Guarantor Financing in a Four‐Party Supply Chain Game with Leadership Influence

Abstract

This study investigates manufacturer guarantor financing (MG) and third‐party logistics (3PL) guarantor financing (LG) in a four‐party supply chain game that features a manufacturer, a 3PL, a capital‐constrained retailer, and a bank. The manufacturer or 3PL can act as the guarantor for the retailer who borrows bank credit. Two different leadership structures are investigated, namely, Nash game and manufacturer leadership Stackelberg game, where the manufacturer and 3PL decide simultaneously and sequentially, respectively. The supply chain under both leadership structures prefers guarantor financing to traditional bank financing when the supply chain is sufficiently cost‐efficient. In the Nash game, however, firms encounter a free‐rider dilemma when choosing between MG and LG, wherein both potential guarantors prefer the other to be the guarantor. This free‐rider dilemma can be resolved in the Stackelberg game. We also observe the follower–guarantor advantage in the Stackelberg game, wherein all firms favor the follower to provide guarantor financing. Our analysis shows that the supply chain under guarantor financing with a longer decision hierarchy (i.e., the Stackelberg game) can be conditionally more effective than that with a shorter one (i.e., the Nash game). By further analyzing different cost structures, pricing mechanism, and retailer’s initial capital, we find that most of our qualitative results remain accurate under more sophisticated conditions. These findings enhance our understanding of the value of guarantor financing in a capital‐constrained supply chain and the impact of leadership structure on financing decisions.

Keywords

guarantor financing bank financing Stackelberg leadership Nash game

Introduction

Financial constraints are prevalent worldwide, especially in developing countries. According to the World Bank Group Enterprise Surveys (The World Bank, 2018), which covers more than 131,000 firms across 139 countries, 26.5% of firms identify “access to finance” as a major constraint. In developing countries, such as those in Sub‐Saharan Africa, the ratio can reach 38.3%. Traditionally, bank financing is considered a solution for many capital‐constrained firms. However, according to the World Bank Group Enterprise Surveys, 79.2% of loans, on average, require collateral. The value of the collateral is 2.06 times the loan value, which is extremely demanding for many startups and fast‐growing firms, especially when they lack creditworthiness. Unfortunately, the lack of financial resources in one firm can jeopardize the performance of the entire supply chain.

To address the issue on creditworthiness, companies utilized guarantor financing as an alternative to traditional bank financing. In guarantor financing, one capital‐abundant supply chain member (e.g., the manufacturer) provides a guarantee for the capital‐constrained firm (e.g., the retailer) to borrow from a bank and repays the firm’s debt if the latter defaults on agreed repayments.

Two major guarantor financing schemes are widely used in developing countries where credit history and collateral are negligible for small firms. The first is referred to as manufacturer guarantor financing (MG), wherein the manufacturer serves as the guarantor. For example, the New Hope Group (NHG) is one of the largest feed producers in China. Thousands of farmers purchase feedstuff from NHG to cultivate dairy cows, chickens, and ducks. Most farmers are small in scale and they struggle to obtain financing because of their lack of collateral. To overcome this financing impediment, NHG started providing guarantor financing to farmers in 2007 when they attempt to obtain loans from banks. By the end of 2014, NHG secured loans of 18 billion yuan, which supported 90,000 farmers. Guarantor financing also rallied NHG’s own sales; for example, sales volume of feedstuff was improved by 1,350,000 tons in 2014 (Puhui, 2015).

The second major guarantor financing is called third‐party logistics (3PL) guarantor financing (LG), in which 3PL firms (e.g., UPS Capital, AIMS Logistics, and Schneider Logistics) step up their financing roles in supply chain procurement (Chen et al. 2018). In recent years, 3PL guarantor financing has swiftly gained popularity in developing countries, such as China. For example, Eternal Asia established a microcredit company in 2009 and further expanded investments to the company in 2013 to provide financial services for small businesses, especially supply chain members, by assuming their default risks (www.cnstock.com, 2013). Since 2015, S. F. Express, the top 3PL company in China, has been providing similar guarantor financing services to customers who employ its logistic services (Wu, 2015).

Guarantor financing has been practiced for years, but it is relatively novel to supply chain finance literature, especially in a four‐party game that includes a manufacturer, a retailer, a 3PL, and a bank. To the best of our knowledge, no other study has theoretically investigated the effectiveness of MG and LG. In particular, no existing study has examined relative efficiency among different guarantor financing schemes that interact with the Stackelberg leadership structure. Moreover, no existing study has characterized the conditions to implement guarantor financing, even if traditional bank financing is viable. This study attempts to address these issues.

Primary Findings and Contributions

To study guarantor financing, we extend the classic selling‐to‐newsvendor model (i.e., a manufacturer selling through a retailer) to a four‐party (supply chain) game composed of a manufacturer, a 3PL firm, a capital‐constrained retailer, and a bank. We mainly consider two game settings, namely, Nash game and manufacturer leadership Stackelberg game. The supply chain game has two time periods. In the first time period, the manufacturer determines the wholesale price, the 3PL firm decides the logistic service rate, and their decision sequence depends on the leadership scenario (i.e., either Nash or Stackelberg game). The capital‐constrained retailer then decides its order quantity, secures a loan from the bank directly or under either MG or LG, and then pays the manufacturer and 3PL for the order and delivery, respectively. Demand is realized in the second time period and the retailer repays the bank. The guarantor will pay the rest of the loan to the bank if the retailer would default.

Four major findings are obtained. First, guarantor financing can outperform traditional bank financing (BF) for the whole supply chain, especially when upstream firm costs are sufficiently low and the financial risk satisfies certain conditions in both game settings. The rationale is that the retailer’s total purchasing cost (wholesale price plus logistic service rate) is higher in guarantor financing, but its loan interest rate is lower, which leads to a higher order quantity than BF. The interest‐rate benefit outweighs the purchasing‐cost drawback when costs of upstream firms are not substantially high.

Second, the willingness to act as guarantor varies under different game leadership structures. In the Nash game, the retailer and the supply chain are indifferent of either the manufacturer or the 3PL guarantor financing. However, the manufacturer and the 3PL prefer the other to be the guarantor (i.e., being a free‐rider in guarantor financing is better than being the guarantor).

The free‐rider dilemma can be resolved if the manufacturer emerges as the Stackelberg leader in the supply chain. In the manufacturer leadership Stackelberg game, all firms prefer the follower to be the guarantor. The supply chain can benefit from the follower–guarantor advantage because the follower–guarantor is constrained by the leader from charging an extremely high‐risk premium. In contrast, the manufacturer as the Stackelberg leader does not charge an extremely high wholesale price for not carrying the guarantor’s financial risk. This follower–guarantor advantage results in a high‐order quantity that benefits all firms in the supply chain.

Third, a Stackelberg leader always prefers to free‐ride the other upstream firm’s guarantor financing service. In contrast, the follower always prefers to be the guarantor due to the follower–guarantor advantage. However, from the guarantor’s perspective, being the follower is not always advantageous. The Stackelberg leadership advantage can outplay the follower–guarantor advantage when the financial risk level (i.e., the retailer’s default risk) is sufficiently low. This outcome can be attributed to the fact that the Stackelberg leader can more easily transfer the pricing pressure to the follower to demand a bigger profit margin for being the guarantor. Nevertheless, if the retailer is more likely to default, the guarantor can gain more benefits from being a follower than being the leader.

Fourth, under LG, all firms and the supply chain can perform better with a longer decision hierarchy (i.e., the Stackelberg game) than that with a shorter decision hierarchy (i.e., the Nash game). This finding contradicts the conventional wisdom that a flatter decision hierarchy is more efficient as confirmed under bank financing. This phenomenon occurs when financial risk is substantially high. In this situation, the manufacturer is more reluctant to increase its wholesale price in the Stackelberg game than in the Nash game, thereby resulting in a lower wholesale price and a higher order quantity in the Stackelberg game.

This study contributes to the extant literature in several aspects. First, this study is the first to analyze and compare different guarantor financing schemes in a four‐party supply chain game. Second, this study provides a theoretical guideline for capital‐constrained retailers on whom to request for the guarantor financing service. Third, this study demonstrates that the follower–guarantor advantage and the Stackelberg leadership advantage can outmatch each other, such that the guarantor does not always prefer to be the Stackelberg leader. Fourth, this study shows that the supply chain with a longer decision hierarchy might not be worse off. This result corroborates the practice of some focal firms, such as manufacturers, of stepping up as the Stackelberg leaders in supply chains.

Literature Review

This study is related to multiple research streams in the growing supply chain finance literature. The first research stream is related to the interface between financial and operational decisions in supply chains. Modigliani and Miller (1958) show that operational and financial decisions can be made separately in a perfect capital market. Brander and Lewis (1986) indicate that the choice of financial structure can affect the market in terms of a limited liability effect and strategic bankruptcy effect in an oligopoly market. A limited liability effect may leverage a firm to a more aggressive stance. Babich and Sobel (2004) explore how to coordinate operational decisions with financial decisions, which include production, sales, and loan size, to maximize the expected present value of the proceeds from an initial public offering. Buzacott and Zhang (2004) incorporate asset‐based financing into production decisions and present how asset‐based financing benefits the bank and the firm. Dada and Hu (2008) consider the inventory procurement problem of a capital‐constrained newsvendor by proposing a nonlinear loan schedule to coordinate the channel under traditional bank financing. Lai et al. (2009) and Chen et al. (2013) respectively employ a mathematical model and laboratory experiments to explore how order modes or inventory financing affects inventory decisions. Zhang (2010) extends the models of Lai et al. (2009) and Chen et al. (2013) to multiproduct settings. Li et al. (2013) consider a single‐product firm that makes the decisions related to production, borrowing, and dividends under demand uncertainty. Deng et al. (2018) study an assembly system with one assembler and multiple heterogeneous suppliers and reveal the conditions when buyer finance is better than bank finance for different parties. These models are not based on guarantor financing given their distinct focuses.

The second research stream is related to trade credit. Trade credit may be an effective alternative when traditional bank financing is limited (Burkart and Ellingsen 2004). Jing et al. (2012) show that when bank and trade credits are viable, the unique equilibrium is trade credit financing if and only if the production cost is relatively low. Kouvelis and Zhao (2012) argue that the retailer will always prefer manufacturer financing to bank financing, and the manufacturer will always finance the retailer. Chod (2016) identifies one of the advantages of trade credit over bank financing, wherein the former mitigates a limited liability effect. Peura et al. (2017) prove that trade credit can soften horizontal price competition in a Bertrand competition framework. Cai et al. (2014) discuss the usage of bank credit and trade credit when the capital‐constrained retailer has incentives to divert the credits to other investments in the presence of moral hazard. They compare four scenarios and analytically and empirically prove that trade credit and bank credit can be either substitutable or complementary. Hu and Yang (2018) discuss trade credit from the perspective of trade credit and explain the phenomenon of coexistence of trade credit and reverse factoring. We refer to Cai et al. (2014) and Lee and Rhee (2011) for more detailed discussions on trade credit.

Manufacturer guarantor financing is similar with trade credit because the manufacturer shoulders the retailer’s default risk in both financing schemes. However, multiple differences exist. First, in trade credit, the manufacturer delays payment from the retailer, whereas in manufacturer guarantor financing, the manufacturer receives the cash from the bank in advance. Second, the manufacturer increases the wholesale price in guarantor financing, whereas it charges interest or equivalent in trade credit financing. Third, compared with trade credit, guarantor financing provides a healthier cash flow for the manufacturer because banks provide the loans, especially when the manufacturer faces many capital‐constrained retailers.

The third research stream is related to the emerging practice of innovative supply chain financing. Wu et al. (2014) explore a buyer‐backed purchase order financing scheme, wherein the buyer supports the supplier in financing through a loan guarantee. Tunca and Zhu (2017) analyze the role and efficiency of buyer intermediation in supplier financing based on the practice of JD, a large Chinese online retailer. Yang et al. (2016) analyze the role of trade credit insurance as a risk management tool for suppliers to guarantee against payment default by buyers. This study supplements the practice database by discussing guarantor financing provided by upstream firms to a capital‐constrained downstream firm.

The fourth research stream is related to 3PL financing. In recent years, 3PL firms played an important role in supply chain procurement. However, only a few studies focused on this subject partly because of the complexity that stems from the long, three‐stage supply chains. One exception is Chen and Cai (2011) who formulate a Stackelberg model, wherein the 3PL firm provides 3PL credit financing services. Compared with the traditional manufacturer trade credit model, the 3PL credit model outperforms the manufacturer trade credit model when the 3PL firm’s marginal profit is greater than that of the manufacturer. Chen et al. (2018) also examine the financing role of a 3PL firm in supply chain procurement and show that delayed payment from the manufacturer to the 3PL firm can enhance firms and, thereby, the whole supply chain’s profit. Unlike 3PL credit financing and delayed payment, 3PL guarantor financing does not require the 3PL to pay cash in advance. Instead, it guarantees repayment to the bank if the retailer defaults.

The remainder of this study is organized as follows. The model is described in Section 2. We analyze three financing schemes in the Nash game and the manufacturer leadership Stackelberg game in Sections 3 and 4, respectively. We discuss three extensions in Section 5 and conclude in Section 6. All proofs are provided in the Appendix: Online Supplements.

Model

This model attempts to explore guarantor financing, wherein a capital‐constrained retailer borrows from the bank under an upstream firm’s financial guarantee to fund its purchase and logistics costs from upstream firms. We extend the classic selling‐to‐newsvendor model (i.e., a manufacturer selling through a retailer) to a four‐party game model by additionally including a 3PL firm and a bank. The manufacturer determines the wholesale price

w

and the 3PL decides the logistic service rate

s

. The capital‐constrained retailer decides its order quantity

q

We assume that the retailer has no capital and must obtain a loan amounting to

(w + s) q

from the bank, with which the retailer pays the manufacturer and the 3PL for the order and delivery, respectively. In our baseline model, we follow the zero‐capital assumption for tractability, which is also adopted by Ning and Babich (2017) and Jing et al. (2012). In Section 5.3, we numerically discuss the case in which the retailer has non‐zero initial capital. The manufacturer and the 3PL are assumed to be endowed with abundant capital. Consistent with Brennan et al. (1988), Dammon and Senbet (1988), Xu and Birge (2004), Jing et al. (2012), and Kouvelis and Zhao (2012), we assume that the financial market is risk neutral and competitive, such that the bank prices the loan with an interest rate

r

to obtain a return equivalent to that with a risk‐free interest rate (denoted by

r_{f}

We assume that the manufacturer and the 3PL have constant positive marginal costs,

c_{m}

and

c_{l}

, respectively. The retailer incurs no other variable costs besides wholesale price

w

and logistic service rate

s

. A more general cost structure is discussed in the Extensions, where our main findings still hold. For simplicity, we assume zero salvage value and no stock‐out penalty.

The classic newsvendor models in extant literature frequently assumed a fixed retail price for tractability. However, this assumption might have ignored the reality that retailers usually adjust their retail prices as purchasing costs change. To better describe pricing dynamics between retail price and purchasing cost, we assume that the products are sold at a markup retail price, that is,

p = k (w + s)

, where

k

is a constant and

k > 1 + r_{f}

such that the retailer is willing to participate in the market. A retailer with high

k

belongs to a high‐margin business, whereas a low

k

denotes a low‐margin business (Biery, 2015). This markup pricing strategy is widely used in practice. As early as the 1930s, Hall and Hitch (1939) documented that many firms set their prices by adding a certain percentage markup to their costs. Recently, Greenslade and Parker (2012) surveyed 693 UK firms (conducted in December 2007 and February 2008) and reported that markup pricing is one of the most important explanations of firm pricing behaviors. Markup pricing strategy is not only extensively discussed in literature, such as in Vilcassim et al. (1999), Silva‐Risso et al. (1999), Buzacott (1975), Eichner (1973), Morrison (1992), and Nishimura et al. (1999), but also confirmed by industry insiders (Bond 2008, Carroll 2012, and Winshang.com 2013). In a survey conducted by Noble and Gruca (1999) among US pricing mangers, 56% of respondents used the cost‐based markup pricing strategy. To provide a different perspective, we also discuss the case of endogenous pricing numerically in Section 5.2 where most of our main findings still hold.

Demand is uncertain and price sensitive. To be specific, the demand is assumed to be the product of expected demand and a positive random variable, which is

D (p) = (α - β p) d

, were

α > 0

β > 0

, and

d

is the positive random variable with a mean value of 1. We further define the expected demand

E [D (p)] = α - β p

E [D]

. Similar demand functions were adopted by Emmons and Gilbert (1998) and Petruzzi and Dada (1999). Stochastic term

d

follows a cumulative distribution

F

with density

f

H (d) = d h (d)

is defined as generalized failure rate, where

h (d) = f (d) / \bar{F} (d)

is the failure rate and

\bar{F} (d) = 1 - F (d)

. We assume that

H (d)

increases with

d

, which is satisfied by most common distributions (e.g., Uniform, Normal, Gamma, and Weibull distribution) (Lariviere 2006). Frequent notations are summarized in Table 1.

Table 1
Notations

Variable name Variable description Variable type

$r_{ti}$ : Bank's interest rate in financing mode
$i = m, l, b$
, (i.e., manufacturer guarantor financing, 3PL guarantor financing, and bank financing, respectively) and leadership type
$t = M, N$
(i.e., manufacturer leadership Stackelberg and Nash game, respectively) Endogenous decision variables

$q_{ti}$
: Retailer's order quantity

$s_{ti}$
: 3PL's logistic service rate

$w_{ti}$
: Manufacturer's wholesale price

$p_{ti}$
: Retail price equal to
$k (w_{ti} + s_{ti})$

$E [D_{ti}]$
: Expected demand equivalent to
$α - β p_{ti}$

$k$
: Retail price markup rate Exogenous parameters

$c_{m}$
: Manufacturer’s unit cost

$c_{l}$
: 3PL’s unit cost

C: Joint unit cost of upstream firms,
$C = c_{m} + c_{l}$

$α$
: Potential market demand

$β$
: Price sensitivity coefficient

$d$
: Stochastic factor of demand

$r_{f}$
: Risk‐free interest rate in the financial market

$π_{ti}^{j}$
: Profit of supply chain member
$j = R, M, L, S C$
(i.e., retailer, manufacturer, 3PL, and whole supply chain, respectively) in financing mode
$i = m, l, b$
, and leadership type
$t = M, N$
Objective function

Variable name	Variable description	Variable type
$r_{ti}$ :	Bank's interest rate in financing mode $i = m, l, b$ , (i.e., manufacturer guarantor financing, 3PL guarantor financing, and bank financing, respectively) and leadership type $t = M, N$ (i.e., manufacturer leadership Stackelberg and Nash game, respectively)	Endogenous decision variables
$q_{ti}$ :	Retailer's order quantity
$s_{ti}$ :	3PL's logistic service rate
$w_{ti}$ :	Manufacturer's wholesale price
$p_{ti}$ :	Retail price equal to $k (w_{ti} + s_{ti})$
$E [D_{ti}]$ :	Expected demand equivalent to $α - β p_{ti}$
$k$ :	Retail price markup rate	Exogenous parameters
$c_{m}$ :	Manufacturer’s unit cost
$c_{l}$ :	3PL’s unit cost
C:	Joint unit cost of upstream firms, $C = c_{m} + c_{l}$
$α$ :	Potential market demand
$β$ :	Price sensitivity coefficient
$d$ :	Stochastic factor of demand
$r_{f}$ :	Risk‐free interest rate in the financial market
$π_{ti}^{j}$ :	Profit of supply chain member $j = R, M, L, S C$ (i.e., retailer, manufacturer, 3PL, and whole supply chain, respectively) in financing mode $i = m, l, b$ , and leadership type $t = M, N$	Objective function

Financing Schemes

Three different financing schemes are studied in this article: manufacturer guarantor financing (MG), 3PL guarantor financing (LG), and traditional bank financing (BF).

Manufacturer Guarantor Financing (MG): The manufacturer serves as guarantor for the retailer to borrow from the bank. If the retailer would default, the manufacturer would cover the potential loss of the bank for lending to the retailer.

3PL Guarantor Financing (LG): The 3PL serves as the guarantor for the retailer to borrow from the bank. If the retailer would default, the 3PL would cover the potential loss of the bank for lending to the retailer.

Traditional Bank Financing (BF): The bank directly lends the money to the retailer and takes the default financial risk by itself.

Traditional bank financing serves as a benchmark in this study. In practice, the retailer, who is usually a small or startup company, cannot access a bank loan by itself due to lack of collateral or credit history. In our model, bank financing for supply chain members serves as a seemingly best scenario because firms can borrow from the bank without a third‐party’s guarantee and the bank market is risk‐free. In practice, the bank would charge extra risk premium, which incurs additional financing cost for the supply chain. Our analysis will show that guarantor financing is practical and could be better than the risk‐free case of bank financing. This finding means that guarantor financing can outperform bank financing for the supply chain, even if the bank is willing to offer loans directly to the retailer.

The main disparity between guarantor financing and traditional bank financing is that the retailer’s default risk is shifted from the bank to the guarantor. After demand realization, the retailer repays

(w + s) (1 + r) q

to the bank if the revenue is sufficient to cover the loan and the interest; otherwise, the retailer repays its entire realized revenue to the bank after declaring bankruptcy and the guarantor then undertakes the loss under guarantor financing.

We use subscript

i = m, l, b

to denote MG, LG, and BF, respectively. We use superscript

j = R, M, L, a n d S C

to represent the retailer, the manufacturer, the 3PL firm, and the supply chain, respectively.

Game Setting

The supply chain game consists of two time periods. In the first time period, the manufacturer determines the wholesale price

w

, and the 3PL decides the logistic service rate

s

. The decision sequence of the manufacturer and the 3PL depends on the leadership type: either Nash game or manufacturer leadership Stackelberg game.

Nash Game: The manufacturer and the 3PL sign contracts with the retailer separately. These two upstream firms do not know the contract price of each other with the retailer. This means that both firms determine wholesale/service price only in anticipation of the other’s best response decision. This case is equivalent to that upstream firms simultaneously decide in a Nash game. This setting represents the basic situation, wherein the manufacturer and the 3PL are identical on decision sequence and information state.

2. Manufacturer Leadership Stackelberg Game: The manufacturer determines the wholesale price first and then the 3PL decides its logistic service rate. In practice, the 3PL can obtain information about the value of the product via insurance service, which makes manufacturer leadership Stackelberg game possible. Oftentimes, the wholesale price information can be also obtained on B2B websites, such as Alibaba (Alibaba.com) and Import Express Wholesale (import‐express.com).

In the second time period, the retailer decides on an optimal order quantity and then borrows a loan from the bank with an interest rate of r with or without guarantor financing. The retailer announces the retail price by adding a markup on purchasing cost. Price‐sensitive stochastic demand D is then realized and the retailer repays the bank. Under guarantor financing, if the retailer would default, the guarantor would cover the potential loss of the bank for lending to the retailer.

We use subscript

t = M, N

to denote manufacturer leadership Stackelberg game and simultaneous Nash game, respectively. The solution to any of the four‐party games can be described in a subgame perfect equilibrium, which is solved backwards.

Profit Functions and Decisions of Firms

The profit functions of firms in the Nash game and Stackelberg game are the same. The difference lies in the decision sequence of the two upstream firms as stated in Section 2.2. We start solving for the bank's decision using backward induction. Under manufacturer and 3PL guarantor financing, the bank does not undertake any financial risk and sets the interest rate to the same value as the risk‐free interest rate, which is

r_{ti} = r_{f}, i = m, l

. Unlike guarantor financing, the retailer’s default financial risk in BF is on the bank. Thus, the bank determines the interest rate

r_{tb}

to earn the same return as in a risk‐free market as follows:

E \min \{p_{tb} \min (D_{tb}, q_{tb}), {(w_{tb} + s_{tb}) (1 + r_{tb}) q}_{tb}\} = (w_{tb} + s_{tb}) (1 + r_{f}) q_{tb} .

Real interest rate

r_{tb}

will be higher than the risk‐free interest rate

r_{f}

because of financial risk. The retailer's objective is to maximize its profit by optimizing

q_{ti}

, given

w_{ti}

and

s_{ti}

. After demand is realized, the retailer collects its revenue and repays the bank. If the realized revenue fails to fully settle the loan (principal plus interest), the retailer repays as much as possible with its revenue. The guarantor then pays the remaining loan in guarantor financing or the bank undertakes the loss in bank financing. Thus, the retailer's profit is described as follows:

π_{ti}^{R} = E \{{[p_{ti} \min (D_{ti}, q_{ti}) - (w_{ti} + s_{ti}) (1 + r_{ti}) q_{ti}]}^{+}\}, i = m, l, b .

We then consider the decisions of upstream firms. In the Nash game, the manufacturer (3PL) optimizes its profit by deciding

w_{N i}

(

s_{N i}

) without the knowledge of 3PL (manufacturer)’s decision, but only with the expectation of

s_{N i}

(

w_{N i}

). In the manufacturer Stackelberg game, the 3PL decides after the manufacturer, in line with Xu and Birge (2004), Lai et al. (2009), and Kouvelis and Zhao (2012), we assume that the revenue of upstream firms collected in the first time‐period can be reinvested in the same financial market at the risk‐free interest rate. Therefore, in manufacturer and 3PL guarantor financing, the manufacturer or 3PL will earn an expected profit of

(w_{ti} - c_{m}) (1 + r_{f}) q_{ti}

(s_{ti} - c_{l}) (1 + r_{f}) q_{ti}

, respectively. In manufacturer (3PL) guarantor financing, the guarantor manufacturer (3PL) incurs a cost equal to

E \{{[(w_{ti} + s_{ti}) (1 + r_{ti}) q_{ti} - p_{ti} \min (D_{ti}, q_{ti})]}^{+}\}

if the retailer defaults in the second period. Considering the time value of money, the manufacturer and the 3PL’s profit can then be written as follows:

π_{ti}^{M} = \{\begin{matrix} (w_{ti} - c_{m}) (1 + r_{f}) q_{ti} \\ - E \{{[(w_{ti} + s_{ti}) (1 + r_{ti}) q_{ti} - p_{ti} \min (D_{ti}, q_{ti})]}^{+}\}, i f i = m; \\ (w_{ti} - c_{m}) (1 + r_{f}) q_{ti}, i f i = b, l . \end{matrix},

π_{ti}^{L} = \{\begin{matrix} (s_{ti} - c_{l}) (1 + r_{f}) q_{ti}, i f i = b, m; \\ (s_{ti} - c_{l}) (1 + r_{f}) q_{ti} \\ - E \{{[(w_{ti} + s_{ti}) (1 + r_{ti}) q_{ti} - p_{ti} \min (D_{ti}, q_{ti})]}^{+}\}, i f i = l . \end{matrix} .

For a meaningful discussion, any supply chain member that participates in the game must have a positive profit.

Financing under Nash Game

In the Nash game, the manufacturer and the 3PL sign contracts with the retailer separately without the knowledge of the pricing decision of the other. This section first analyzes decisions under bank financing and two kinds of guarantor financing in the Nash game and then compares these different financing schemes to underline the benefit of guarantor financing.

Equilibrium Analysis of Bank Financing

We start with the retailer’s best response function for any given wholesale price and logistic service rate by solving the game backwards. By combining the bank’s profit function in Equation (1) with the retailer’s profit function in Equation (2), we transfer the retailer’s problem in BF to the following:

\max_{q_{N b} \geq 0} E \{p_{N b} \min (D_{N b}, q_{N b}) - (w_{N b} + s_{N b}) (1 + r_{f}) q_{N b}\},

which is equivalent to the risk‐free case without capital constraint. This finding is consistent with Xu and Birge (2004), Jing et al. (2012), and Kouvelis and Zhao (2012). The retailer’s optimal order quantity is solved as follows:

\bar{F} (\frac{q_{N b}^{*}}{E [D_{N b}]}) = \frac{1 + r_{f}}{k} .

We define

\hat{q_{b}} \equiv \frac{q_{N b}^{*}}{E [D_{N b}]}

, which is denoted as the demand satisfaction level under bank financing. More potential consumers can be served and the stockout probability is lower when

\hat{q_{b}}

is larger. Risk‐free interest rate

r_{f}

represents the capital opportunity cost, and

k

denotes the retailer’s profit margin level in the retail industry. Thus, given the same upstream prices, the retailer orders more if the capital opportunity cost is low (i.e., the outside return of capital is limited compared with purchasing products) or the retailer belongs to a high‐margin business. This result is consistent with the fact that high‐margin businesses often produce high‐quality products and offer excellent services to their customers.

By solving the game backwards, we can obtain the firms’ optimal decisions and profits, as summarized in Lemma 1:

Lemma 1

Upstream firms can earn positive profit and are willing to participate in the Nash game with bank financing if

C < \frac{α}{β k}

. Given

C < \frac{α}{β k}

, the optimal decisions and corresponding profits of supply chain members with bank financing in the Nash game are listed in the following table.

$q_{N b}^{*}$	$w_{N b}^{*}$	$s_{N b}^{*}$
$E [D_{N b}^{*}] \hat{q_{b}}$	$\frac{α - β k C}{3 β k} + c_{m}$	$\frac{α - β k C}{3 β k} + c_{l}$
$π_{N b}^{R}$	$π_{N b}^{M}$	$π_{N b}^{L}$
$p_{N b}^{} E [D_{N b}^{}] A_{1}$	$\frac{(1 + r_{f}) \hat{q_{b}}}{β k} E^{2} [D_{N b}^{*}]$	$\frac{(1 + r_{f}) \hat{q_{b}}}{β k} E^{2} [D_{N b}^{*}]$

where

E [D_{N b}^{*}] = \frac{α - β k C}{3}

A_{1} \equiv \int_{0}^{\hat{q_{b}}} ε f (ε) d ε

Condition

C < \frac{α}{β k}

is necessary for a positive expected demand because

p < \frac{α}{β}

is required to ensure a positive expected demand and p cannot be less than

kC

. Under bank financing, this condition is sufficient to warrant a positive expected demand and profit for every supply chain member.

Through Lemma 1, we learn that the manufacturer and the 3PL earn the same marginal profits (

w_{N b} - c_{m} = s_{N b} - c_{l}

) and total profits (

π_{N b}^{M} = π_{N b}^{L}

) as they are in the same position in terms of decisions and information state.

For the retailer,

A_{1}

can be interpreted as the retailer’s profitable demand ratio under bank financing, which measures the portion of demand used to generate profits for the retailer (revenue minus cost). The term

A_{1}

increases with

k

, but decreases with

r_{f}

. It is intuitive that a retailer profits from a higher retail mark‐up ratio, but suffers from a higher risk‐free interest rate conditional on the same purchasing cost.

Equilibrium Analysis of Guarantor Financing

The derivation procedure of equilibrium solution under guarantor financing is similar to that under bank financing. We summarize the optimal decisions and profits of all supply chain members in Lemma 2. We only list the manufacturer guarantor financing case because the optimal decisions and profits are symmetrical under 3PL guarantor financing.

Lemma 2

Upstream firms can earn positive profit and are willing to participate in the Nash game with bank financing if

C < \frac{α}{β k g}

. The optimal decisions and corresponding profits of supply chain members under manufacturer guarantor financing in the Nash game is listed in the following table.

$q_{N m}^{*}$	$w_{N m}^{*}$	$s_{N m}^{*}$
$E [D_{N m}^{*}] \hat{q}$	$\frac{α g + β k (g - 2) C}{β k (g + 2)} + c_{m}$	$\frac{α - β k g C}{β k (g + 2)} + c_{l}$
$π_{N m}^{R}$	$π_{N m}^{M}$	$π_{N m}^{L}$
$p_{N m}^{} E [D_{N m}^{}] A_{2}$	$\frac{A}{β k} E^{2} [D_{N m}^{*}]$	$\frac{(1 + r_{f}) \hat{q}}{β k} E^{2} [D_{N m}^{*}]$

where

\hat{q}

satisfies

\bar{F} (\hat{q}) = \frac{1 + r_{f}}{k} \bar{F} (\frac{1 + r_{f}}{k} \hat{q})

A \equiv (1 + r_{f}) \hat{q} - k \int_{0}^{\frac{1 + r_{f}}{k} \hat{q}} (\frac{1 + r_{f}}{k} \hat{q} - ε) f (ε) d ε

A_{2} \equiv \int_{\frac{1 + r_{f}}{k} \hat{q}}^{\hat{q}} ε f (ε) d ε

g \equiv \frac{(1 + r_{f}) \hat{q}}{A}

, and

E [D_{N m}^{*}] = \frac{α - β k g C}{g + 2}

Similar to that under bank financing,

\hat{q}

is defined as the demand satisfaction level of the retailer under guarantor financing. We obtain

\bar{F} (\hat{q}) < \bar{F} (\hat{q_{b}})

, which suggests that guarantor financing yields a lower stock‐out probability than traditional bank financing, because the guarantor partially undertakes the risk of overstock such that the retailer is more likely to provide a higher level of service.

The term

A

measures the expected level of revenue of upstream firms minus the potential default loss. Given the same wholesale price and logistic service price, upstream firms with a higher

A

earn more profits. From the perspective of financial risk, the expected cost of risk is

(w_{N m}^{*} + s_{N m}^{*}) E [D_{N m}] ((1 + r_{f}) \hat{q} - A)

. Hereafter, we define

g \equiv \frac{(1 + r_{f}) \hat{q}}{A}

as an indicator of financial risk. A larger

g

entails a higher expected cost of financial risk. Based on Lemma 2, it is easy to verify that

g \geq 1

. Note that

g = 1

holds only when there is no demand uncertainty (i.e., no financial risk). Accordingly, there is no difference between BF, MG, and LG when

g = 1

. So, for interesting discussion in the following, we only consider the case with uncertainty, that is

g > 1

. The term

A_{2}

is the retailer’s profitable demand ratio under guarantor financing.

A_{2}

increases with

k

, but decreases with

r_{f}

The necessary condition for a positive expected demand stated in BF,

C < \frac{α}{β k}

, is not sufficient to ensure positive profits of all supply chain members under guarantor financing. The sufficient condition to earn positive profit for every supply chain member under guarantor financing,

C < \frac{α}{β k g}

, is stricter than that under bank financing. This result is understandable because the supply chain under guarantor financing takes more risk as the retailer orders more. Moreover,

\frac{α}{β k g}

decreases with

r_{f}

. Under guarantor financing, the supply chain itself undertakes financing risk when the retailer defaults, of which the potential loss increases with the risk‐free interest rate. Under bank financing, condition

C < \frac{α}{β k}

is independent of the risk‐free interest rate

r_{f}

. For a meaningful discussion, we assume that condition

C < \frac{α}{β k g}

is satisfied such that BF and guarantor financing schemes are profitable.

Comparison between Bank Financing and Guarantor Financing

This subsection compares firms’ performances between guarantor financing and bank financing in the Nash game. The comparison results are listed in Theorem 1.

Theorem 1

In the Nash game,

w_{N m}^{*} > w_{N b}^{*} > w_{N l}^{*}

s_{N l}^{*} > s_{N b}^{*} > s_{N m}^{*}

w_{N m}^{*} + s_{N m}^{*} = w_{N l}^{*} + s_{N l}^{*} > w_{N b}^{*} + s_{N b}^{*}

q_{N m} = q_{N l} \geq q_{N b}

if and only if

C \leq \frac{α [3 \hat{q} - (g + 2) \hat{q_{b}}]}{β k [3 g \hat{q} - (g + 2) \hat{q_{b}}]}

;

The retailer prefers MG or LG to BF (i.e.,

π_{N b}^{R} \leq π_{N m}^{R}

π_{N b}^{R} \leq π_{N l}^{R}

) if and only if

A_{1} < A_{2}

g < \frac{9 - 8 A_{1} / A_{2} + 3 \sqrt{9 - 8 A_{1} / A_{2}}}{4 A_{1} / A_{2}}

and

C \leq \hat{C_{1}}

, where

\hat{C_{1}}

is the positive solution of

Q_{1} α^{2} - Q_{2} α β k \hat{C_{1}} - Q_{2} β^{2} k^{2} {\hat{C_{1}}}^{2} = 0

Q_{1} \equiv 9 A_{2} A [(1 + r_{f}) \hat{q} + A] - 2 A_{1} {[(1 + r_{f}) \hat{q} + 2 A]}^{2}

and

Q_{2} \equiv 9 A_{2} {(1 + r_{f})}^{2} {\hat{q}}^{2} - A_{1} {[(1 + r_{f}) \hat{q} + 2 A]}^{2}

;

The manufacturer (the 3PL) prefers MG (LG) to BF (i.e.,

π_{N b}^{M} \leq π_{N m}^{M}

π_{N b}^{L} \leq π_{N l}^{L}

), if and only if

g < \frac{9 - 4 \hat{q_{b}} / \hat{q} + 3 \sqrt{9 - 8 \hat{q_{b}} / \hat{q}}}{2 \hat{q_{b}} / \hat{q}}

and

C \leq \frac{α [3 - (g + 2) \sqrt{g \hat{q_{b}} / \hat{q}}]}{β k [3 g - (g + 2) \sqrt{g \hat{q_{b}} / \hat{q}}]}

;

The manufacturer (the 3PL) prefers LG (MG) to BF (i.e.,

π_{N b}^{M} \leq π_{N l}^{M}

π_{N b}^{L} \leq π_{N m}^{L}

), if and only if

C \leq \frac{α [3 - (g + 2) \sqrt{\hat{q_{b}} / \hat{q}}]}{β k [3 g - (g + 2) \sqrt{\hat{q_{b}} / \hat{q}}]}

Theorem 1 shows the significance of guarantor financing given that the supply chain members may still prefer guarantor financing even if the capital‐constrained retailer can borrow directly from the bank.

A trade‐off exists between guarantor financing and BF for the retailer. On the one hand, the retailer benefits from a lower interest rate in guarantor financing (interest‐rate effect) than in BF. On the other hand, the price of upstream firms depends on the operation cost and the financial risk cost (for serving as a guarantor). The guarantor increases the price for carrying the retailer’s default risk (i.e.,

w_{N m}^{*} > w_{N b}^{*}

s_{N l}^{*} > s_{N b}^{*}

); thus, the retailer’s total purchasing cost is higher in guarantor financing than in BF (guarantor‐cost effect), that is,

w_{N m}^{*} + s_{N m}^{*} = w_{N l}^{*} + s_{N l}^{*} > w_{N b}^{*} + s_{N b}^{*}

. When the total supply chain operations costs are sufficiently small (i.e.,

C \leq \frac{α [3 \hat{q} - (g + 2) \hat{q_{b}}]}{β k [3 g \hat{q} - (g + 2) \hat{q_{b}}]}

), the guarantor‐cost effect is undermined. Under this condition, the guarantor‐cost effect is outpaced by the interest‐rate effect, so the retailer orders more under guarantor financing with lower combined costs.

The retailer prefers guarantor financing when

A_{2} > A_{1}

g < \frac{9 - 8 A_{1} / A_{2} + 3 \sqrt{9 - 8 A_{1} / A_{2}}}{4 A_{1} / A_{2}}

, and

C \leq \hat{C_{1}}

. The first condition

A_{2} > A_{1}

requires that the profitable demand ratio of guarantor financing should be appealing compared with that of bank financing. This condition is more likely to hold if a given demand distribution is skewed to the right (a larger demand is more likely). As explained previously, the second and the third conditions ensure that the guarantor‐cost effect is remarkably curtailed, such that the positive interest‐rate effect stands out.

The potential guarantor also faces a trade‐off. On the one hand, the guarantor increases wholesale price or service price to enhance its profit margin (profit‐margin effect). On the other hand, the guarantor undertakes the retailer’s default risk and faces potential losses (financial‐risk effect). The guarantor‐cost effect is offset by interest‐rate effect when the supply chain cost is low; consequently, the retailer orders a large quantity. This outcome boosts the positive profit‐margin effect to overshadow the negative financial‐risk effect and, therefore, the guarantor benefits from providing guarantor financing.

The non‐guarantor in guarantor financing is forced to reduce price (profit margin loss), but it enjoys the benefit of guarantor financing (order quantity boost) without taking the guarantee risk. The sufficient condition for the non‐guarantor to enjoy guarantor financing is less constrained than the guarantor’s given that it does not shoulder the retailer’s default risk like the guarantor or the bank.

Comparison between Guarantor Financing Schemes

Given the importance of guarantor financing compared with bank financing, we are curious about which supply chain member is preferable as guarantor. A comparison of the two guarantor financing schemes achieves the following results:

Theorem 2

In the Nash game, we obtain the following:

w_{N l}^{*} < w_{N m}^{*}

s_{N l}^{*} > s_{N m}^{*}

E [D_{N l}^{*}] = E [D_{N m}^{*}]

, and

q_{N l}^{*} = q_{N m}^{*}

The retailer and the whole supply chain are indifferent to the manufacturer or the 3PL being the guarantor (i.e.,

π_{N l}^{R} = π_{N m}^{R}

and

π_{N l}^{L} + π_{N l}^{M} = π_{N m}^{L} + π_{N m}^{M}

), and

The 3PL and the manufacturer prefer the other firm to be the guarantor (i.e.,

π_{N l}^{L} < π_{N m}^{L}

and

π_{N l}^{M} > π_{N m}^{M}

The upstream firm charges a higher price/rate when acting as guarantor (i.e.,

w_{N l}^{*} < w_{N m}^{*}

and

s_{N l}^{*} > s_{N m}^{*}

). Nevertheless, the expected demand is the same between MG and LG, and the retailer orders the same quantity because the leadership positions of the manufacturer and the 3PL in the MG and LG schemes are symmetric. Thus, the retailer and the whole supply chain achieve the same profits under the manufacturer and 3PL guarantor financing schemes (i.e.,

π_{N l}^{R} = π_{N m}^{R}

and

π_{N l}^{L} + π_{N l}^{M} = π_{N m}^{L} + π_{N m}^{M}

Combining the observations in Theorems 1 and 2, we can summarize the firms’ preference sequences of the three financing schemes in Corollary 1.

Corollary 1

In the Nash game, the firms’ preference sequences of the three financing schemes are as follows:

For the retailer, 1) when

A_{1} < A_{2}

g < \frac{9 - 8 A_{1} / A_{2} + 3 \sqrt{9 - 8 A_{1} / A_{2}}}{4 A_{1} / A_{2}}

and

C \leq \hat{C_{1}}

π_{N l}^{R} = π_{N m}^{R} \geq π_{N b}^{R}

; 2) otherwise,

π_{N b}^{R} > π_{N l}^{R} = π_{N m}^{R}

For the manufacturer, 1) when

C > \frac{α [3 - (g + 2) \sqrt{\hat{q_{b}} / \hat{q}}]}{β k [3 g - (g + 2) \sqrt{\hat{q_{b}} / \hat{q}}]}

π_{N b}^{M} > π_{N l}^{M} > π_{N m}^{M}

；2) when

C \leq \frac{α [3 - (g + 2) \sqrt{g \hat{q_{b}} / \hat{q}}]}{β k [3 g - (g + 2) \sqrt{g \hat{q_{b}} / \hat{q}}]}

and

g < \frac{9 - 4 \hat{q_{b}} / \hat{q} + 3 \sqrt{9 - 8 \hat{q_{b}} / \hat{q}}}{2 \hat{q_{b}} / \hat{q}}

π_{N l}^{M} > π_{N m}^{M} \geq π_{N b}^{M}

; 3) otherwise,

π_{N l}^{M} \geq π_{N b}^{M} > π_{N m}^{M}

For the 3PL, 1) when

C > \frac{α [3 - (g + 2) \sqrt{\hat{q_{b}} / \hat{q}}]}{β k [3 g - (g + 2) \sqrt{\hat{q_{b}} / \hat{q}}]}

{π_{N b}^{L} > π}_{N m}^{L} > π_{N l}^{L}

; 2) when

C \leq \frac{α [3 - (g + 2) \sqrt{g \hat{q_{b}} / \hat{q}}]}{β k [3 g - (g + 2) \sqrt{g \hat{q_{b}} / \hat{q}}]}

and

g < \frac{9 - 4 \hat{q_{b}} / \hat{q} + 3 \sqrt{9 - 8 \hat{q_{b}} / \hat{q}}}{2 \hat{q_{b}} / \hat{q}}

π_{N m}^{L} > π_{N l}^{L} \geq π_{N b}^{L}

; 3) otherwise,

π_{N m}^{L} \geq π_{N b}^{L} > π_{N l}^{L}

The explanation of trade‐offs for firms in Corollary 1 is similar to those for Theorems 1 and 2. We hereby employ Figure 1 to further illustrate the firms’ preference sequences between BF, MG, and LG. Notably, throughout this article,

α = 100

β = 1

c_{l} = 10

k = 1.3

r_{f} = 0.02

, and the demand

d

follows a gamma distribution with a mean of 1 and variance of 0.5 [i.e.,

d \sim G a (2, 2)

], unless mentioned otherwise. To highlight the profit difference in Figure 1, the horizontal axis starts from

c_{m} = 20

instead of

c_{m} = 0

FIGURE 1

Profits of Different Financing Schemes under Nash Game with Respect to the Manufacturer’s Cost

c_{m}

It is intuitive that the firms’ profits decrease with the manufacturer’s production cost

c_{m}

, as illustrated in Figure 1. As explained previously, when the upstream firms’ costs are sufficiently low, the negative guarantor‐cost effect for the retailer is considerably curtailed such that the retailer order more under guarantor financing. In this situation, the guarantor’s financial‐risk effect is outpaced by the combination of profit‐margin effect and the increase in order quantity and, thus, guarantor financing is more attractive.

It is natural that firms’ profits will be affected by the retail price markup rate (

k

). One can argue that a more powerful retailer (a higher

k

) takes more profits from other firms, which has been verified in our numerical experiments using different values of

k

. Although our qualitative insights remain the same, we find that the range, in which guarantor financing outperforms bank financing for all firms, shrinks as benefit of guarantor financing is undermined by a higher

k

However, there exists a dilemma on which firm should provide guarantor financing. In particular, there is a trade‐off between the revenue margin and potential default loss. On the one hand, the guarantor can charge a higher price to earn a higher revenue margin. On the other hand, being the guarantor incurs a potential loss caused by the retailer’s default risk. Provided that the order quantity is the same under both guarantor financing types, in the trade‐off, the risk premium of the guarantor is edged out by the free‐rider benefit of allowing the other firm to serve as guarantor; therefore, both the manufacturer and the 3PL prefer the other firm to provide guarantor financing.

To illustrate this free‐riding dilemma in more details, we temporarily relax our original assumption here by assuming whether to be the guarantor is a decision of the manufacturer and the 3PL. In line with the practice, we also assume that only one firm, either the manufacturer or the 3PL, can be the guarantor. As a result, whoever not to be the guarantor will be a free‐rider in the guarantor financing game. The equilibrium outcome of the Nash game is summarized in the following table. To be consistent with the context, bank financing is available if no guarantee service is provided. Note that if bank financing is not available, the supply chain members will have more incentives to coordinate the supply chain.

${(π}_{N l}, π_{N m})$		Manufacturer
${(π}_{N l}, π_{N m})$		Guarantor	Free‐rider
3PL	Guarantor	$(N A, N A)$	$(\frac{A}{β k} E^{2} [D_{N m}^{}], \frac{(1 + r_{f}) \hat{q}}{β k} E^{2} [D_{N m}^{}])$
Free‐rider	$(\frac{(1 + r_{f}) \hat{q}}{β k} E^{2} [D_{N m}^{}], \frac{A}{β k} E^{2} [D_{N m}^{}])$	$(\frac{(1 + r_{f}) \hat{q_{b}}}{β k} E^{2} [D_{N b}^{}], \frac{(1 + r_{f}) \hat{q_{b}}}{β k} E^{2} [D_{N b}^{}])$

According to Theorem 1, there exist conditions when guarantor financing outperforms bank financing for every supply chain member. Under the conditions, we can see that the game has two pure‐strategy Nash equilibria, which is a typical battle of the sexes game where each firm prefers the other to be the guarantor. Theoretically and practically, either equilibrium is possible considering that either firm might have been used to provide guarantee insurance to the retailer or is more likely to do so due to a stronger supply chain relationship with the retailer. If neither upstream firm would do so, they will lose the extra profits. Nevertheless, coordination can make one of the Nash equilibria be Pareto dominant.

One potential coordination solution is that the retailer can transfer an extra side payment to one of the upstream firms for the guarantee insurance. For example, the retailer can transfer the side payment to the 3PL. Compared to the case without guarantor, the retailer can earn additional

p_{N m}^{*} E [D_{N m}^{*}] A_{2} - p_{N b}^{*} E [D_{N b}^{*}] A_{1}

under 3PL guarantor financing, so the retailer is willing to provide at most

p_{N m}^{*} E [D_{N m}^{*}] A_{2} - p_{N b}^{*} E [D_{N b}^{*}] A_{1}

side payment to the 3PL. In contrast, with a side payment no less than

\frac{(1 + r_{f}) \hat{q} - A}{β k} E^{2} [D_{N m}^{*}]

(i.e., the difference of the 3PL’s profits between being a guarantor and being a free‐rider), the 3PL being a guarantor can earn at least the same profit as being a free‐rider and is willing to provide such guarantee. Because

p_{N m}^{*} E [D_{N m}^{*}] A_{2} - p_{N b}^{*} E [D_{N b}^{*}] A_{1} > \frac{(1 + r_{f}) \hat{q} - A}{β k} E^{2} [D_{N m}^{*}]

will conditionally hold in our model, which can be verified by numerical experiments, all firms can benefit from the retailer’s coordination. For example, when

p_{N m}^{*} E [D_{N m}^{*}] A_{2} - p_{N b}^{*} E [D_{N b}^{*}] A_{1} > \frac{(1 + r_{f}) \hat{q} - A}{β k} E^{2} [D_{N m}^{*}]

holds, the retailer can provide a side payment valued at

\frac{(1 + r_{f}) \hat{q} - A}{β k} E^{2} [D_{N m}^{*}]

to the 3PL or the manufacturer; as a result, the upstream firm being the guarantor can earn the same profit as being a free‐rider and, thus, is willing to provide the guarantee service.

In reality, a focal firm, such as the manufacturer, may often emerge as the leader of the supply chain. Our next section proposes that such free‐riding dilemma will disappear in the manufacturer leadership Stackelberg game without extra coordination effort.

Financing under the Manufacturer Leadership Stackelberg Game

Due to the similarity of analysis between the Nash game and Stackelberg game, for a concise exposition, we skip the derivation details, but focus on the comparisons between different financing schemes and leadership structures to demonstrate how a change in leadership influences the game. We list the following equilibrium results for reference:

Lemma 3

We obtain the following results under the manufacturer leadership Stackelberg game:

1. Given

C < \frac{α}{β k}

, the optimal decisions and profits of supply chain members with bank financing are listed as follows

$q_{M b}^{*}$	$w_{M b}^{*}$	$s_{M b}^{*}$
$E [D_{M b}^{*}] \hat{q_{b}}$	$\frac{α - β k C}{2 β k} + c_{m}$	$\frac{α - β k C}{4 β k} + c_{l}$
$π_{M b}^{R}$	$π_{M b}^{M}$	$π_{M b}^{L}$
$p_{M b}^{} E [D_{M b}^{}] A_{2}$	$\frac{2 (1 + r_{f}) \hat{q_{b}}}{β k} E^{2} [D_{M b}^{*}]$	$\frac{(1 + r_{f}) \hat{q_{b}}}{β k} E^{2} [D_{M b}^{*}]$

2. Given

C < \frac{α}{β k g}

, the optimal decisions and profits of supply chain members with manufacturer guarantor financing are listed as follows

$q_{M m}^{*}$	$w_{M m}^{*}$	$s_{M m}^{*}$
$E [D_{M m}^{*}] \hat{q}$	$\frac{α g - β k C}{β k (g + 1)} + c_{m}$	$\frac{α - β k g C}{2 β k (g + 1)} + c_{l}$
$π_{M m}^{R}$	$π_{M m}^{M}$	$π_{M m}^{L}$
$p_{M m}^{} E [D_{M m}^{}] A_{2}$	$\frac{(1 + r_{f}) \hat{q} + A}{β k} E^{2} [D_{M m}^{*}]$	$\frac{(1 + r_{f}) \hat{q}}{β k} E^{2} [D_{M m}^{*}]$

3. Provided

C < \frac{α}{β k g}

, the optimal decisions and profits of supply chain members with 3PL guarantor financing are listed as follows

$q_{M l}^{*}$	$w_{M l}^{*}$	$s_{M l}^{*}$
$E [D_{M l}^{*}] \hat{q}$	$\frac{α - β k g C}{2 β k g} + c_{m}$	$\frac{α (3 g - 2) - β k g (2 - g) C}{4 β k g} + c_{l}$
$π_{M l}^{R}$	$π_{M l}^{M}$	$π_{M l}^{L}$
$p_{M l}^{} E [D_{M l}^{}] A_{2}$	$\frac{2 A}{β k} E^{2} [D_{M l}^{*}]$	$\frac{A}{β k} E^{2} [D_{M l}^{*}]$

We have

E [D_{M b}^{*}] = \frac{α - β k C}{4}

E [D_{M m}^{*}] = \frac{α - β k g C}{2 (g + 1)}

E [D_{M l}^{*}] = \frac{α - β k g C}{4}

Comparison between Bank Financing and Guarantor Financing

One may wonder whether the benefit of guarantor financing still holds for the retailer when the latter has access to bank financing in a Stackelberg game. The following comparison result confirms such benefit.

Theorem 3

We compare LG and MG to BF in the manufacturer leadership Stackelberg game.

w_{M m}^{*} > w_{M b}^{*} > w_{M l}^{*}

s_{M l}^{*} > s_{M b}^{*} > s_{M m}^{*}

w_{M m}^{*} + s_{M m}^{*} > w_{M l}^{*} + s_{M l}^{*} > w_{M b}^{*} + s_{M b}^{*}

q_{M m} \geq q_{M b}

if and only if

C \leq \frac{α [2 \hat{q} - (g + 1) \hat{q_{b}}]}{β k [2 g \hat{q} - (g + 1) \hat{q_{b}}]}

, and

q_{M l} \geq q_{M b}

if and only if

C \leq \frac{α (\hat{q} - \hat{q_{b}})}{β k (g \hat{q} - \hat{q_{b}})}

;

The retailer prefers LG to BF (i.e.,

π_{M b}^{R} \leq π_{M l}^{R}

) if and only if

A_{1} < A_{2}

and

C \leq \hat{C_{2}}

; the retailer prefers MG to BF (i.e.,

π_{M b}^{R} \leq π_{M m}^{R}

) if and only if

A_{1} < A_{2}

g < \frac{4 - 3 A_{1} / A_{2} + 2 \sqrt{4 - 3 A_{1} / A_{2}}}{3 A_{1} / A_{2}}

and

C \leq \hat{C_{3}}

;

The manufacturer prefers MG to BF (i.e.,

π_{M b}^{M} \leq π_{M m}^{M}

) if and only if

C \leq \frac{α [\sqrt{2} - \sqrt{(g^{2} + g) \hat{q_{b}} / \hat{q}}]}{β k [\sqrt{2} g - \sqrt{(g^{2} + g)} \hat{q_{b}} / \hat{q}]}

; The manufacturer prefers LG to BF (i.e.,

π_{M b}^{M} \leq π_{M l}^{M}

) if and only if

C \leq \frac{α (1 - \sqrt{g \hat{q_{b}} / \hat{q}})}{β k (g - \sqrt{g \hat{q_{b}} / \hat{q}})}

;

The 3PL prefers LG to BF (i.e.,

π_{M b}^{L} \leq π_{M l}^{L}

) if and only if

C \leq \frac{α (1 - \sqrt{g \hat{q_{b}} / \hat{q}})}{β k (g - \sqrt{g \hat{q_{b}} / \hat{q}})}

; the 3PL prefers MG to BF (i.e.,

π_{M b}^{L} \leq π_{M m}^{L}

) if and only if

C \leq \frac{α [2 - (g + 1) \sqrt{\hat{q_{b}} / \hat{q}}]}{β k [2 g - (g + 1) \sqrt{\hat{q_{b}} / \hat{q}}]}

The threshold

\hat{C_{2}}

is the positive solution of

3 A^{2} Q_{5} α^{2} - 2 A Q_{6} α β k \hat{C_{2}} - Q_{7} β^{2} k^{2} {\hat{C_{2}}}^{2} = 0

, where

Q_{5} \equiv A_{2} - A_{1}

Q_{6} \equiv (1 + r_{f}) \hat{q} A_{2} - A A_{1}

Q_{7} \equiv {(1 + r_{f})}^{2} {\hat{q}}^{2} A_{2} - A^{2} A_{1}

. Moreover,

\hat{C_{3}}

is the positive solution of

- Q_{8} α^{2} + 2 Q_{9} α β k \hat{C_{3}} + Q_{9} β^{2} k^{2} {\hat{C_{3}}}^{2} = 0

, where

Q_{8} \equiv 3 {((1 + r_{f}) \hat{q} + A)}^{2} A_{1} - 4 A (2 (1 + r_{f}) \hat{q} + A) A_{2}

Q_{9} \equiv {((1 + r_{f}) \hat{q} + A)}^{2} A_{1} - 4 {(1 + r_{f})}^{2} {\hat{q}}^{2} A_{2}

\hat{C_{3}} < \hat{C_{2}}

Theorem 3 indicates that the advantage of guarantor financing relative to bank financing still conditionally holds. In other words, supply chain members may still favor guarantor financing even though bank financing is viable. We compare the thresholds and find that the condition for benefitting from a guarantor role in the Stackelberg game is less strict than that in a Nash game. In other words, guarantor financing becomes more attractive to upstream firms under the Stackelberg game. As the guarantor, the manufacturer can charge a higher risk premium and hence exploits its leadership position better than that in the Nash game. As the follower, the 3PL suffers for being forced by the leader to charge a lower service price, but the guarantor role enables the 3PL to earn extra revenue.

Follower–Guarantor Advantage

We also investigate which upstream firm is preferable to be the guarantor under the manufacturer leadership Stackelberg game.

Theorem 4

We compare LG to MG in the manufacturer leadership Stackelberg game.

w_{M l}^{*} < w_{M m}^{*}

s_{M l}^{*} > s_{M m}^{*}

E [D_{M l}^{*}] > E [D_{M m}^{*}]

, and

q_{M l}^{*} > q_{M m}^{*}

;

Every supply chain member prefers LG to MG (i.e.,

π_{M m}^{R} < π_{M l}^{R}

π_{M m}^{L} < π_{M l}^{L}

π_{M m}^{M} < π_{M l}^{M}

π_{M m}^{SC} < π_{M l}^{SC}

The preferences of all supply chain members are unexpectedly the same under the Stackelberg game, which means that all supply chain members earn more profit from LG than from MG. Hence, the manufacturer leadership Stackelberg game consolidates all firm preferences on the guarantor type and avoids the free‐rider dilemma in the Nash game.

In the Stackelberg game, the guarantor, which is the manufacturer or the 3PL, charges a higher price for the risk it takes (i.e.,

w_{M l}^{*} < w_{M m}^{*}

s_{M l}^{*} > s_{M m}^{*}

). When the 3PL (follower) acts as the guarantor in LG, it is constrained by the leader from charging an exceedingly high‐risk premium. In contrast, when the manufacturer (leader) acts as the guarantor in MG, the manufacturer possesses the first mover advantage and can then command a higher risk premium. Thus, the joint value of the wholesale price and logistic service rate is higher in MG than in LG (i.e.,

w_{M l}^{*} + s_{M l}^{*} < w_{M m}^{*} + s_{M m}^{*}

). Consequently, the higher risk premium in MG undermines the order quantity (demand effect) and corrodes the total supply chain profit than that in LG. The retailer’s preference for LG is straightforward because the retailer’s profit predominantly relies on the demand level. Accordingly, the retailer prefers LG to MG.

The preference of the manufacturer is also intuitive. Besides the free‐rider benefit discussed in Theorem 2 of Nash game that exempts the manufacturer from the guarantor financial risk, the manufacturer enjoys a higher order quantity when the 3PL is the guarantor. Therefore, the manufacturer also prefers LG to MG.

However, the preference of the 3PL is unexpected, which means that the 3PL favors the guarantor role over the free‐rider role. Three facets exist in the trade‐off. Under the manufacturer leadership Stackelberg game, order quantity is higher in LG than in MG (demand effect). Coupled with a higher revenue margin (revenue‐margin effect) for being the guarantor, the 3PL can overcome the potential default risk loss (financial‐risk effect) to achieve a better performance in LG than in MG.

This finding contradicts the conventional notion that a capital‐constrained retailer might want to seek guarantor financing from the dominant firm (i.e., the Stackelberg leader), given the classic Stackelberg leadership advantage. Theorem 4 cautions on the existence of the “follower–guarantor advantage” for the supply chain in guarantor financing. In this advantage, all members of the supply chain benefit from the follower acting as the guarantor. Thus, the capital‐constrained firm should instead request a dominated player to serve as the guarantor to prevent the Stackelberg leader from charging a higher risk premium for acting as the guarantor. The intertwining forces of these two advantages provide a new perspective on the role of supply chain leadership in guarantor financing. This aspect will be further discussed in the following subsection.

To provide a full picture of firms’ preference of the three financial schemes in the manufacturer Stackelberg game, we summarize the comparisons in Theorems 3 and 4 into the following corollary.

Corollary 2

In the manufacturer leadership Stackelberg game, the firms’ preference sequences of the three financing schemes are as follows:

For the retailer, 1) when

A_{1} < A_{2}

g < \frac{4 - 3 A_{1} / A_{2} + 2 \sqrt{4 - 3 A_{1} / A_{2}}}{3 A_{1} / A_{2}}

and

C \leq \hat{C_{3}}

π_{M l}^{R} > π_{M m}^{R} \geq π_{M b}^{R}

; 2) when

A_{1} < A_{2}

C \leq \hat{C_{2}}

and as long as

g \geq \frac{4 - 3 A_{1} / A_{2} + 2 \sqrt{4 - 3 A_{1} / A_{2}}}{3 A_{1} / A_{2}}

C > \hat{C_{3}}

holds, then

π_{M l}^{R} \geq π_{M b}^{R} > π_{M m}^{R}

; 3) otherwise,

π_{M b}^{R} > π_{M l}^{R} > π_{M m}^{R}

For the manufacturer, 1) when

C \leq \frac{α [\sqrt{2} - \sqrt{(g^{2} + g) \hat{q_{b}} / \hat{q}}]}{β k [\sqrt{2} g - \sqrt{(g^{2} + g)} \hat{q_{b}} / \hat{q}]}

π_{M l}^{M} > π_{M m}^{M} \geq π_{M b}^{M}

; 2) when

\frac{α [\sqrt{2} - \sqrt{(g^{2} + g) \hat{q_{b}} / \hat{q}}]}{β k [\sqrt{2} g - \sqrt{(g^{2} + g)} \hat{q_{b}} / \hat{q}]} < C \leq \frac{α (1 - \sqrt{g \hat{q_{b}} / \hat{q}})}{β k (g - \sqrt{g \hat{q_{b}} / \hat{q}})}

π_{M l}^{M} \geq π_{M b}^{M} > π_{M m}^{M}

; 3) otherwise,

π_{M b}^{M} > π_{M l}^{M} > π_{M m}^{M}

For the 3PL, 1) when

C \leq \frac{α [2 - (g + 1) \sqrt{\hat{q_{b}} / \hat{q}}]}{β k [2 g - (g + 1) \sqrt{\hat{q_{b}} / \hat{q}}]}

π_{M l}^{L} > π_{M m}^{L} \geq π_{M b}^{L}

; 2) when

\frac{α [2 - (g + 1) \sqrt{\hat{q_{b}} / \hat{q}}]}{β k [2 g - (g + 1) \sqrt{\hat{q_{b}} / \hat{q}}]} < C \leq \frac{α (1 - \sqrt{g \hat{q_{b}} / \hat{q}})}{β k (g - \sqrt{g \hat{q_{b}} / \hat{q}})}

π_{M l}^{L} \geq π_{M b}^{L} > π_{M m}^{L}

; 3) otherwise,

π_{M b}^{L} > π_{M l}^{L} > π_{M m}^{L}

The above threshold values can be further summarized in the following table

preference firm	LG> MG≥BF	LG ≥ BF>MG	BF> LG>MG
Retailer	$A_{1} < A_{2}$ , $g < \frac{4 - 3 A_{1} / A_{2} + 2 \sqrt{4 - 3 A_{1} / A_{2}}}{3 A_{1} / A_{2}}$ and $C \leq \hat{C_{3}}$	$A_{1} < A_{2}$ , $C \leq \hat{C_{2}}$ and as long as $g \geq \frac{4 - 3 A_{1} / A_{2} + 2 \sqrt{4 - 3 A_{1} / A_{2}}}{3 A_{1} / A_{2}}$ or $C > \hat{C_{3}}$ holds	Otherwise.
Manufacturer	$C \leq \frac{α [\sqrt{2} - \sqrt{(g^{2} + g) \hat{q_{b}} / \hat{q}}]}{β k [\sqrt{2} g - \sqrt{(g^{2} + g)} \hat{q_{b}} / \hat{q}]}$	$\frac{α [\sqrt{2} - \sqrt{(g^{2} + g) \hat{q_{b}} / \hat{q}}]}{β k [\sqrt{2} g - \sqrt{(g^{2} + g)} \hat{q_{b}} / \hat{q}]} < C \leq \frac{α (1 - \sqrt{g \hat{q_{b}} / \hat{q}})}{β k (g - \sqrt{g \hat{q_{b}} / \hat{q}})}$	$C > \frac{α (1 - \sqrt{g \hat{q_{b}} / \hat{q}})}{β k (g - \sqrt{g \hat{q_{b}} / \hat{q}})}$
3PL	$C \leq \frac{α [2 - (g + 1) \sqrt{\hat{q_{b}} / \hat{q}}]}{β k [2 g - (g + 1) \sqrt{\hat{q_{b}} / \hat{q}}]}$	$\frac{α [2 - (g + 1) \sqrt{\hat{q_{b}} / \hat{q}}]}{β k [2 g - (g + 1) \sqrt{\hat{q_{b}} / \hat{q}}]} < C \leq \frac{α (1 - \sqrt{g \hat{q_{b}} / \hat{q}})}{β k (g - \sqrt{g \hat{q_{b}} / \hat{q}})}$	$C > \frac{α (1 - \sqrt{g \hat{q_{b}} / \hat{q}})}{β k (g - \sqrt{g \hat{q_{b}} / \hat{q}})}$

preference

firm

LG> MG≥BF

LG ≥ BF>MG

BF> LG>MG

Retailer

A_{1} < A_{2}

g < \frac{4 - 3 A_{1} / A_{2} + 2 \sqrt{4 - 3 A_{1} / A_{2}}}{3 A_{1} / A_{2}}

and

C \leq \hat{C_{3}}

A_{1} < A_{2}

C \leq \hat{C_{2}}

and as long as

g \geq \frac{4 - 3 A_{1} / A_{2} + 2 \sqrt{4 - 3 A_{1} / A_{2}}}{3 A_{1} / A_{2}}

C > \hat{C_{3}}

holds

Otherwise.

Manufacturer

C \leq \frac{α [\sqrt{2} - \sqrt{(g^{2} + g) \hat{q_{b}} / \hat{q}}]}{β k [\sqrt{2} g - \sqrt{(g^{2} + g)} \hat{q_{b}} / \hat{q}]}

\frac{α [\sqrt{2} - \sqrt{(g^{2} + g) \hat{q_{b}} / \hat{q}}]}{β k [\sqrt{2} g - \sqrt{(g^{2} + g)} \hat{q_{b}} / \hat{q}]} < C \leq \frac{α (1 - \sqrt{g \hat{q_{b}} / \hat{q}})}{β k (g - \sqrt{g \hat{q_{b}} / \hat{q}})}

C > \frac{α (1 - \sqrt{g \hat{q_{b}} / \hat{q}})}{β k (g - \sqrt{g \hat{q_{b}} / \hat{q}})}

3PL

C \leq \frac{α [2 - (g + 1) \sqrt{\hat{q_{b}} / \hat{q}}]}{β k [2 g - (g + 1) \sqrt{\hat{q_{b}} / \hat{q}}]}

\frac{α [2 - (g + 1) \sqrt{\hat{q_{b}} / \hat{q}}]}{β k [2 g - (g + 1) \sqrt{\hat{q_{b}} / \hat{q}}]} < C \leq \frac{α (1 - \sqrt{g \hat{q_{b}} / \hat{q}})}{β k (g - \sqrt{g \hat{q_{b}} / \hat{q}})}

C > \frac{α (1 - \sqrt{g \hat{q_{b}} / \hat{q}})}{β k (g - \sqrt{g \hat{q_{b}} / \hat{q}})}

The explanation for Corollary 2 is similar to those for Corollary 1, Theorem 3, and Theorem 4 and, thus, skipped for a concise exposition.

Interaction between Guarantor and Leadership Roles

The previous section introduces the follower–guarantor advantage, wherein the supply chain benefits from letting the follower be the guarantor. However, is being the guarantor always beneficial to the follower? To address this concern, we hypothetically propose two new Stackelberg games only in this subsection for further discussion: MG and LG with 3PL Stackelberg leadership. The inclusion of these two games facilitates our discussion, but is not required (without these two games, the same qualitative conclusion can be achieved by equating firms’ operational costs for fair comparison). In practice, a 3PL firm may emerge as the leader of a supply chain, such as the orchestrator role of Eternal Asia in its one‐stop logistic system (Chen et al. 2018). Besides the original MG and LG with manufacturer leadership, we present a symmetric framework that allows the comparison of the firm’s profits among different leadership roles (leader vs. follower) and guarantor roles (guarantor vs. non‐guarantor/free‐rider). This analysis also enables the comparison between leadership advantage and the follower–guarantor advantage.

Theorem 5

Comparison of different combinations of leadership roles and guarantor roles results in the following outcomes.

Guarantor role preference:

Leader: the free‐rider role is better than the guarantor role (i.e.,

π_{M l}^{M} > π_{M m}^{M}

);

Follower: the guarantor role is better than the free‐rider role (i.e.,

π_{L l}^{M} < π_{L m}^{M}

Leadership role preference:

Free‐rider: the leadership role is better than the follower role;

Guarantor: the leadership role is better if

g \leq 3

(i.e.,

π_{M m}^{M} (= π_{L l}^{L}) \geq π_{L m}^{M} (= π_{M l}^{L})

); otherwise, the follower role is better (i.e.,

π_{M m}^{M} (= π_{L l}^{L}) < π_{L m}^{M} (= π_{M l}^{L})

Theorem 5 delivers several intriguing messages about firms’ preferences for different leadership roles and guarantor roles. First, if a firm starts as a leader/follower, its preference of a guarantor role is clear. The Stackelberg leader is known to have the leadership advantage, whereas the follower has the follower–guarantor advantage. In contrast, the guarantor is required to shoulder the retailer’s default risk. The leader will never choose to be a guarantor by forfeiting the benefit of the free‐rider role (Item 1.a), because the leader cannot benefit from the follower–guarantor advantage. In contrast, the follower constantly benefits from claiming the follower–guarantor advantage (Item 1.b). The follower may benefit from the free‐rider role, but such a benefit alone (without the leadership advantage) cannot exceed the follower–guarantor advantage. Conversely, a non‐guarantor free‐rider cannot claim the follower–guarantor advantage and, thus, strongly prefers to acquire the leadership advantage (Item 2.a). Consequently, as shown by the analysis, the dual role of leader and free‐rider is often the most attractive choice, whereas simultaneously being the follower and free‐rider is the worst.

The second major message indicates that the guarantor must choose between the leadership advantage and the follower–guarantor advantage (Item 2.b). This message is more intriguing than the first one. In particular, as the Stackelberg leader, the guarantor enjoys the dominant power, but is deprived of the follower–guarantor advantage. In contrast, a follower–guarantor misses the Stackelberg leadership advantage. We find that the guarantor’s preference of a leadership role depends on the financial risk level. The Stackelberg leader has a large room to increase its own profit margin when the guarantor’s financial risk level is sufficiently low (i.e.,

g \leq 3

). In this situation, the Stackelberg leadership advantage is more apparent because the leader can more easily transfer the pricing pressure to the follower; thus, the guarantor prefers to be the leader. However, as the financial risk level elevates (i.e.,

g > 3

), the follower–guarantor refuses to compromise by marking down the price, but instead increases its charge as the leader increases the price. As a result, the leader is considerably restricted from exploiting the follower, thereby alleviating pressure on the follower–guarantor. Moreover, allowing the guarantor to assume the follower role generates a higher order quantity for the supply chain (see Theorem 4 and its discussion). In the trade‐off, the follower–guarantor advantage stands out and, thus, the guarantor prefers to be the follower.

The guarantor’s choice of the leadership role also reflects the battle between the leadership advantage and the follower–guarantor advantage. Theorem 5 reveals that these two advantages vary over the financial risk level (i.e., the retailer’s default level). Overall, if the demand variance is high or the retailer is likely to default, the guarantor would benefit more as the follower than as the leader.

Comparison of Leadership Structures

This subsection conceptually supposes that the firm can choose between different leadership structures by comparing the Nash game with the Stackelberg game to explore whether a shorter decision hierarchy is better than a longer decision hierarchy. There is a perception that a shorter decision hierarchy is more efficient than a longer one. Similarly, in the context of this study, because of its shorter decision hierarchy, a Nash game might be considered more efficient than a Stackelberg game for the whole supply chain. We find this perception is true under bank financing, but the outcome could be reversed under guarantor financing.

Leadership Structure Analysis under Bank Financing

This subsection shows that the comparison result in bank financing without a guarantor’s influence fits the conventional wisdom where a supply chain with a shorter decision hierarchy is more efficient than that with a longer decision hierarchy. The comparison result is given as follows.

Theorem 6

In bank financing, the retailer, 3PL, and supply chain prefer the Nash game (i.e.,

π_{M b}^{R} < π_{N b}^{R}

π_{M b}^{L} < π_{N b}^{L}

and

π_{M b}^{S C} < π_{N b}^{S C}

), whereas the manufacturer prefers the manufacturer leadership game (i.e.,

π_{N b}^{M} < π_{M b}^{M}

The underlying rationale for Theorem 6 lies in the triple marginalization in the manufacturer Stackelberg leadership game and the manufacturer leadership advantage. First, typically, a supply chain with a shorter supply chain decision hierarchy (in the Nash game) is more efficient than a longer one (in the Stackelberg game), because the triple marginalization of a longer supply chain decision hierarchy leads to a lower order quantity than that of a shorter decision hierarchy.

Second, being a Stackelberg leader in bank financing is the best choice for upstream firms (

π_{N b}^{M} < π_{M b}^{M}

), whereas being a follower is the worst (

π_{M b}^{L} < π_{N b}^{L}

). In other words, the first mover has advantages over the second mover. Different from that in a Nash game, the Stackelberg leader can transfer the price pressure to the follower at the follower’s expense.

Overall, the retailer, the 3PL, and the whole supply chain all suffer from the aforementioned worse triple marginalization, and the retailer and the 3PL further suffer from the follower disadvantages in the Stackelberg game. In other words, the whole supply chain benefits from a shorter decision hierarchy (i.e., the Nash game is preferred over the Stackelberg game). In contrast, for the manufacturer, the leadership advantage outpaces the triple marginalization harm, so the manufacturer leadership Stackelberg game is preferred.

Leadership Structure Analysis under Manufacturer Guarantor Financing

We obtain the following result by comparing the performance of MG between the Nash game and the manufacturer leadership Stackelberg game.

Theorem 7

In MG, the retailer, 3PL, and supply chain prefer the Nash game (i.e.,

π_{M m}^{R} < π_{N m}^{R}

π_{M m}^{L} < π_{N m}^{L}

π_{M m}^{S C} < π_{N m}^{S C}

), whereas the manufacturer prefers the manufacturer leadership Stackelberg game (i.e.,

π_{N m}^{M} < π_{M m}^{M}

The conventional wisdom in Theorem 6 sustains under MG. In other words, the triple marginalization and follower disadvantages in the Stackelberg game continue to hurt the retailer, the 3PL, and the whole supply chain; as a result, they all prefer the Nash game. In fact, as the Stackelberg leader and guarantor, the manufacturer bears the retailer’s potential default risk and, thus, charges a higher risk premium than that under the Nash game and earn more profit, but at the expense of other firms. Correspondingly, the negative impact of triple marginalization is worsened, as the leadership advantage is widened, in the Stackelberg game, such that the state of the retailer, the 3PL, and the whole supply chain is aggravated as compared to that under BF, whereas the manufacturer is better off. As a result, the profit gaps between the Nash game and the Stackelberg game for all firms are indeed even wider under MG than under BF.

Leadership Structure Analysis under 3PL Guarantor Financing

We compare the performance of LG under different leadership structures and achieve the following result.

Theorem 8

In LG, if

g < 2

, the retailer, 3PL, and supply chain prefer the Nash game (i.e.,

π_{M l}^{R} < π_{N l}^{R}

π_{M l}^{L} < π_{N l}^{L}

and

π_{M b}^{S C} < π_{N b}^{S C}

); otherwise, they prefer the manufacturer leadership Stackelberg game. The manufacturer constantly prefers the manufacturer leadership game (i.e.,

π_{M l}^{M} \geq π_{N l}^{M}

Unlike the findings in BF and MG, Theorem 8 reveals a different outcome that under LG the retailer, 3PL, and supply chain conditionally prefer a longer decision hierarchy (i.e., Stackelberg game) than a shorter one (i.e., Nash game). Note that the triple marginalization and follower disadvantages persist in the Stackelberg game. However, the trade‐off is subtler when the 3PL provides the guarantor financing. For the manufacturer, the leadership advantage continues to dominate the triple marginalization harm, so it still prefers the manufacturer leadership Stackelberg game. But, for the 3PL, the retailer, and the whole supply chain, the follower–guarantor advantage starts to play a critical role under certain conditions.

In particular, if the financial risk is sufficiently low (i.e.,

g < 2

), the 3PL, as the risk bearer, becomes willing to boost the demand by reducing its service rate when the manufacturer increases wholesale price. In the Stackelberg game, the manufacturer has incentives to utilize its first‐mover advantage to charge a higher wholesale price. In contrast, in the Nash game, the manufacturer and the 3PL’s pricing strategies are more balanced and constrained due to their simultaneous moves. When financial risk is sufficiently low (i.e.,

g < 2

), the benefit of a higher order surpasses the disadvantage of a lower profit margin. Thus, the 3PL is willing to compromise and reduce its service rate to maintain a decent sales volume. Nevertheless, the manufacturer takes advantage of the 3PL and charges an exceedingly high wholesale price such that the joint wholesale and logistic price is higher than in the Nash game (i.e.,

w_{N l}^{*} + s_{N l}^{*} < w_{M l}^{*} + s_{M l}^{*}

). Thus, the retail price is higher, but the order quantity is lower in the Stackelberg game than in the Nash game. Consequently, the retailer and 3PL prefer the Nash game to the Stackelberg game. Although the manufacturer enjoys a higher profit margin in the Stackelberg game, the manufacturer’s gain cannot compensate for the joint loss of the retailer and the 3PL. Ultimately, the entire supply chain also prefers the Nash game.

When financial risk is substantially high (i.e.,

g \geq 2

), the 3PL prefers a lower order quantity to subdue the retailer’s default risk. In this situation, the manufacturer is aware of that the 3PL will not compromise by reducing, but instead increasing its service rate along with the manufacturer’s wholesale price. Given that the order quantity, and thus all firm profits, will be highly sensitive to the wholesale price increase, the manufacturer, as the Stackelberg leader, becomes highly cautious about raising its wholesale price, because the wholesale price increase will remarkably jeopardize the retailer’s order quantity. Overall, the manufacturer becomes more sensitive to its own wholesale price increase in the Stackelberg game than in a Nash game. As a result, the wholesale price in the Stackelberg game becomes lower than that in the Nash game (i.e.,

w_{N l}^{*} \geq w_{M l}^{*}

), and the joint wholesale and logistic price is lower in the Stackelberg game than in the Nash game (i.e.,

w_{N l}^{*} + s_{N l}^{*} \geq w_{M l}^{*} + s_{M l}^{*}

). These outcomes lead to a lower retail price and higher order quantity in the manufacturer leadership Stackelberg game than in the Nash game. Therefore, the retailer, 3PL, and whole supply chain, as well as the manufacturer, benefit from the manufacturer’s leadership and, hence, prefer a longer decision hierarchy.

Figure 2 further illustrates the impact of the financial risk level on the supply chain’s preference of leadership structure. The parameter values are the same as those in Figure 1 except that we set

c_{m} = 5

. We use demand variance as the horizontal axis in Figure 2, because

g

measures the level of financial risk and increases with demand variance. We also assume the demand

d

follows a gamma distribution with a mean of 1. Overall, the supply chain profit decreases with the variance. If the variance is sufficiently large, the supply chain profit under the Stackelberg game dominates that of the Nash game; otherwise, the Nash game dominates. The benefit of a longer decision hierarchy increases as the variance level grows.

FIGURE 2

Supply Chain Profits under LG with Respect to Demand Variance Given

c_{m} = 5

Extensions

We explore three extensions, namely, impact of cost structure, impact of retail pricing rules, and impact of initial capital.

Impact of Cost Structure with Economies of Scale

We generalize the cost structure of upstream firms to examine the impact of cost structure. We assume that the manufacturer’s and the 3PL’s total operational costs are in the form of

C_{i} (q) = \frac{e_{i}}{2} q^{2} + c_{i} q

i = m, l

, following the recommendations in Ha et al. (2011). The cost function consists of a linear term

c_{i} q

and a quadratic term

\frac{e_{i}}{2} q^{2}

, where

e_{i}

denotes (dis)economies of scale. Diseconomies of scale emerge when

e_{i} > 0

. A larger

e_{i}

reflects greater diseconomies of scale. On the contrary, when

e_{i} < 0

, the manufacturer or the 3PL benefits from economies of scale. When

e_{i} = 0

, the manufacturing cost or the logistic cost is degenerated into the linear form, which is used in the main body of this article.

We further assume that

C_{i}^{'} (q) > 0

, which means

e_{i} q + c_{i} > 0

, such that the marginal cost will not be negative. The retailer’s decision is

q = E [D] \hat{q_{b}}

under BF, and

q = E [D] \hat{q}

under both guarantor financings. Therefore, the condition of

e_{i} q + c_{i} > 0

becomes

e_{i} > - \frac{c_{i}}{E [D] \hat{q_{b}}}

and

e_{i} > - \frac{c_{i}}{E [D] \hat{q}}

for any

E [D]

in BF and guarantor financing, respectively. Conditions

e_{i} > - \frac{c_{i}}{α \hat{q_{b}}}

and

e_{i} > - \frac{c_{i}}{α \hat{q}}

are required for positive demand because the upper bound of

E [D]

α

. We assume

e_{i} > - \frac{c_{i}}{α \hat{q}}

, which is a stricter constraint, for the remaining portion of this subsection.

By comparing MG/LG and BF under both leadership structures, we find that guarantor financing is more attractive than bank financing when upstream firms are sufficiently cost‐efficient (i.e., upstream firms do not have severe diseconomies of scale). The result is consistent with our former finding that MG/LG can be better than BF when

C

is sufficiently small.

Our analysis shows that the preference between MG and LG by different firms remains unchanged in the Nash game. In the Stackelberg game, the 3PL’s preference relies on the (dis)economy scale level, although the preferences of the manufacturer, the retailer, and the whole supply chain remain independent of the (dis)economy scale level. We state these findings in Theorem 9, which is an extension of Theorem 4.

Theorem 9 Extension of Theorem 4

Under the manufacturer leadership Stackelberg game:

w_{M l}^{*} < w_{M m}^{*}

s_{M l}^{*} > s_{M m}^{*}

E [D_{M l}^{*}] > E [D_{M m}^{*}]

, and

q_{M l}^{*} > q_{M m}^{*}

The manufacturer, the retailer, and the whole supply chain prefer LG to MG (i.e.,

π_{M m}^{R} < π_{M l}^{R}

π_{M m}^{M} < π_{M l}^{M}

π_{M m}^{S C} < π_{M l}^{S C}

);

For the 3PL, given

e_{l}

, a threshold value of

\hat{e_{m}} (e_{l}) > 0

exists such that if

e_{m} \leq \hat{e_{m}} (e_{l})

, then the 3PL prefers LG to MG (i.e.,

π_{M l}^{L} \geq π_{M m}^{L})

; otherwise, the 3PL prefers MG (i.e.,

π_{M l}^{L} < π_{M m}^{L}

Theorem 9 reveals that the 3PL does not always prefer LG to MG under manufacturer leadership. This result is different from the case of the linear cost structure without considering the (dis)economy scale level. The rationale behind this finding is as follows. As previously discussed, the 3PL faces a three‐facet trade‐off for serving as the guarantor in LG. The three facets include positive demand effect, positive revenue effect, and negative financial‐risk effect. When the manufacturer has substantial diseconomies of scale (i.e.,

e_{m} > \hat{e_{m}} (e_{l})

), the manufacturer charges a disproportionally high wholesale price, thereby squeezing the profit margin from the 3PL. Consequently, the revenue effect and the demand effect are undermined and fail to offset the default risk loss (financial‐risk effect). As a result, the 3PL prefers not to be the guarantor when

e_{m} > \hat{e_{m}} (e_{l})

We also find that the retailer and the 3PL’s preferences of different supply chain decision structures are affected by the (dis)economies of scales, although the manufacturer’s preference remains unchanged, as displayed below.

Theorem 10 Extension of Theorem 8

Consider LG.

The retailer, the 3PL, and the supply chain prefer the Nash game (i.e.,

π_{M l}^{R} < π_{N l}^{R}

π_{M l}^{L} < π_{N l}^{L}

and

π_{M b}^{S C} < π_{N b}^{S C}

) if

g (1 - β k \hat{q} e_{l}) < 2

; otherwise, they prefer the manufacturer leadership game.

The manufacturer prefers the manufacturer leadership game (i.e.,

π_{M l}^{M} \geq π_{N l}^{M}

The comparison between Theorem 10 and Theorem 8 shows that the impact of (dis)economies of scales on the retailer’s and 3PL’s preferences is not qualitative, but only quantitative. The difference is that, if the follower 3PL in the Stackelberg game has economies of scale (

e_{l} < 0

), the manufacturer leadership Stackelberg game would likely outperform the Nash game; otherwise, the opposite becomes true. Cost efficiency provides the 3PL with additional confidence to increase the price and deter the leader from setting a high price. This finding means that the 3PL begins to price high in response to the high price of the manufacturer when

g (1 - β k \hat{q} e_{l}) \geq 2

as compared to

g \geq 2

when no economies of scale is present (i.e.,

e_{l} = 0

We then conclude that extending the baseline model with a unit cost to a more general cost structure does not significantly alter our main findings. Our main qualitative findings (e.g., the observations on the follower–guarantor advantage and the higher performance in the supply chain with a longer decision hierarchy than in the supply chain with a shorter decision hierarchy) continue to hold as long as the upstream firms’ (dis)economies of scale are contained.

Impact of Retail Pricing Rules

Our baseline model assumes a markup retail price scheme to reflect the reality that the retailer will likely adjust its retail price as the wholesale price varies. Markup retail price can be considered an intermediate case between an exogenous retail price and an endogenous one. Nevertheless, numerically we observe that markup retail pricing can conditionally outperform the endogenous retail pricing; this occurs because a fixed markup prevents the upstream firms from increasing their prices, such that the triple marginalization problem is softened. This numerical finding might partially justify the use of markup retail pricing in practice. To study the influence of different pricing assumptions, we discuss guarantor financing practices under exogenous and endogenous pricing schemes. We start with the exogenous price as follows.

Impact of Exogenous Pricing

Given that reality is uncertain and evolves quickly, the assumption of an exogenous retail price is an extreme scenario. The condition where any retailer will never increase its retail price is hardly true even if the wholesale price skyrockets. Nevertheless, the exogenous retail price assumption has been used often in newsvendor models for tractability. Thus, we examine firm performance in guarantor financing with an exogenous retail price.

Corollary 3

If the retail price is exogenous, the Stackelberg leader under manufacturer leadership in MG and LG sets the prices as high as possible and obtains all unreserved supply chain profit, whereas in the Nash game, the two upstream firms share the profit. In both game settings, the retailer only earns the reserved profit.

The result of Corollary 3 is expected because similar results have been reported in the extant literature on newsvendor Stackelberg games. Given that the retail price is fixed, the expected demand is constant. Under manufacturer leadership, the Stackelberg leader has the incentive to hike the price/rate and extracts all extra unreserved profit from the followers. If the other upstream firm is a guarantor, the firm will earn positive revenues, but its expected profit is zero due to the financial risk cost incurred by the retailer’s potential default. At the bottom of the supply chain, the retailer is powerless, especially when the retailer borrows from the bank under either guarantor financing scheme. Therefore, the retailer is indifferent between MG and LG. This result further confirms that not every retailer can benefit from guarantor financing, particularly if it is deprived of the power of retail pricing.

Impact of Endogenous Pricing

We then examine endogenous retail pricing. We firstly consider the assumption that price decision is made before demand realization. In this scenario, the retailer must simultaneously decide on the retail price and order quantity, which is analytically intractable. Employing the same numerical experiment setting as former ones, we can demonstrate that under the manufacturer leadership Stackelberg game, all supply chain members, except 3PL, prefer LG to MG (see Figure 3). The 3PL prefers to be the guarantor only if

c_{m}

is sufficiently small (i.e., the Stackelberg leader is cost efficient); otherwise, the manufacturer will extract the profit margin from the 3PL and make LG less attractive for the 3PL.

FIGURE 3

Profits of Different Financing Schemes under Endogenous Pricing and Manufacturer Leadership Stackelberg Game with Respect to the Manufacturer’s Cost

c_{m}

Similar to that in our markup pricing setting, guarantor financing is more attractive than bank financing for the upstream firms and supply chain when the upstream firms are cost efficient. However, the retailer performs better in bank financing because the former gains more discretion under the assumption of endogenous pricing than under exogenous pricing.

We also explore a different endogenous pricing setting by allowing the retailer to decide the retail price after demand realization, an assumption like that in Kouvelis, et al. (2018). Although the firms’ preferences are very similar to those when the retailer decides the retail price before demand realization, either the manufacturer or the 3PL might prefer to be the guarantor, because they can earn a higher profit margin when the retailer can price after demand realization. Given the complexity of this extension, this topic on endogenous retail pricing—either before or after demand realization—remains a top research priority.

Impact of Initial Capital

For tractability, we assume zero initial capital for the retailer in our baseline model. For a model that includes a manufacturer, a 3PL, a retailer, and a bank, the analysis instantly becomes intractable when the retailer’s initial capital is positive. As Kouvelis and Zhao (2012) show, optimal decisions are mostly determined by implicit functions for a two‐party supply chain model with a positive initial capital. A similar analytical impasse occurs in the two‐firm supply chain models analyzed by Jing et al. (2012). To shed light on the impact of the initial capital on our main results, we only employ numerical experiments to examine the impact of the initial capital on the optimal solutions of different financing schemes.

Figure 4 demonstrates how the firms’ profits change over the initial capital level in the manufacturer leadership Stackelberg game. First, the profits of the manufacturer under MG and the 3PL under LG initially increase and then decrease with initial capital within certain ranges. We explain the phenomenon from the trade‐off between the aforementioned financial‐risk effect, revenue margin, and demand effect. Notably, the retailer may become more risk‐seeking by ordering more when its initial capital is lower, because its loss is limited up to its entire initial capital. As shown in Figure 5a, the retailer tends to order less as the initial capital grows and, thus, the financial‐risk effect, revenue margin, and demand effect for the guarantor in either LG or MG diminish. Initially, the financial risk can be more substantially subdued when the initial capital is lower (vs. when the initial capital is higher). Thus, the benefit of financial risk reduction effectively outplays the reduced revenue‐margin effect [e.g., see the wholesale price (

w

) of MG in Figure 5b] and demand effect, such that the guarantor’s profit first increases as the retailer’s initial capital increases. However, when the initial capital level is sufficiently high, the financial risk reduction is slowed down, whereas the revenue margin and demand are continuously given away (i.e., lower prices and smaller order). Thus, the guarantors’ profits decrease as the initial capital continuously increases. For the non‐guarantor in MG and LG, the retailer needs less financial support from upstream firms as the initial capital grows and, thus, surplus from free‐riding guarantor financing diminishes.

FIGURE 4

Profits of Different Financing Schemes under Manufacturer Leadership Stackelberg Game with Respect to the Initial Capital Given

c_{m} = 5

FIGURE 5

Firms’ Optimal Decisions under Different Financing Schemes under Manufacturer Leadership Stackelberg Game with Respect to the Initial Capital Given

c_{m} = 5

We previously find that LG is always better than MG for every supply chain member when the retailer has no initial capital. The result still holds when the initial capital is not sufficiently high. On the other extreme, when the initial capital is abundant (no capital‐constrained), the retailer no longer requires external financing; therefore, no difference exists between these three different financing schemes. When the retailer’s initial capital is slightly insufficient (i.e., the retailer is only slightly capital constrained), the retailer does not borrow immediately because of the non‐zero risk‐free interest rate. In this situation, the retailer prefers to order a slightly decreased quantity under no financing to avoid any potential financing cost.

However, we observe that under the manufacturer leadership Stackelberg game, MG can outperform LG for every supply chain member when the initial capital is at the moderate level where the retailer still needs financing. Given that the manufacturer’s financial risk in MG is considerably diminished due to the substantial initial capital, the manufacturer is willing to reduce the wholesale price in exchange for an increased order quantity from the retailer [see Figures 5a and b]. Given that the manufacturer is the Stackelberg leader, its impact in MG is greater than that in LG. This observation indicates that the follower–guarantor advantage scales down as the initial capital builds up. In this situation, MG outperforms LG for all firms.

Conclusions

This study investigates two schemes of guarantor financing, namely, MG and LG, in a four‐party supply chain game that features a manufacturer, a 3PL, a capital‐constrained retailer, and a bank. We first analyze the MG, LG, and BF in a game where the two upstream firms decide simultaneously (Nash game). We find that the supply chain members may still choose MG or LG over BF, provided that the BF is viable for the retailer. However, a free‐rider dilemma exists in the Nash game, wherein both upstream firms prefer the other firm to be the guarantor. We then observe that such free‐rider dilemma can be resolved if the manufacturer emerges as the Stackelberg leader in a Stackelberg game between the two upstream firms, such that all firms prefer the follower assuming the guarantor role. This finding reveals the follower–guarantor advantage, which can thoroughly compensate for the Stackelberg follower disadvantage. This finding means that the follower may prefer to bear the financial risk as the guarantor. In fact, the follower–guarantor advantage can be so strong that a guarantor might decline the Stackelberg leadership despite being given such an opportunity. Our extended discussion shows that the follower–guarantor advantage decreases with the retailer’s initial capital level. Overall, the Stackelberg leadership raises the probability that guarantor financing outperforms bank financing. Moreover, by comparing the firms’ performances among the two game settings (i.e., Nash vs. Stackelberg), we observe that a supply chain with a longer decision hierarchy (i.e., Stackelberg) can conditionally outshine that with a shorter decision hierarchy (i.e., Nash). This outcome exhibits the benefit of having a focal firm as the Stackelberg leader in the supply chain.

Our results deliver several managerial implications. First, both guarantor financing and bank financing can outperform each other. Thus, a capital‐constrained firm should be further selective in financing scheme than retaining bank financing alone. Second, when multiple upstream firms, such as the manufacturer and the 3PL, are present, the capital‐constrained retailer should consider requesting for guarantor financing from the less dominant firm along the supply chain (i.e., the follower) instead of the most dominant firm (i.e., leader). Finally, under guarantor financing, a longer decision hierarchy can be better than a shorter one. Therefore, encouraging a focal firm to serve as the Stackelberg leader of the supply chain may benefit all firms.

This study is the first attempt to address manufacturer guarantor financing and 3PL guarantor financing in a four‐party game with two types of leadership structures. However, this study has some limitations due to the analytical complexity caused by the four‐party interaction. First, we assume common knowledge about demand uncertainty as applied in most related literature. Nevertheless, exploring these financing schemes is attractive from the perspective of the incentive theory. For example, demand uncertainty becomes incomplete information to upper stream firms. Second, we adopt markup retail pricing because of its popularity in practice and tractability, but our future research priority is to investigate the impact of other potential retail pricing strategies. Third, consistent with extant literature, we assume that the retailer will maximize its profit. In practice, the retailer may decide to maximize the probability of making a predetermined profit. If profit maximization is correlated with maximizing the probability of meeting a profit target, we suspect that our qualitative results will hold. Fourth, we realize that our outcomes can differ if the demand function is converted from a multiplicative form to an additive form. Although the analysis of an additive form becomes extremely challenging in our model setting, this direction remains a priority for our future research. Fifth, our main analysis assumes zero initial capital for tractability. However, we find that the initial capital level can influence the firms’ advantages in extensions. Therefore, this issue must be addressed in a tractable, but potentially simplified model in the future. Finally, examining the impact of salvage value, stock‐out penalty, and supply chain coordination is another priority topic for further investigation.

Footnotes

Acknowledgments

The authors are listed in reverse alphabetical order and contributed equally to this work. They are co‐first authors and co‐corresponding authors. We thank the department editor, Eric Johnson, the senior editor, and the two anonymous referees for their very insightful and constructive comments. Weihua Zhou acknowledges the support from the National Natural Science Foundation of China (Grants no. U1509221, 71571160) and National Key R&D Program of China (No. 2019YFB1404901). Gangshu Cai acknowledges the support from the National Natural Science Foundation of China through grant #71629001 and Santa Clara University research scholarship programs.

ORCID

Weihua Zhou

References

Babich

Sobel

M. J.

. 2004. Pre‐IPO operational and financial decisions. Management Sci. 50(7): 935–948.

Biery

M. E.

2015. These industries generate the highest profit margins. Sageworks Stats. Available at https://www.forbes.com/sites/sageworks/2015/09/06/these-industries-generate-the-highest-profit-margins/#181e408e310c (accessed date June 10, 2017).

Bond

R. L.

2008. Finding the right price for your retail product. Entrepreneur. Available at https://www.entrepreneur.com/article/193986 (accessed date June 10, 2017.

Brander

J. A.

Lewis

T. R.

. 1986. Oligopoly and financial structure: The limited liability effect. Am. Econ. Rev. 76(5): 956–970.

Brennan

M. J.

Maksimovic

Zechner

. 1988. Vendor financing. J. Finan. 43(5): 1127–1141.

Burkart

Ellingsen

. 2004. In‐kind finance: A theory of trade credit. Am. Econ. Rev. 94(3): 569–590.

Buzacott

1975. Economic order quantities with inflation. Oper. Res. Quart. 26(3): 553–558.

Buzacott

J. A.

Zhang

R. Q.

. 2004. Inventory management with asset‐based financing. Management Sci. 50(9): 1274–1292.

Cai

G. G.

Chen

Xiao

. 2014. The roles of bank and trade credits: Theoretical analysis and empirical evidence. Prod. Oper. Manag. 23(4): 583–598.

10.

Carroll

2012. How Fashion Brands Set Prices. Forbes. Available at https://www.forbes.com/sites/matthewcarroll/2012/02/22/how-fashion-brands-set-prices/2/#7c0e60df4fb4 (accessed date June 10, 2017).

11.

Chen

Kök

A. G.

Tong

J. D.

. 2013. The effect of payment schemes on inventory decisions: The role of mental accounting. Management Sci. 59(2): 436–451.

12.

Chen

Cai

G. G.

Song

. 2018. The cash flow advantages of 3PLs as supply chain Orchestrators. Manuf. Serv. Oper. Manag. 22(2): 154–171.

13.

Chen

Cai

G. G.

. 2011. Joint logistics and financial services by a 3PL firm. Eur. J. Oper. Res. 214(3): 579–587.

14.

Chod

2016. Inventory, risk shifting, and trade credit. Management Sci. 63(10): 3207–3225.

15.

Dada

. 2008. Financing newsvendor inventory. Oper. Res. Lett. 36(5): 569–573.

16.

Dammon

R. M.

Senbet

L. W.

. 1988. The effect of taxes and depreciation on corporate investment and financial leverage. J. Finan. 43(2): 357–373.

17.

Deng

Cai

. 2018. Financing multiple heterogeneous suppliers in assembly systems: Buyer finance vs. bank finance. Manuf. Serv. Oper. Manag. 20(1): 53–69.

18.

Eichner

A. S.

1973. A theory of the determination of the mark‐up under oligopoly. Econ. J. 83(332): 1184–1200.

19.

Emmons

Gilbert

S. M.

. 1998. The role of returns policies in pricing and inventory decisions for catalogue goods. Management Sci. 44(2): 276–283.

20.

Greenslade

J. V.

Parker

. 2012. New insights into price‐setting behaviour in the UK: Introduction and survey results. Econ. J. 122(558): F1–F15.

21.

A. Y.

Tong

Zhang

. 2011. Sharing demand information in competing supply chains with production diseconomies. Management Sci. 57(3): 566–581.

22.

Hall

R. L.

Hitch

C. J.

. 1939. Price theory and business behaviour. Oxf. Econ. Pap. 2: 12–45.

23.

Jing

Chen

Cai

G. G.

. 2012. Equilibrium financing in a distribution channel with capital constraint. Prod. Oper. Manag. 21(6): 1090–1101.

24.

Kouvelis

Zhao

. 2012. Financing the newsvendor: Supplier vs. bank, and the structure of optimal trade credit contracts. Oper. Res. 60(3): 566–580.

25.

Kouvelis

Turcic

Zhao

. 2018. Supply chain contracting in environments with volatile input prices and frictions. Manuf. Serv. Oper. Manag. 20(1): 130–146.

26.

Lai

Debo

L. G.

Sycara

. 2009. Sharing inventory risk in supply chain: The implication of financial constraint. Omega 37(4): 811–825.

27.

Lariviere

M. A.

2006. A note on probability distributions with increasing generalized failure rates. Oper. Res. 54(3): 602–604.

28.

Lee

C. H.

Rhee

B.‐D.

. 2011. Trade credit for supply chain coordination. Eur. J. Oper. Res. 214(1): 136–146.

29.

Shubik

Sobel

M. J.

. 2013. Control of dividends, capital subscriptions, and physical inventories. Management Sci. 59(5): 1107–1124.

30.

Modigliani

Miller

M. H.

. 1958. The cost of capital, corporation finance and the theory of investment. Am. Econ. Rev. 48(3): 261–297.

31.

Morrison

C. J.

1992. Unraveling the productivity growth slowdown in the United States, Canada and Japan: The effects of subequilibrium, scale economies and markups. Rev. Econ. Statisti. 74(3): 381–393.

32.

Ning

Babich

. 2017. R&D investments in the presence of knowledge spillover and debt financing: Can risk shifting cure free riding? Manuf. Serv. Oper. Manag. 20(1): 97–112.

33.

Nishimura

K. G.

Ohkusa

Ariga

. 1999. Estimating the mark‐up over marginal cost: A panel analysis of Japanese firms 1971–1994. Int. J. Ind. Organ. 17(8): 1077–1111.

34.

Noble

P. M.

Gruca

T. S.

. 1999. Industrial pricing: Theory and managerial practice. Market. Sci. 18(3): 435–454.

35.

Peura

Yang

S. A.

Lai

. 2017. Trade credit in competition: A horizontal benefit. Manuf. Serv. Oper. Manag. 19(2): 263–289.

36.

Petruzzi

N. C.

Dada

. 1999. Pricing and the newsvendor problem: A review with extensions. Oper. Res. 47(2): 183–194.

37.

Puhui . 2015. Introduction of Puhui Rural Credit Guarantee Company. Available at http://www.puhuichina.com /comcontent_detail/&FrontComContent_list01-001ContId=8&comContentId=8.html (accessed date December 21, 2015).

38.

Silva‐Risso

J. M.

Bucklin

R. E.

Morrison

D. G.

. 1999. A decision support system for planning manufacturers' sales promotion calendars. Market. Sci. 18(3): 274–300.

39.

The World Bank . 2018. Enterprise surveys. Available at http://www.enterprisesurveys.org (accessed date May 18, 2018).

40.

Tunca

T. I.

Zhu

. 2017. Buyer intermediation in supplier finance. Management Sci. 64(12): 5631–5650.

41.

Vilcassim

N. J.

Kadiyali

Chintagunta

P. K.

. 1999. Investigating dynamic multifirm market interactions in price and advertising. Management Sci. 45(4): 499–518.

42.

Winshang.com . 2013. The business model of HeiLanHome: A revenue‐sharing mechanism supporting industry chain alliance. Available at http://news.winshang.com/html/019/9280.html (accessed date July 20, 2017).

43.

2015. SF is speeding up financial layout. Available at http://www.eeo.com.cn/2015/ 0508/275736.shtml, in Chinese, [Accessed: February 10, 2016].

44.

Huang

Chiang

D.‐ M.‐H.

. 2014. Support SME suppliers through buyer‐backed purchase order financing. Available at SSRN 2462521.

45.

. www.cnstock.com . 2013. Eternal Asia plans to expand investment on wholly‐owned subsidiary micro‐credit company. Available at http://www.cnstock.com/gonggaojd/zkyg/201305/2583454.htm (accessed date February 10, 2016).

46.

Birge

J. R.

. 2004. Joint production and financing decisions: Modeling and analysis. Available at SSRN 652562.

47.

Yang

S. A.

Chen

C. J.

Bakshi

. 2016. Cancelability in trade credit insurance. Available at SSRN 2735907.

48.

Zhang

2010. The multi‐product newsboy problem with supplier quantity discounts and a budget constraint. Eur. J. Oper. Res. 206(2): 350–360.