financing system and the centralized BCF system satisfy πas (qas∗ ) = πbs (qbs∗ ) and F (qas∗ ) = F (qbs∗ ) = When F ka (qa∗ )) =
(1−λa )(1+rf ) , 1+ra
w(1+rf ) . p
πar (qa∗ ) ≥ πbr (qb∗ ), and πap (qa∗ ) ≥ πbp (qb∗ ) are concurrently satisfied,
the active EB platform financing can achieve SCF coordination. However, the conservative EB platform financing may not be able to achieve SCF coordination. (See Appendix A.6 for the proof ). From Propositions 1 and 2, the optimal order quantity in the centralized BCF system is big16
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ger than that in the decentralized BCF system, i.e., qbs∗ > qb∗ , and thus the participants’ profits satisfy πbs (qbs∗ ) > πbr (qb∗ ) + πbp (qb∗ ). Specifically, qb∗ , qbs∗ , πbr (qb∗ ), πbp (qb∗ ) and πbs (qbs∗ ) are all fixed. Then Corollary 1 and Proposition 4 show that, when F ka (qa∗ )) =
(1+rf )(1−λa ) 1+ra
(which
is the coordination condition on order quantity) is satisfied by setting suitable ra and λa , the active EB platform can entice the online retailer to replicate the optimal order quantity deci-
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sion of the centralized BCF system or the centralized active EB platform financing system, i.e., qa∗ = qas∗ = qbs∗ > qb∗ , where qa∗ , qas∗ , qbs∗ and qb∗ are all fixed. Moreover, when the coordination condition on order quantity F ka (qa∗ )) =
(1+rf )(1−λa ) 1+ra
is satisfied, the total profit of the decentral-
ized active EB platform financing system is equal to that of the centralized active EB platform financing system or the centralized BCF system, but bigger than that of the decentralized BCF
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system, i.e., πar (qa∗ ) + πap (qa∗ ) = πas (qas∗ ) = πbs (qbs∗ ) > πbr (qb∗ ) + πbp (qb∗ ), where πas (qas∗ ), πbs (qbs∗ ), πbr (qb∗ ) and πbp (qb∗ ) are all fixed, and πar (qa∗ ) and πap (qa∗ ) are both functions of ra and λa . Therefore, when F ka (qa∗ )) =
(1+rf )(1−λa ) 1+ra
is satisfied, πar (qa∗ ) ≥ πbr (qb∗ ) and πap (qa∗ ) ≥ πbp (qb∗ ) (which are the coordina-
tion conditions on participants’ profits) are easy to be satisfied under active EB platform financing by setting suitable λa and ra . It implies that the active EB platform financing can achieve SCF
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coordination.
Under conservative EB platform financing, for the constant value of m and λa = λf , we can find (1+rf )(1−λa ) is satisfied. However, this unique solution 1+ra πap (qa∗ ) ≥ πbp (qb∗ ) simultaneously. This suggests that the
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the unique solution of ra when F ka (qa∗ )) = of ra may not satisfy πar (qa∗ ) ≥ πbr (qb∗ ) and
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conservative EB platform financing may not be able to achieve SCF coordination. Proposition 4 implies that the active EB platform financing performs better than the conser-
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vative case in terms of SCF coordination. When the active EB platform financing achieves SCF coordination, not only the optimal order quantity under active EB platform financing can be equal to that in the centralized SCF system, but also the participants’ profits under active EB platform fi-
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nancing can be larger than (or at least equal to) those under BCF. Therefore, the coordinated active EB platform financing can improve the supply chain efficiency and is attractive for all participants.
6. Numerical Studies and Extensions We now present numerical studies and extensions in order to clarify our findings and to illustrate the impacts of different parametric variables on SCF coordination. We maintain the following assumptions over the numerical studies: (i) demand x follows an exponential distribution with a 17
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mean of 100, which is similar to the parameter set by Buzacott and Zhang (2004), and (ii) the base values of the parameters are p = 1.6, w = 0.4, λf = 0.25, rf = 0.1, and m = 7. 6.1. Coordination Analysis Based on Different EB Platform’s Loan Interest Rates Proposition 5. Given p, w, m, rf , λf , and the IFR assumption, when the coordination condition dπar (qa∗ ) dra
> 0, and
dπap (qa∗ ) dra
(1+rf )(1−λa ) , 1+ra
is satisfied,
1−λa 1+ra
is a fixed value,
< 0. (See Appendix A.7 for the proof ).
a = − 1−λ 1+ra < 0,
λa
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0.4
0.35
0.3
0.25
0.2
0.15
ra(λa=0.25)=0.376
0
0.1
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EB platform’s usage fee rate λa when the
coordination condition on order quantity is satisfied
0.5
dλa dra
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on order quantity, F ka (qa∗ )) =
0.2 0.3 EB platform’s loan interest rate ra
0.4
0.5
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Figure 2: EB platform’s usage fee rate under different ra
40
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Participants’ profits when the coordination condition on order quantity is satisfied
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πpa(q*a)
38
πra(q*a)
πpb(q*b)
πrb(q*b)
36 34 32 30
coordination zone
28 26 24 22 20
ra(λa=0.25)=0.376 0
0.1
0.2 0.3 EB platform’s loan interest rate ra
0.4
0.5
Figure 3: Participants’ profits under different ra
If the coordination condition on order quantity F ka (qa∗ )) =
(1+rf )(1−λa ) 1+ra
is satisfied and m is a
constant, we obtain qa∗ = qas∗ = qbs∗ , which are all fixed. Then, there exists a one-to-one mapping 18
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between ra and λa from the equation F ka (qa∗ )) =
(1+rf )(1−λa ) . 1+ra
Provided m = 7 and ra changes
from 0 to 0.5 with an increment of 0.02, Figure 2 shows a linear decreasing trend between λa and ra , which suggests that
dλa dra
is a fixed negative number. In Figure 3, the respective profits of
the online retailer and the EB platform in the decentralized BCF system, πbr (qb∗ ) and πbp (qb∗ ), are independent of the EB platform’s loan interest ra . However, the respective profits of the online
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retailer and the EB platform in the decentralized active EB platform financing system, πar (qa∗ ) and πap (qa∗ ), are tightly tied to the EB platform’s loan interest ra . Here, πar (qa∗ ) increases in ra , whereas πap (qa∗ ) decreases in ra . Moreover, for any ra ∈ [0.4, 0.46], πar (qa∗ ) ≥ πbr (qb∗ ) and πap (qa∗ ) ≥ πbp (qb∗ ) are satisfied simultaneously, and thus SCF coordination can be achieved. If ra < 0.4, we obtain πar (qa∗ ) < πbr (qb∗ ), which implies that the online retailer would prefer BCF to EB platform financing.
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Similarly, if ra > 0.46, we obtain πap (qa∗ ) < πbp (qb∗ ), which implies that the EB platform will not choose EB platform financing. Specifically, when the active EB platform sets his loan interest rate at ra (λa = λf = 0.25) = 0.376, he cannot achieve SCF coordination. This suggests that the conservative EB platform may not be able to achieve SCF coordination.
From Proposition 4, when the coordination condition on order quantity is satisfied, we get
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πar (qa∗ ) + πap (qa∗ ) > πbr (qb∗ ) + πbp (qb∗ ), in which πbr (qb∗ ) and πbp (qb∗ ) are both fixed. Considering that with ra increasing the monotonicity of πar (qa∗ ) is different from that of πap (qa∗ ) in Proposition 5, we can
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further deduce that πar (qa∗ ) ≥ πbr (qb∗ ) and πap (qa∗ ) ≥ πbp (qb∗ ) can be satisfied simultaneously at some
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reasonable points of ra . This has been confirmed by the numerical study in Figure 3. 6.2. Coordination Analysis Based on Different EB Platform’s Usage Fee Rates
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Proposition 6. Given p, w, m, rf , λf , and the IFR assumption, when the coordination condition on order quantity, F ka (qa∗ )) = < 0, and
is satisfied,
1+ra 1−λa
> 0. (See Appendix A.8 for the proof ).
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dπar (qa∗ ) dλa
dπap (qa∗ ) dλa
(1+rf )(1−λa ) , 1+ra
19
is a fixed value,
dra dλa
1+ra = − 1−λ < 0, a
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coordination condition on order quantity is satisfied
0.9 ra
EB platform¡¯s loan interest rate ra when the
0.8 0.7 0.6 0.5
0.3 0.2 0.1 0
λa=λf=0.25 0
0.05
0.1
0.15 0.2 0.25 0.3 0.35 EB platform’s usage fee rate λa
0.4
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0.4
0.45
πpa(q*a) 40
35
30
πpb(q*b)
πrb(q*b)
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25
20
15
0
0.05
0.1
λa=λf=0.25
0.15 0.2 0.25 0.3 0.35 EB platform’s usage fee rate λa
0.4
0.45
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10
πra(q*a)
coordination zone
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Participants’ profits when the coordination condition on order quantity is satisfied
45
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Figure 4: EB platform’s loan interest under different λa
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Figure 5: Participants’ profits under different λa
Provided the online retailer’s initial working capital m = 7, if F ka (qa∗ )) =
(1+rf )(1−λa ) 1+ra
is
satisfied, we demonstrate the relations among the participants’ profits, EB platform’s loan interest
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rate and the EB platform’s usage fee rate. A linear decreasing trend between λa and ra suggests dλa dra
is a fixed negative number in Figure 4. Figure 5 shows that the online retailer’s profit in
the decentralized EB platform financing system πar (qa∗ ) decreases in the usage fee rate λa , and has only one intersection point with the πbr (qb∗ ) curve. The EB platform’s profit in the decentralized EB platform financing system πap (qa∗ ) increases in the usage fee rate λa , and has an intersection point with the πbp (qb∗ ) curve. For any λa ∈ [0.20, 0.22], both πar (qa∗ ) ≥ πbr (qb∗ ) and πap (qa∗ ) ≥ πbp (qb∗ ) are satisfied simultaneously, and SCF coordination can be achieved. Note that if λa < 0.20 or λa > 0.22, either πar (qa∗ ) < πbr (qb∗ ) or πap (qa∗ ) < πbp (qb∗ ) occurs, and the online retailer or the EB 20
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platform will not select EB platform financing. Moreover, the fluctuation of πar and πap in Figure 5 is bigger than that in Figure 3, with either λa or ra ranging from 0 to 0.46. The reason may be that
∂qa∗ ∂λa
≤
∂qa∗ ∂ra
< 0 and λa has a greater impact on qa∗ from Proposition 2. Specifically, if
λa = λf = 0.25, we have πar (qa∗ ) < πbr (qb∗ ), which means the conservative EB platform may not be
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able to achieve SCF coordination. When the coordination condition on order quantity is satisfied, with λa increasing the monotonicity of πar (qa∗ ) is different from that of πap (qa∗ ) in Proposition 6. We have known that the participants’ profits satisfy πar (qa∗ ) + πap (qa∗ ) > πbr (qb∗ ) + πbp (qb∗ ) in Proposition 4. Additionally, πbr (qb∗ ) and πbp (qb∗ ) are both fixed. Thus, πar (qa∗ ) > πbr (qb∗ ) and πap (qa∗ ) > πbp (qb∗ ) can be satisfied simultaneously at some reasonable points of λa , which has been confirmed by the numerical study in Figure
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5.
6.3. EB Platform Financing Coordination Under a Two Part-tariff Contract In the case of a two-part tariff, after the EB platform financing is granted, the capital-constrained
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online retailer should not only pay the usage fee during the sales period and repay the loan debt at the end of sales period, but also pay a lump sum payment at the beginning of the sales period, as it is frequently asked as an entry fee to transactions in the real marketplace (Lee and Rhee, 2010).
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The entry fee is independent of the order quantity and is a decision variable of the EB platform. Let A denote the entry fee. Then, the participants’ profits are πar = (1 − λa )p[s(qa ) − s(ka )] − m(1 +
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rf ) − A(1 + rf ) and πap = (1 − λa )ps(ka ) + λa ps(qa ) − (wqa − m)(1 + rf ) + A(1 + rf ).
Because A is independent of the order quantity qa , the SCF coordination conditions are still (1−λa )(1+rf ) , 1+ra
πar (qa∗ ) ≥ πbr (qb∗ ), and πap (qa∗ ) ≥ πbp (qb∗ ). It can be further concluded that,
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F ka (qa∗ )) =
after we add the entry fee as an EB platform’s decision variable, both the active EB platform and
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the conservative EB platform can achieve SCF coordination. Previous numerical studies have demonstrated the existence of coordination zone when the ini-
tial working capital m is a constant. However, when m is within a certain range, we could find many groups of (ra , λa ) that can achieve SCF coordination. Are there some significant characteristics of these coordination points? Is there a threshold of m that determines whether the online retailer applies for an outside financing or refuses any outside financing? We will answer these questions next. In the following numerical studies, the parameters values of no special instructions are consistent with those of previous sections.
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6.4. The Relations among the Usage Fee Rate, the Loan Interest Rate, and the Initial Working Capital When the Weak Coordination Conditions and the BCF Equilibrium are Satisfied Proposition 7. Given p, w, rf , λf , and the IFR assumption, for m within a certain range, then, (1+rb )2 h(kb (qb∗ )) drb dm = − (1−λf )p[1−H(kb (qb∗ ))] < 0. (1+rf )(1−λa ) and πar (qa∗ ) = πbr (qb∗ ) (7.2) when the weak coordination conditions, F ka (qa∗ )) = 1+ra (1+rf )[H(ka (qa∗ ))−λa ] (1+ra ) dra ∗ a are satisfied, dλ dm = p[1−H(ka (qa∗ ))][s(qa∗ )−s(ka (qa∗ ))] , dm = − (1−λa )p[1−H(ka (qa∗ ))] [h(ka (qa ))(1 + ra ) + H(ka (qa∗ ))−λa dka (qa∗ ) (1+ra ) . = − (1−λa )p[1−H(k ∗ dm s(qa∗ )−s(ka (qa∗ )) (1 + rf )], and a (qa ))] dka (qa∗ ) dH(ka (qa∗ )) (1) We have dm < 0 and < 0; dm
(2) If λa < H(ka (qa∗ )) < 1, we have
dλa dm
w(1+rf ) (1−λf )p ,
> 0 and
(3) If 0 < H(ka (qa∗ )) < λa , we have (i)
dλa dm
dra dm
are satisfied,
< 0;
< 0, (ii) dra dm
< 0 for h(ka (qa∗ ))(1 + ra )[s(qa∗ ) −
> 0 for h(ka (qa∗ ))(1 + ra )[s(qa∗ ) − s(ka (qa∗ ))] −
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s(ka (qa∗ ))] − [λa − H(ka (qa∗ ))](1 + rf ) > 0, and (iii)
dra dm
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(7.1) when the BCF equilibrium, F (qb∗ ) =
[λa − H(ka (qa∗ ))](1 + rf ) < 0. (See Appendix A.9 for the proof ).
1.8
r
a
r
b
M
1.4
λf
1.2 1
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Loan interest rate when the weak coordination conditions and the BCF equilibrium are satisfied
1.6
λa
1.6 1.4 1.2 1
0.8
0.6
0.6
0.4
0.4
PT
0.8
0.2
0.2
0
−0.12 0
2
4
0 6
−0.12 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Online retailer’s initial working capital m
Figure 6: ra ,λa , rb , and rf under different m
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EB platform’s usage fee rate when the weak coordination conditions and the BCF equilibrium are satisfied
1.8
22
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0.76
0.76 0.7
0.64
0.64
0.58
0.58
0.52
0.52
0.46
0.46
0.4
0.4
0.34
0.34
0.28
0.28
0.22
0.22
0.16
0.16
0.1
0.1
0.04 −0.02
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Generalized failure rate when the weak coordination conditions are satisfied
λa
EB platform’s usage fee rate when the weak coordination conditions are satisfied
H(ka(q*a))
0.7
0.04
0
2
4
6
8
10
12 14 16 18 20 22 24 26 28 Online retailer’s initial working capital m
30
32
34
36
38
−0.02 40
We refer to F ka (qa∗ )) =
(1+rf )(1−λa ) 1+ra
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Figure 7: H(ka (qa∗ )) and λa under different m
and πar (qa∗ ) = πbr (qb∗ ) as the weak coordination conditions.
Accordingly, when m is within a certain range from 0 to 40, we obtain many groups of (λa , ra ) that satisfy the weak coordination conditions. From Proposition 1, after we find the optimal order
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quantity qb∗ under BCF, we can find rb from the equation (1 − λf )ps(kb (qb∗ )) = (wqb∗ − m)(1 + rf ) for any given m. Figure 6 has depicted the impacts of the initial working capital on the loan interest
equilibrium are satisfied.
ED
and on the EB platform’s usage fee rate, when the weak coordination conditions and the BCF
We could find that rb decreases with m in Proposition 7.1, which has been confirmed by the
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numerical study in Figure 6. That ka (qa∗ ) and H(ka (qa∗ )) both decrease with m in Proposition 7.2 has also been confirmed by the numerical study in Figure 7. In Figure 6 and Figure 7, we could
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find that, when m < 20 (i.e., H(ka (qa∗ )) > λa ), λa increases with m. However, when m > 20 (i.e., H(ka (qa∗ )) < λa ), λa decreases with m. This monotonicity of λa is consistent with Proposition 7.2.
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In Figure 6, we also find that ra always decreases with m. For m ∈ [20, 40] (i.e., H(ka (qa∗ )) < λa ) in
our numerical study, h(ka (qa∗ ))(1 + ra )[s(qa∗ ) − s(ka (qa∗ ))] − [λa − H(ka (qa∗ ))](1 + rf ) is always bigger than 0. Thus, this monotonicity of ra is also consistent with Proposition 7.2. In Proposition 7.2, if λa < H(ka (qa∗ )) < 1,
dλa dm
> 0; otherwise,
dλa dm
< 0. It suggests that, with
the initial working capital m increasing within a certain range, the EB platform’s usage fee rate λa does not remain constant. From the monotonicity analysis of ra in Proposition 7.2, we also find that, the EB platform’s loan interest rate also changes with the initial working capital. It could be concluded that the EB platform should adjust the variables of λa and ra together in order to 23
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achieve SCF coordination. If λa < H(ka (qa∗ )) < 1, λa monotonically increases with m in Proposition 7.2. Thus, under the condition of λa < H(ka (qa∗ )) < 1 there exists at most one intersection point of λa and λf . When m = 8, we could find this intersection point in Figure 6. Only in this intersection point, can the
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conservative EB platform financing achieve SCF coordination. However, in Figure 6, within a large range of the online retailer’s initial working capital m ∈ (2, 24), the reasonable active EB platform’s coordinated financing contracts, where ra and λa are both in the range of [0,1], could be designed. This again demonstrates that, in order to achieve SCF coordination, the EB platform should be active to adjust variables of λa and ra together.
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From the Proposition 7.2 and Figure 6, there exist very few intersection points of λa and λf where the conservative EB platform financing could achieve SCF coordination. It will stunt the development of the financing business. However, the active EB platform can offer many groups of (ra , λa ) to design his coordinated financing contracts, which can be applicable to different conditions of the online retailer’s initial working capital. In other words, the online retailer will have a variety
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of coordinated financing options. It will facilitate the operation of the active EB platform financing. The active EB platform is more flexible to achieve SCF coordination than the conservative one.
ED
6.5. Online Retailer’s Selection: Using an Outside Financing, or Refusing Any Outside Financing When the online retailer refuses any outside financing, she uses her own initial working capital
PT
to purchase and the EB platform’s usage fee rate is λf . Let qnl denote the online retailer’s order r . quantity and the expected profit of the online retailer is denoted by πnl
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r = Proposition 8. When the online retailer refuses any outside financing, her expected profit is πnl −1 w(1+rf ) ( (1−λf )p ), the optimal −1 w(1+rf ) F ( (1−λf )p ). Moreover,
(1 − λf )ps(qnl ) − wqnl (1 + rf ) with a constraint of wqnl ≤ m. If m < wF
AC
∗ = order quantity is qnl
m w.
∗ = Otherwise, the optimal order quantity is qnl
r (q ∗ ) ≤ π r (q ∗ ). when the active EB platform financing can achieve SCF coordination, we have πnl a a nl
(See Appendix A.10 for the proof ). ∗ and corresponding maximum For any given m, we can find the optimal order quantity qnl r (q ∗ ) from Proposition 8. With m increasing from 0 to infinity, the online expected profit πnl nl r (q ∗ = F retailer’s maximum expected profit is πnl nl
−1 w(1+rf ) ( (1−λf )p )),
which is identical to her maximum
expected profit under BCF in Proposition 1. Moreover, from Proposition 4, when the coordination
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r (q ∗ = F of the active EB platform financing is achieved, we could deduce that πnl nl
−1 w(1+rf ) ( (1−λf )p ))
=
πbr (qb∗ ) ≤ πar (qa∗ ). Therefore, we could conclude that, in order to obtain a bigger profit, the online retailer will prefer using the coordinated active EB platform financing to refusing any outside
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financing, even if she has sufficient initial working capital.
7. Conclusion
In this paper, we present two financing options (EB platform financing and BCF) for an online retailer in order to study the interaction of the financing and operating decisions. Our model offers some analyses of the equilibrium and coordination of EB platform financing from a supply
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chain perspective. In the decentralized EB platform financing system, we obtain the Stackelberg equilibrium and local optimal parameters. Furthermore, we analyze the SCF coordination problem by comparing the decentralized EB platform financing system, the decentralized BCF system, and the centralized BCF system (or the centralized EB platform financing system) to find the global optimal parameters and to design a properly structured financing contract.
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The Stackelberg equilibrium analysis of the EB platform financing reveals that the EB platform’s loan interest rate and the EB platform’s usage fee rate both have an impact on the online retailer’s
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order quantity decision, as well as the participants’ profits. From the coordination analysis, we conclude that EB platform financing performs better than BCF does in terms of SCF efficiency
PT
improvement, and that the active EB platform financing can achieve SCF coordination, but the conservative EB platform financing may not. When SCF coordination is achieved, as shown in Proposition 4, the optimal order quantity in the decentralized EB platform financing system is
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equal to that in the centralized BCF system, and the participants’ profits in the decentralized EB platform financing system are not less than those in the decentralized BCF system. Furthermore,
AC
in order to achieve SCF coordination, the EB platform should adjust variables of the EB platform’s loan interest rate and the EB platform’s usage fee rate together. When the coordination condition on order quantity is satisfied, if the online retailer’s initial working capital is a constant, the EB platform’s loan interest rate is linearly decreasing in the EB platform’s usage fee rate. When the weak coordination conditions are satisfied, the active EB platform can offer many coordinated financing contracts, which are applicable to different initial working capital conditions, while the conservative EB platform could only design very few coordinated financing contracts. Compared with the decentralized BCF system, the EB platform’s profit under the coordinated 25
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active EB platform financing could be improved by inducing a larger online retailer’s order quantity. Thus, the EB platform is willing to finance the online retailer, even though he needs to invest a lot of capital as the loan-granting quota. The online retailer also prefers the coordinated active EB platform financing to BCF, because her profit under the coordinated active EB platform financing is larger than (or at least equal to) that in the decentralized BCF system. Moreover, the optimal
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order quantity under the coordinated active EB platform financing is bigger than that in the decentralized BCF system, which is conducive to improving the customer service level.
By formulating a model, in which the online retailer refuses any outside financing, and comparing her maximum profits in different cases, we analyze the impact of the initial working capital on the online retailer’s financing selection. We conclude that, in order to obtain a bigger profit,
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the online retailer will always prefer using the coordinated active EB platform financing to refusing any outside financing, even though she has sufficient initial working capital. We have various future research directions to explore. First, this study establishes that the active EB platform financing can achieve SCF coordination and is better than BCF in terms of
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the participants’ profits and the order quantity. However, the conservative EB platform may not achieve SCF coordination. This raises the question that why not all EB platforms set their usage fee rate and loan interest rate at the same time, or why not all EB platforms merge their lending
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business and leasing business. We might conjecture that isolating risks from different businesses, online retailer’s operating information asymmetry, and market imperfections, such as bankruptcy
PT
costs and taxes, have a relatively large impact on the EB platform’s decision. We will study these issues in future works. Second, we also intend to analyze the issue of lending competition among
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multiple EB platforms, which could be studied by a special Stackelberg game with multiple leaders and one follower. Third, we can also apply our research method to analyze the equilibrium and coordination of a new SCF form, which is operated on a peer-to-peer lending platform. Fourth,
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the capital-constrained online retailer’s dynamic inventory equilibrium problem in the case of a multi-period setting is also worth studying. Acknowledgements The authors thank the editor and two anonymous reviewers of this paper for their constructive suggestions and comments. This research was funded by the National Natural Science Foundation of China (No. 71701182, No. 71372187 and No. 71772115). Appendix:
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A.1. Proof of Lemma 1 The online retailer’s actual repayment at the end of selling season is the smaller one of the agreed repayment and the sales revenue after subtracting the EB platform’s usage fee, i.e., min[(wq−m)(1+ r), (1 − λ) p min(x, q)] = (1 − λ) p min[ (wq−m)(1+r) , min(x, q)], where (wq − m)(1 + r) is the agreed (1−λ) p
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repayment, and (1 − λ) p is the sales revenue per unit after subtracting the EB platform’s usage fee. Additionally, w(1 + r) is the purchasing cost per unit, which includes the loan interest. We obtain that w(1+r) ≤ (1−λ) p and
(wq−m)(1+r) (1−λ) p
≤ q. Thus, we have (1−λ) p min[ (wq−m)(1+r) , min(x, q)] = (1−λ) p
(1 − λ) p min[ (wq−m)(1+r) , x]. This implies that if x ≥ (1−λ) p
(wq−m)(1+r) , (1−λ)p
the online retailer’s actual
repayment can be (wq − m)(1 + r), and she can repay the full loan debt; If x <
(wq−m)(1+r) , (1−λ)p
the
online retailer cannot repay the full loan debt and must transfer all sales revenue (1 − λ)px to the
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. The online retailer’s lender. Hence, the demand threshold with no bankruptcy is k = (wq−m)(1+r) (1−λ)p Rk expected actual repayment is (1 − λ)pE[min(x, k)] = (1 − λ)p 0 F (x)dx = (1 − λ)ps(k). A.2. Proof of Proposition 1 Substituting kb =
(wqb −m)(1+rb ) (1−λf )p
into Eq.(1), we have (1 − λf ) p E{min[kb , min(x, qb )]} = (wqb −
M
m)(1+rf ). Because of kb ≤ qb , it follows (1−λf ) p E{min[kb , min(x, qb )]} = (1−λf ) p E{min[kb , x]}. Hence, Eq.(1) can be further converted to (1−λf )ps(kb ) = (wqb −m)(1+rf ). From the participants’
ED
profits expressions in subsection 3.2, under BCF the online retailer’s expected profit is πbr = E{(1 − λf )pmin(x, qb ) − min[(wqb − m)(1 + rb ), (1 − λf )pmin(x, qb )]}+ − m(1 + rf )
PT
= (1 − λf )pE{min(x, qb ) − min[kb , x]} − m(1 + rf ) = (1 − λf )p{s(qb ) − s(kb )} − m(1 + rf )
CE
= (1 − λf )ps(qb ) − wqb (1 + rf ).
(A.1)
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The EB platform can only obtain the usage fee and thus his expected profit is πbp = λf ps(qb ).
(A.2)
πbb = (1 − λf )ps(kb ) − (wqb − m)(1 + rf ) = 0.
(A.3)
The bank’s expected profit is
Taking the first-order and second-order derivative of πbr with respect to qb , we obtain that the optimal order quantity qb∗ satisfies F (qb∗ ) = F (kb (qb∗ ))
w(1 + rf ) w(1 + rb ) = . (1 − λf )p (1 − λf )p 27
(A.4)
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A.3. Proof of Proposition 2 From Proposition 1, πbs = πbr + πbp + πbb = ps(qbs ) − wqbs (1 + rf ). After taking the first-order and
second-order derivative of πbs with respect to qbs , we deduce that the second-order derivative is not w(1+rf ) . p
great than 0, and thus F (qbs∗ ) =
Since F (qb∗ ) =
w(1+rf ) (1−λf )p
in Proposition 1, it can be deduced
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that, only if λf = 0, can we obtain qb∗ = qbs∗ . However, the case of λf = 0 is almost impossible in the real economy. Therefore, the SCF coordination cannot be achieved under BCF. A.4. Proof of Proposition 3
Under active EB platform financing, the market demand threshold with no bankruptcy is ka = (wqa −m)(1+ra ) . (1−λa )p
According to Lemma 1, we obtain ka ≤ qa .
is
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From the participants’ profits expressions in subsection 3.2, the online retailer’s expected profit
πar = E{(1 − λa )pmin(x, qa ) − min[(wqa − m)(1 + ra ), (1 − λa )pmin(x, qa )]}+ − m(1 + rf ) = (1 − λa )p[s(qa ) − s(ka )] − m(1 + rf ),
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and the EB platform’s expected profit is
(A.5)
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πap = (1 − λa )ps(ka ) + λa ps(qa ) − (wqa − m)(1 + rf ). Proof of Proposition 3.1: Let Ω =
w(1+ra ) (1−λa )p
(A.6)
for simplicity. According to Lemma 1, we know
PT
that w(1 + ra ) ≤ (1 − λa )p and Ω ≤ 1. After we drive the first and second derivative of πar dπar dqa
CE
f (qa )]. When
dπar dqa
2 r
= (1 − λa )p[F (qa ) − ΩF (ka )] and ddqπ2a = (1 − λa )p[Ω2 f (ka ) − a 2 r = 0, we get F (qa∗ ) = F ka (qa∗ ) Ω. Thus, we further deduce that ddqπ2a |qa =qa∗ =
with respect to qa , we get that
a
(1 − λa )pF (qa∗ )[Ωh(ka (qa∗ )) − h(qa∗ )]. Since Ω ≤ 1, qa∗ ≥ ka (qa∗ ) and the IFR assumption, we get d2 πar dqa2
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Ωh(ka (qa∗ )) − h(qa∗ ) ≤ 0 and
≤ 0. Then, the optimal order quantity satisfies F (qa∗ ) = F ka (qa∗ ) Ω.
(A.7)
After taking partial derivative of Eq.(A.7) with respect to ra on both sides, we obtain
1−H(ka (qa∗ )) (1+ra )(Ωh(ka (qa∗ ))−h(qa∗ )) .
to ra on both sides, We further obtain 1− Ω=
w(1+ra ) (1−λa )p
Then, we take the partial derivative of ka (qa∗ ) =
∂ka (qa∗ ) ∂ra (wqa∗ −m)
=Ω
w
∂qa∗ ∂ra
+
wqa∗ −m (1−λa )p
=
Ω−ka (qa∗ )h(qa∗ ) (1+ra )(Ωh(ka (qa∗ ))−h(qa∗ ))
(wqa∗ −m)(1+ra ) (1−λa )p
≤ 1, ka ≤ qa and the IFR assumption. 28
=
with respect
∗ −m) (wqa 1− h(qa∗ ) w (1+ra )(Ωh(ka (qa∗ ))−h(qa∗ )) (wqa∗ −m)(1+ra ) ∗
=Ω
h(qa∗ ) ≤ 1−H(ka (qa∗ )), because of H(ka (qa∗ )) =
∂qa∗ ∂ra
(1−λa )p
.
h(ka (qa )),
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Similar to Chen and Wang (2012), Yan and Sun et al. (2016) and Gao and Fan et al.(2018), we also use proof by contradiction to imply deduce that
∂ka (qa∗ ) ∂ra
≥ 0, owing
and the IFR assumption. Let
∂qa∗ ∂ra
< 0. Assume that the inequality of
∂qa∗ ∂ra
∗ to Ωh(ka (qa∗ )) − h(qa∗ ) < 0 , 1 − (wqaw−m) h(qa∗ ) ≤ 1 ∗ ∂ka (qa∗ ) |ra =r0 = 0, it follows (wqa (rw0 )−m) h(qa∗ (r0 )) ∂ra
≥ 0 holds, we
− H(ka (qa∗ )) ≤ 0 = 1. Note that
0 ≤ ra ≤ 1 , we have three cases to discuss for any ra .
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∗ (wqa∗ (ra )−m) h(qa∗ (ra )) < (wqa (rw0 )−m) h(qa∗ (r0 )) = 1, because w ∗ ∗ a (qa ) Therefore, 1 − (wqa (rwa )−m) h(qa∗ (ra )) > 0, and thus ∂k∂r < 0, a
(1)If r0 ≥ 1, for any ra < 1, we have
of
∂qa∗ ∂ra
≥ 0 and the IFR assumption.
which is inconsistent with the assumption. (2)If 0 < r0 < 1, we have
∂ka (qa∗ ) ∂ra
< 0 for any ra ∈ (0, r0 ). It is also a contradiction to the
(3)If r0 = 0, we get ∂qa∗ ∂ra |ra =r0 ≤
0 and it is a contradiction to the assumption.
Therefore, the inequalities of wqa∗ −m h(qa∗ ) w
∗
0, and then, 0 = 1 − wqa (rw0 )−m h(qa∗ (r0 )) ≤ 1 − H(ka (qa∗ , r0 )).
∂qa∗ ∂ra
< 0 and
≤ 1.
Similarly, we derive
∂qa∗ ∂λa
=
Because of 1 + ra ≥ 1 − λa , it
∂ka (qa∗ ) ∂ra
< 0 hold, which further lead to H(ka (qa∗ )) <
∗ ∗ 1−H(ka (qa∗ )) ∂ka (qa∗ ) a (qa )h(qa ) = (1−λaΩ−k ∂λa (1−λa )(Ωh(ka (qa∗ ))−h(qa∗ )) < 0 and )(Ωh(ka (qa∗ ))−h(qa∗ )) ∗ ∂qa∗ ∂ka (qa∗ ) ∂ka (qa∗ ) a is obvious that ∂λ ≤ ∂q < 0. ∂ra < 0 and ∂λa ≤ ∂ra a
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Hence,
∂ka (qa∗ ) |ra =r0 = ∂ra
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assumption.
< 0.
ED
Proof of Proposition 3.2: When F (qa∗ ) = F ka (qa∗ ) Ω, after taking partial derivative of
Eq.(A.6) with respect to ra , we obtain
PT
∂ka (qa∗ ) ∂πap (qa∗ ) ∂q ∗ = (1 − λa )pF ka (qa∗ ) + [λa pF (qa∗ ) − w(1 + rf )] a = 0, ∂ra ∂ra ∂ra
(A.8)
CE
which can be converted to
(A.9)
AC
(1 − λa )pF ka (qa∗ ) [Ω − ka (qa∗ )h(qa∗ )] + [λa pF (qa∗ ) − w(1 + rf )][1 − H(ka (qa∗ ))] = 0.
When F (qa∗ ) = F ka (qa∗ ) Ω, after taking partial derivative of Eq.(A.6) with respect to λa , we
obtain
∂ka (qa∗ ) ∂πap (qa∗ ) ∂q ∗ = ps(qa∗ ) − ps(ka (qa∗ )) + (1 − λa )pF ka (qa∗ ) + [λa pF (qa∗ ) − w(1 + rf )] a = 0. ∂λa ∂λa ∂λa
(A.10)
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From Eq.(A.10)-Eq.(A.8)=0, we get 1 1 Ω − ka (qa∗ )h(qa∗ ) − ) ( ∗ ∗ Ωh(ka (qa )) − h(qa ) 1 − λa 1 + ra 1 − H(ka (qa∗ )) 1 1 + [λa pF (qa∗ ) − w(1 + rf )] − ) = 0. ( Ωh(ka (qa∗ )) − h(qa∗ ) 1 − λa 1 + ra
ps(qa∗ ) − ps(ka (qa∗ )) + (1 − λa )pF ka (qa∗ ))
(A.11)
Substituting Eq.(A.9) into Eq.(A.11), we obtain ps(qa∗ ) − ps(ka (qa∗ )) = 0, i.e., qa∗ = ka (qa∗ ). This p
p
∗
∗
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a (qa ) a (qa ) = 0 and ∂π∂λ = 0 be satisfied simultaneously. means that, only when qa∗ = ka (qa∗ ), can ∂π∂r a a ∗ −m)(1+r ) w(1+r ) (wq a a a Then, owing to F (qa∗ ) = F ka (qa∗ ) (1−λa )p and ka (qa∗ ) = , if qa∗ = ka (qa∗ ), we can obtain (1−λa )p
that the EB platform’s optimal decision variables satisfy w(1 + ra∗ ) = (1 − λ∗a )p and m(1 + ra∗ ) = 0. Therefore, after we get the above Stackelberg equilibrium, the EB platform’s expected profit
expression (i.e., Eq.(A.6)) can be convert to πap (ra∗ , λ∗a , qa∗ ) = ps(qa∗ )−(wqa∗ −m)(1+rf ), and the online
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retailer’s expected profit expression (i.e., Eq.(A.5)) can be convert to πar (ra∗ , λ∗a , qa∗ ) = −m(1 + rf ) which is not larger than 0. A.5. Proof of Corollary 1
ED
A.6. Proof of Proposition 4
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a) s∗ From Proposition 2 and Proposition 3, we know that F (qa∗ ) = F ka (qa∗ ) w(1+r (1−λa )p and F (qb ) = (1+ra ) w(1+rf ) . If F ka (qa∗ ) (1−λ < 1 + rf , it follows F (qa∗ ) < F (qbs∗ ), i.e., qa∗ > qbs∗ . Similarly, if p a) (1+ra ) (1+ra ) F ka (qa∗ ) (1−λ > 1 + rf , we get qa∗ < qbs∗ , and, if F ka (qa∗ ) (1−λ = 1 + rf , we get qa∗ = qbs∗ . a) a) From Proposition 3, we obtain πas = πar + πap = ps(qas ) − wqas (1 + rf ), which is similar to the
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expression of πbs in Proposition 2. After taking derivative, we get qas∗ = qbs∗ and F (qbs∗ ) = F (qas∗ ) =
w(1 + rf ) . p
(A.12)
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From Propositions 1 and 2, we obtain qbs∗ > qb∗ and πbs (qbs∗ ) > πbr (qb∗ ) + πbp (qb∗ ). Then, from Corollary 1 and Eq.(A.12), if F (ka (qa∗ )) =
(1−λa )(1+rf ) , 1+ra
which is the coordination condition on
AC
order quantity, is satisfied, we yield qa∗ = qas∗ = qbs∗ > qb∗ and πar (qa∗ ) + πap (qa∗ ) = πas (qas∗ ) = πbs (qbs∗ ) > πbr (qb∗ ) + πbp (qb∗ ), where qa∗ , qas∗ , qbs∗ , qb∗ , πas (qas∗ ), πbs (qbs∗ ), πbr (qb∗ ), and πbp (qb∗ ) are all fixed, and πar (qa∗ ) and πap (qa∗ ) are both functions of ra and λa . According to Proposition 1, qb∗ is the optimal order quantity in the decentralized BCF system. In order to entice the participants to support this coordination strategy, we can get the coordination conditions on participants’ profits, πar (qa∗ ) ≥ πbr (qb∗ ) and πap (qa∗ ) ≥ πbp (qb∗ ). If F (ka (qa∗ )) =
(1−λa )(1+rf ) 1+ra
is satisfied, we have obtained that πar (qa∗ ) + πap (qa∗ ) > πbr (qb∗ ) + πbp (qb∗ ),
where the parameters of qa∗ , qb∗ , πbr (qb∗ ), and πbp (qb∗ ) are all fixed, and πar (qa∗ ) and πap (qa∗ ) are both 30
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functions of ra and λa . Hence, under active EB platform financing, we can figure out the feasible region of (λa , ra ), for any constant value of m, from the coordination conditions F (ka (qa∗ )) = (1−λa )(1+rf ) , 1+ra
πar (qa∗ ) ≥ πbr (qb∗ ) and πap (qa∗ ) ≥ πbp (qb∗ ). However, under conservative EB platform
financing, for any constant value of m and λa = λf , we can find an unique solution of ra from and πap (qa∗ )
unique solution of ra may not be able to satisfy πar (qa∗ ) ≥ πbr (qb∗ ) time. Therefore, the active EB platform financing can achieve
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(1−λa )(1+rf ) , but this 1+ra p ∗ ≥ πb (qb ) at the same
F (ka (qa∗ )) =
SCF coordination, while the conservative EB platform financing may not be able to. A.7. Proof of Proposition 5
From Proposition 4, if the coordination condition on order quantity, F
wqa∗ −m 1+ra p 1−λa )
= (1 +
AN US
a) ∗ s∗ ∗ rf ) (1−λ (1+ra ) , is satisfied, we can deduce that qa = qa and qa is fixed. Then, if m is a constant, this
1−λa 1−λa 1+ra . Thus, 1+ra (1−λa )(1+rf ) with = 1+ra
coordination condition on order quantity is an equation with only one variable, is fixed. Furthermore, the first derivative of the right-hand side of F (ka (qa∗ )) dλa
respect to ra is −(1 + rf ) dra
(1+ra )+(1−λa ) (1+ra )2
If m is a constant and F (ka (qa∗ )) = is fixed. Then,
dπar (qa∗ ) dra
=
ED
A.8. Proof of Proposition 6 From Appendix A.7, we yield
dra dλa
> 0 and
dπap (qa∗ ) dra
=
1−λa ∗ ∗ 1+ra p[s(ka (qa )) − s(qa )]
1+ra = − 1−λ < 0. Then, if qa∗ = qas∗ and a
= −p[s(qa∗ ) − s(ka (qa∗ ))] < 0 and
PT
a = − 1−λ 1+ra < 0.
is satisfied, we have obtained that qa∗ = qas∗ and
1−λa ∗ ∗ 1+ra p[s(qa ) − s(ka (qa ))]
0.
dπar (qa∗ ) dλa
(1−λa )(1+rf ) 1+ra
dλa dra
M
1−λa 1+ra
= 0, so that
dπap (qa∗ ) dλa
1−λa 1+ra
<
is fixed, it follows
= p[s(qa∗ ) − s(ka (qa∗ ))] > 0.
A.9. Proof of Proposition 7
CE
Proof of Proposition 7.1: From Eq.(A.4), we obtain that qb∗ is fixed and F (kb (qb∗ )) = Appendix A.2. Then, we derive the first derivative of F (kb (qb∗ )) =
AC
to m,
1+rf 1+rb
1+rf 1+rb
in
on both sides with respect
1 + rf drb f (kb (qb∗ )) drb [−(1 + rb ) + (wqb∗ − m) ]= , (1 − λf )p dm (1 + rb )2 dm 1 + rf drb (wqb∗ − m)f (kb (qb∗ )) f (kb (qb∗ ))(1 + rb ) [ − ] = , (1 − λf )p (1 + rb )2 dm (1 − λf )p [kb (qb∗ )f (kb (qb∗ )) − F (kb (qb∗ ))]
f (kb (qb∗ ))(1 + rb )2 drb = , dm (1 − λf )p
(1 + rb )2 h(kb (qb∗ )) drb =− . dm (1 − λf )p(1 − H(kb (qb∗ )))
According to Appendix A.4, H(kb (qb∗ )) is less than 1, so that 31
(A.13) drb dm
< 0.
ACCEPTED MANUSCRIPT
Proof of Proposition 7.2: From Proposition 4, if F ka (qa∗ )) =
(1+rf )(1−λa ) , 1+ra
we get qa∗ = qas∗ =
qbs∗ , which are all fixed. Then, πar (qa∗ ) = πbr (qb∗ ) can be convert to πar (qa∗ ) − πbr (qb∗ ) = 0, and the first derivative of the equation πar (qa∗ ) − πbr (qb∗ ) = 0 with respect to m is
Taking derivative of F ka (qa∗ )) =
1 + rf dλa dka (qa∗ ) + (1 − λa )F (ka (qa∗ )) =− . dm dm p
(1+rf )(1−λa ) 1+ra
and ka (qa∗ ) =
(wqa∗ −m)(1+ra ) (1−λa )p
to m, respectively, it follows that
(A.14)
on both sides with respect
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[s(qa∗ ) − s(ka (qa∗ ))]
1 + rf dka (qa∗ ) dλa dra = [(1 + ra ) + (1 − λa ) ]. 2 ∗ dm (1 + ra ) f (ka (qa )) dm dm
(A.15)
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dka (qa∗ ) 1 dλa dra = [(wqa∗ − m)(1 + ra ) + (wqa∗ − m)(1 − λa ) − (1 − λa )(1 + ra )]. 2 dm (1 − λa ) p dm dm
(A.16)
We can observe equations, Eq. (A.14), Eq.(A.15) and Eq.(A.16), have three unknown variables dra dλa dm , dm
and
dka∗ dm ,
and, by substituting F ka (qa∗ )) =
can deduce that
and Eq.(A.15) into Eq.(A.14), we
(1 + rf )2 (1 − λa )2 dλa (1 + rf )2 (1 − λa )3 dra 1 + rf ] + =− . ∗ 2 f (ka (qa ))(1 + ra ) dm f (ka (qa∗ ))(1 + ra )3 dm p
M
[s(qa∗ ) − s(ka (qa∗ )) +
(1+rf )(1−λa ) 1+ra
ED
Therefore,
PT
(1 + rf )2 (1 − λa )2 dλa dra f (ka (qa∗ ))(1 + ra )3 f (ka (qa∗ ))(1 + ra )3 ∗ ∗ =− [s(q ) − s(k (q )) + ] − . a a a dm (1 + rf )2 (1 − λa )3 f (ka (qa∗ ))(1 + ra )2 dm p(1 + rf )(1 − λa )3 By substituting Eq.(A.16) and F ka (qa∗ )) =
(1+rf )(1−λa ) 1+ra
into Eq.(A.14), we can obtain
(wqa∗ − m)(1 + rf ) dλa (1 + rf )(wqa∗ − m)(1 − λa ) dra λa (1 + rf ) ] + =− . p dm p(1 + ra ) dm p
CE
[s(qa∗ ) − s(ka (qa∗ )) +
(A.17)
(A.18)
AC
Substituting Eq.(A.17) into Eq.(A.18), we yield (1 + rf )[H(ka (qa∗ )) − λa ] dλa = . dm p[1 − H(ka (qa∗ ))][s(qa∗ ) − s(ka (qa∗ ))]
(A.19)
Because of H(ka (qa∗ )) < 1 and qa∗ > ka (qa∗ ) in Proposition 3 and Lemma 1, it can be deduced that if λa < H(ka (qa∗ )) < 1,
dλa dm
> 0; if H(ka (qa∗ )) < λa ,
dλa dm
< 0.
By substituting Eq.(A.19) into (A.17) and simplifying formulas, we have (H(ka (qa∗ )) − λa )(1 + rf ) dra (1 + ra ) ∗ =− [h(k (q ))(1 + r ) + ]. a a a dm (1 − λa )p[1 − H(ka (qa∗ ))]) s(qa∗ ) − s(ka (qa∗ )) 32
(A.20)
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If λa < H(ka (qa∗ )) < 1, we have
dra dm
< 0. If H(ka (qa∗ )) < λa ,
s(ka (qa∗ ))] − [λa − H(ka (qa∗ ))](1 + rf ) > 0, and
dra dm
[λa − H(ka (qa∗ ))](1 + rf ) < 0.
dra dm
< 0 for h(ka (qa∗ ))(1 + ra )[s(qa∗ ) −
> 0 for h(ka (qa∗ ))(1 + ra )[s(qa∗ ) − s(ka (qa∗ ))] −
∗
(1+ra ) a (qa ) = − (1−λa )p[1−H(k <0 We can further deduce that dkdm ∗ a (q ))] a
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A.10. Proof of Proposition 8 When the online retailer refuses any outside financing and uses her own initial working capital to purchase, her opportunity cost of the procurement is wqnl (1 + rf ). Therefore, after deducting the opportunity cost, we can deduce that her expected profit is
with a constraint of wqnl ≤ m.
r with respect to q , We take the derivative of πnl nl r dπnl dqnl
r dπnl dqnl
= (1 − λf )pF (qnl ) − w(1 + rf ). When
r −1 w(1+rf ) −1 w(1+r ) dπnl ( (1−λf )p ). Therefore, if qnl < F ( (1−λf f)p ), we have dqnl > 0 and r −1 w(1+rf ) dπnl r decreases in q . According ≤ 0 and πnl qnl ≥ F ( (1−λf )p ), we have dqnl nl
= 0, it follows qnl = F
r increases in q . If πnl nl
(A.21)
AN US
r = (1 − λf )ps(qnl ) − wqnl (1 + rf ), πnl
m < wF
−1 w(1+rf ) ( (1−λf )p ),
M
r and the constraint of wq ≤ m , we can further infer that if to this monotonicity analysis of πnl nl ∗ = the optimal order quantity is qnl
ED
∗ = optimal order quantity is qnl
−1 w(1+r ) F ( (1−λf f)p ).
m w,
and that if m ≥ wF
−1 w(1+rf ) ( (1−λf )p ),
the
Therefore, with m increasing from 0 to infinity, the
r (q ∗ = F online retailer’s maximum expected profit is πnl nl
−1 w(1+rf ) ( (1−λf )p )).
PT
If the online retailer refuses any outside financing, her expected profit expression Eq.(A.21) r (q ∗ = F and maximum expected profit πnl nl
−1 w(1+rf ) ( (1−λf )p ))
are both identical to those under BCF
CE
in proposition 1 and proposition 4. Hence, if the active EB platform financing can achieve SCF −1 w(1+rf ) ( (1−λf )p ))
= πbr (qb∗ ) ≤ πar (qa∗ ).
AC
r (q ∗ = F coordination, we could conclude that πnl nl
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