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Int. J. Production Economics 124 (2010) 331–339

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Int. J. Production Economics

0925-52

doi:10.1

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(B.-D. R1 Ch

Busines2 By

Adminis

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

Coordination contracts in the presence of positive inventory financing costs

Chang Hwan Lee 1, Byong-Duk Rhee �,2

Ajou University, 5 Woncheon-dong, Yeongtong-gu, Suwon 443-749, Republic of Korea

a r t i c l e i n f o

Article history:

Received 1 June 2009

Accepted 7 November 2009Available online 26 November 2009

Keywords:

Inventory financing costs

Trade-credit

Supply chain coordination

Newsvendor framework

73/$ - see front matter & 2009 Elsevier B.V. A

016/j.ijpe.2009.11.028

esponding author. Tel.: +82 31 219 3630; fax

ail addresses: chlee@ajou.ac.kr (C.H. Lee), brh

hee).

ang Hwan Lee is a professor of operation

s Administration.

ong-Duk Rhee is a professor of market

tration.

a b s t r a c t

Numerous studies have offered diverse contractual forms of alliance, in which the supply chain partners

coordinate their decisions for greater joint performance in an entirely self-interested way. Prior

literature implicitly assumes free inventory financing. However, this assumption is questionable in the

real marketplace. Firms frequently finance their working capital from a variety of credit sources, such as

banks, and incur positive costs of the funds for inventory. Surprisingly, the impact of inventory financing

costs on supply chain coordination has not been sufficiently investigated by supply chain academics. We

address this issue by explicitly assuming capital-constrained agents and positive inventory financing

costs. Specifically, we consider four extensively discussed coordination mechanisms for investigation: (i)

all-unit quantity discount, (ii) buybacks, (iii) two-part tariff, and (iv) revenue-sharing. We show that,

under the assumption of positive inventory financing costs, these contracts fail to achieve joint profit

maximization if each agent relies on direct financing from a financial institution. Positive financing costs

call for trade-credit in order to subsidize the retailer’s costs of inventory financing. Using trade-credit in

addition to the contracts, the supplier fully coordinates the supply chain for the largest joint profits.

Moreover, positive inventory financing costs make revenue-sharing less profitable than the other three

contracts. We present three different schemes for coordination in a decentralized supply chain, using

buybacks, quantity discount, and two-part tariff, respectively. We also derive the optimal trade-credit

rate not only for the supplier’s profit, but also for joint supply chain profit.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

A supplier and its retailers are independent decision makers.Facing different cost structures and uncertainties in either thewholesale or retail market, each makes strategic decisions inorder to maximize its own individual profit. Such independentprofit maximizations often lead to poor performance of the entiresupply chain. Integrating supply chain agents through merger andacquisition eliminates this problem, and improves performance inthe supply chain (Langabeer and Seifert, 2003). However, it is anarduous task to align the localized objectives of channelparticipants with diverse self-serving orientations (Brouwerset al., 2005). Moreover, this form of vertical integration frequentlyresults in structural inflexibilities and organizational rigidities.

Numerous studies have recently focused on a more flexibleform of contractual alliance, in which the supply chain partnerscoordinate their decisions for greater performance in an entirelyself-interested way. These include buybacks (Pasternack, 1985;

ll rights reserved.

: +82 31 219 1616.

ee@ajou.ac.kr

s management at School of

ing at School of Business

Emmons and Gilbert, 1998; Lee, 2001; Tsay, 2001; Lee and Rhee,2007); quantity discounts (Dada and Srikanth, 1987; Weng, 1995;Chen et al., 2001; Cachon, 2003); revenue-sharing (Cachon, 2003,Cachon and Lariviere, 2005); two-part tariffs (Lariviere, 1999);quantity flexibility contracts (Tsay, 1999); wholesale and retailprice protections (Lee et al., 2000); target-level sales rebates(Taylor, 2002); capacity reservation contracts (Barnes-Schusteret al., 2002), etc. These coordination schemes enable the retailerto share the demand uncertainty with its supplier, and make theretailer order the optimal quantity for joint profit maximization.In other words, the schemes coordinate the channel participants’decisions by alleviating the problem in the agents’ independentprofit maximizations, and yield the same profit as in a fullyintegrated supply chain. See Lariviere (1999) and Cachon (2003)for detailed discussions of diverse coordination contracts.

The prior literature on supply chain coordination does not takeinventory financing costs into account, and implicitly assumesthat a supplier and its retailers incur zero cost of the funds forinventory. This assumption is problematic in the real marketplace.Firms frequently finance their working capital from a variety ofcredit sources, such as banks, and incur positive financing costs.Lenders ask interest for the loans they grant, facing uncertaintyregarding what may happen if the loan is in default. Even in thecase of zero credit default risk, such as with US Treasury bills,lenders still demand positive financing costs, not only due tomarket risk, which is the risk of changes in the overall financial

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market, but also due to liquidity risk, which is the risk of beingunable to sell the collateral for cash without significant cost.

Immediate questions are, then, (i) whether the contracts inprevious studies still fully coordinate the supply chain for jointprofit maximization, in the presence of positive inventoryfinancing costs; (ii) if not, how the terms for coordination shouldbe adjusted and what other incentives should be offered toretailers; and (iii) whether one type of contract is better for thesupply chain than others. We attempt to address these issues byexplicitly assuming capital-constrained agents and positiveinventory financing costs.

Inventory financing costs have been assumed as a part ofinventory holding cost in an EOQ framework. Beranek (1967)assumes a firm’s financial arrangement for inventory and derivesthe optimal lot size. Harley and Higgins (1973) assume trade-credit financing and examine both lot size and payment time ofthe trade-credit to minimize total inventory costs. Harley andHiggins’ model has been extended to cases of discounted cashflow (Rachamadugu, 1989), inventories subject to deterioration(Aggarwal and Jaggi, 1995), and possible shortage of inventory(Jamal et al., 1997). Unlike prior studies that analyze the retailer’sstocking policies at positive inventory financing costs, Gupta andWang (2009) have recently taken the supplier’s viewpoint as well,and conjecture that finance charges for inventory may be used toimprove efficiency in the supply chain. They show the effect offinance charges on the supplier’s as well as the retailer’s profits bychanging the size of the finance charge in numerical simulations.

Supply chain academics have recently examined inventoryfinancing costs in a Newsvendor framework. Babich et al. (2008)study a firm’s dual financing strategy. A firm has multiplesuppliers offering limited amounts of trade-credit. It can alsofinance working capital from a bank. The firm will go bankrupt ifthe return (cash flow) from the business activity is less than thetotal of the loans. Given differing interest rates from the suppliersand the bank, Babich et al. (2008) investigate the firm’sprocurement with financing decisions on loan amount from eachsource. Li et al. (2009) questions the assumption of credit default.In contrast to Babich et al. (2008) and Li et al. (2009) assume thata firm can continue its operation by paying a default penalty evenafter the firm defaults on the loan for a particular businessactivity. Hence, going into terminal bankruptcy is also the firm’sstrategic decision. Under this more realistic assumption, Li et al.(2009) examine a multi-period dynamic model, in which a firmcan finance working capital from external as well as internalsources.

Dada and Hu (2008) is the closest to our model. They present acapital-constrained Newsvendor model, in which a vendor haslimited internal capital and needs funds from a bank to financeadditional procurement. They assume that the bank is strategicand coordinates the vendor’s order quantity as a Stackelbergleader. Using a game theoretic approach, Dada and Hu (2008)design a lending rate to induce the capital-constrained vendor toorder the optimal quantity for joint profits of the bank andvendor.

Unlike Gupta and Wang (2009), this paper explicitly considersinventory financing cost as a strategic tool. As in Dada and Hu(2008), a retailer is a capital-constrained Newsvendor and needsfunds from external sources to finance inventory. This paper,however, differs from Dada and Hu (2008) in that we focus oninner supply chain coordination issues. Specifically, we shed lighton inventory financing costs from a supplier’s perspective. Asupplier is strategic and coordinates the supply chain as aStackelberg leader. We show a supplier’s trade-credit as a toolfor supply chain coordination. In the presence of positiveinventory financing costs, previous coordination contracts fail toachieve joint profit maximization if each agent relies on direct

financing. A supplier’s sharing demand uncertainty is not enoughfor its retailers to order the optimal quantity for the entire supplychain. In addition, positive financing costs call for trade-credit inorder to subsidize the retailer’s costs of inventory financing. Usingthese multiple schemes, the supplier fully coordinates the supplychain for the largest joint profits.

Specifically, we consider four extensively discussed coordina-tion mechanisms for investigation: (a) all-unit quantity discount,(b) buybacks, (c) two-part tariff, and (d) revenue-sharing. Inquantity discount and two-part tariff, the supply chain is alwaysbetter off with trade-credit. The supply chain is also better offwith trade-credit even in a buyback contract, if the supplier is acredit-worthy borrower or has a sufficiently large internal capital.These three contracts yield the identical joint profits with trade-credit. On the other hand, positive inventory financing costs makerevenue-sharing less profitable than the other three contracts. Wepresent three different coordination schemes, using buybacks,quantity discount, and two-part tariff, respectively, for a decen-tralized supply chain. We also derive the optimal trade-creditrate not only for the supplier’s profit, but also for joint supplychain profit.

The rest of the paper is organized as follows. Section 2 presentsa model that captures the transactions between a supplier and itsindependent retailer given the retailer’s two financing options:trade-credit and direct financing. Sections 3 and 4 examinetwo benchmark cases of fully integrated supply chain with trade-credit and direct financing, respectively, and derive the conditionsunder which trade-credit yields greater joint supply chainprofit. Section 5 derives three optimal coordination schemesfor a decentralized supply chain. The last section summarizesthe findings and implications, and suggests areas for futureresearch.

2. The model

We follow the standard approach in the ‘‘Selling to theNewsvendor’’ framework (Lariviere and Porteus, 2001). Thesupply chain consists of two risk-neutral agents, a supplier anda retailer. The product has a short life cycle, such as consumerelectronics, fashion apparel, or other perishable goods. Consumerprice p is constant during the short sale period, and the demand isstochastic at p. Specifically, demand y follows continuousdistribution F(y) with density f (y), where F(y=0)=0. The retailerprocures quantity q given the demand, only once at the beginningof the sale period. We assume zero market value of leftover items.We also omit the long-run impact of poor product availability onrevenue (such as loss in goodwill). The supplier incurs a constantmarginal cost c in procurement, and wholesales the product to theretailer at w per unit.

We assume capital-constrained agents, following Dada and Hu(2008) and Li et al. (2009). The supplier and retailer haveinsufficient amounts of internal capital, ks and kr, respectively,reserved a priori for this particular business. The retailer’s invoice,wq, is greater than its internal funds, kr, and needs additionalworking capital, Br=wq�kr, from external sources: (1) directlyfrom a financial institution (direct financing) or (2) through thesupplier’s trade-credit, if it is granted (trade-credit financing).Specifically, in the case of direct financing, the retailer pays theinvoice, wq, to the supplier with the internal capital, kr, and thecapital borrowed from a financial institution, Br, at the beginningof the sale period. The retailer remits the loan, Br, with interestback to the lender at the end of the period. In the case of trade-credit financing, the retailer uses the internal capital, kr, first forthe payment, and finances Br from the supplier by delaying therest of the payment until the end of the sale period. The retailer

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tries to minimize financial costs and borrows only additionalcapital Br from the external source.

When granting trade-credit, the supplier receives kr from theretailer immediately upon delivery. The supplier, then, borrowsadditional working capital Bs=cq�k from a financial institution, inaddition to its internal capital, ks, where k=ks+kr, for theprocurement of order quantity q. The retailer pays the loan, Br,with credit interest to the supplier at the end of the sale period.The supplier then remits its loan amount, Bs, with interest, to itslender upon receipt of the payment, Br, from the retailer. Financialinstitutions do not limit the loan amounts Bs and Br. However, ifthe agents increase the loan amount for a larger q given themarket demand F(y), the overstock increases the probability ofnot paying the full obligation back in the case of the obligor’s(either the supplier’s or retailer’s) bankruptcy. Then, it increasesthe expected loss that the lender suffers, and makes the lender aska higher interest rate on the loan.

The above assumptions imply that trade-credits or loans willbe settled at the end of the sale period in a one-time paymentschedule. Interest compounds only once during the sale period.Let Rj

i denote the interest rate charged by firm j to firm i, wherefirm j is either a financial institution (j= f) or the supplier (j=s), andfirm i is either the supplier (i=s) or the retailer (i=r). Unlike priorliterature on stocking policies at given inventory financing costs(Harley and Higgins, 1973; Gupta and Wang, 2009), we assumethat lenders endogenously determine interest for the loan theygrant. We follow Oliver et al. (2006) in assuming that financialinstitutions ask at least the risk-free interest rate, such as the rateon US Treasury bills. In addition, they face a credit risk that theloan will be in default due to uncertainties of the obligor’sfinancial activities, which should be taken into account in theinterest rate. Let fs and fr denote the supplier’s and retailer’sprobabilities of credit default, respectively. These probabilities arefirm-specific as in the standard approach in finance, such asCreditMetrics, CreditRisk+, Moody’s KMV, etc. (Crouhy et al.,2000; Gordy, 2000; Kealhofer, 2003). KMV has been the mostwidely adopted, and estimates a firm’s default probability withthree components: (a) the market value of the firm’s entire assets,(b) asset volatility, and (c) the firm’s total liabilities. If a firmcarries a range of product lines and items, as in many cases in themarketplace, its default probability relies not only on cash flowsfrom the firm’s numerous economic activities, but also on diversefinancial decisions, such as financial leverage, asset and liabilitystructures, coverage ratio, asset volatility over time, and otherstrategic decisions such as dissolution of the firm (Li et al., 2009).In fact, many firms with profitable operations receive a below-investment credit rate or go bankrupt due to poor financialdecisions, as in the case of Oxford Health Plan (Palepu et al.,2000).3 A firm’s default occurs due to factors outside theoperation of a particular product, and cash flow from the producttransaction, then, does not make a significant impact on the firm-specific default probability, unless the firm does not have assets ascollateral and the transaction makes up the lion’s share of thefirm’s revenue. Therefore, given the nature of the product in aNewsvendor framework, we assume that the firm’s defaultprobability is independent from the demand for a product witha short life cycle.

3 Firms have differing financial strategies in managing their capital structure-

s—how a firm finances its overall operations and growth by using different

sources of funds, such as long-term debt, specific short-term debt, common equity,

and preferred equity. A firm may fall into bankruptcy due to insufficient cash flows

from the economic activities, untimely cash demand, poor decisions in maintain-

ing minimum working capital, ill-managed capital structure, or insufficient assets

to generate an emergency cash loan.

Let Dji denotes the fraction that lender j cannot collect

from the defaulted loan to the obligor i, i.e., Dji ¼

ðdefault lossÞ=ðloan amountÞ, where default loss is the differencebetween the loan amount and the recovery amount from thedefault. Financial institutions strive to protect their loans againstthe obligor’s default and commonly impose a ‘‘Purchase MoneySecurity Agreement’’ (PMSA, also known as purchase moneysecurity interest) on the loan, which makes the lender the firstclaimant of the collateral related to the loan in the case of default.We then assume that the financial institution secures the returnfrom the relevant business activity before other creditors’ claimsin the case of credit default. Therefore, the recovery amount isdefined as min [loan amount, return from the business activity].

Given risk-free interest rate y, the opportunity cost of a loan tofirm i is the interest at rate Rj

i, which makes the risk-free value ofthe loan equal the expected value of the loan under default risk fi,i.e., 1þy¼ ð1þRj

iÞ½ð1�fiÞþfið1�DjiÞ� (Oliver et al., 2006). There-

fore, we derive

Rji ¼ ðyþfiD

jiÞ=ð1�fiD

jiÞ; ð1Þ

where j= f for a financial institution, and i=s and r for the supplierand retailer, respectively. Note that interest rate Rj

i increases withdefault probability fi, the fraction of default loss Dj

i, and risk-freeinterest rate y. In a special case when the firm has neither assetsas collateral nor other liabilities, and has cash flow only fromselling the product, it goes bankrupt only if the return from thebusiness activity is less than the loan amount, and pays 1�Dj

iofthe loan amount back to the lender (Babich et al., 2008). Then, weobtain 1þy¼ ð1þRj

iÞð1�DjiÞ for the special case, which leads to

Rji ¼ ðyþDj

iÞ=ð1�DjiÞ.

We consider four different types of contracts under theassumption of positive inventory financing costs:

(a)

all-unit quantity discount: wholesale price decreases as theretailer orders more (Cachon, 2003),

(b)

buyback contract: the supplier sets a buyback value for unsoldunits (Pasternack, 1985),

(c)

two-part tariff : the retailer makes a fixed lump sum paymentto the supplier (Lariviere, 1999),

(d)

revenue-sharing contract: the retailer’s lump sum payment isproportional to actual sales (Cachon and Lariviere, 2005).

The supplier coordinates the retailer’s decisions on order quantityas a Stackelberg leader using one of the above four contracts. Thesupplier may also grant trade-credit at rate Rs

r . There arenumerous types of quantity discount. We consider all-unitquantity discount for investigation (Cachon, 2003) in whichwholesale price w(q) is applied to all the units ordered. In thecase of two-part tariff, the retailer pays wholesale price w for aunit of the product, and also a lump sum payment at thebeginning of the sale period, as it is frequently asked as an entryfee to transactions in the real marketplace (Lariviere, 1999). Onthe other hand, the payment is made at the end in the case ofrevenue-sharing (Cachon and Lariviere, 2005). A buyback contractrequires the supplier to buy leftover items back at rate b at theend of the sale period. As we assumed previously, financialinstitutions have a higher priority in recovering value fromdefault under the PMSA than supply chain partners. For example,if the retailer takes trade-credit under a buyback contract, thefinancial institution recovers the loan amount from the supplier’sdefault first. The rest of the payment from the retailer is then usedfor buyback rebate.

Given the above assumptions, the game sequence is as follows.The supplier offers a wholesale price and the terms of coordina-tion scheme and trade-credit, first. After observing the supplier’s

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offer, the retailer makes two decisions: (i) order quantity q and (ii)financing option for the invoice, either trade-credit or directfinancing. Under the legally binding agreement (e.g., signing thename confirming the receipt of the products), the retailer ought tomake the wholesale payment to the supplier for the deliveredgoods. Note that the four contracts coordinate the supply chaincompletely under the assumption of free inventory financing.

3. Integrated supply chain with trade-credit financing

We first examine the case of the integrated supply chain, inwhich both the supplier’s and retailer’s decisions are fullycoordinated under the central control of the distribution systemfor joint profit maximization. Suppose that the supplier borrowsBs from a financial institution, and finances the retailer’s purchase,i.e., trade-credit financing. Let PT

t ðqÞ be the expected joint profit inthe supply chain with trade-credit at the end of the sale period,

PTt ðqÞ ¼ pSðqÞ�Bsð1þRf

sÞ�kð1þyÞ; ð2Þ

where S(q) is the expected sales, SðqÞ ¼R q

0 yf ðyÞdyþq½1�FðqÞ�. Thesupplier pays cq, which is financed with external capital Bs andinternal capital k, for the procurement at the beginning of the saleperiod. Thus, it remits Bsð1þRf

sÞ to the lender, and also incurs theexpense k(1+y), which is the time value of the internal capital atthe end of the period. Note that the expected joint profit takesboth agents’ default risks into account. The supply chainmaximizes the expected joint profit, PT

t ðqÞ, given interest rate Rfs

in the financial market. As shown in Eq. (1), financial institutionsdetermine interest rate Rf

s by considering not only risk-free rate yand the supplier’s default probability fs, but also the fraction ofdefault loss, Df

s , which relies on the retailer’s default probability fr

and default loss in the trade-credit. Therefore, we examine Dfs first

to analyze the joint supply chain profit.If the supplier does not default, the financial institution

recovers the full loan amount, Bs, from the supplier at the endof the sale period, regardless of the retailer’s payment.4 In thiscase, there is no default loss, i.e., Df

s ¼ 0. If the supplier goesbankrupt, however, Df

s varies over the types of coordinationcontracts that the firm offers to the retailer. We classify the fourcontracts into two groups, depending on whether the collateral isgreater than the loan amount, Bs. The lender secures the amountfrom the relevant business as collateral. It includes not only theloan amount in the trade-credit, but also a lump sum paymentfrom the retailer if relevant. The first group contains the contractsin which the collateral is always greater than the supplier’s loanamount, Bs. The first three contracts, i.e., quantity discount,buybacks, and two-part tariff, belong to this group, called‘‘Contract 1.’’

In all-unit quantity discount, wholesale price w(q) decreases asorder quantity q increases, but should be higher than thesupplier’s marginal procurement cost c, i.e., w(q)4c. Thus, Bs

(=cq�k) is less than the collateral, which is the loan amount inthe trade-credit, Br (=w(q)q�kr). Under two-part tariff, thecollateral includes wq�kr as well as lump sum fee g, which isfixed, regardless of actual sales y. Obviously, the supplier has to atleast recover its procurement costs by setting wq+g4cq, whichleads to wq�kr+g4cq�k (=Bs). In a buyback contract, thefinancial institution is the first claimant of the supplier’s loanamount, Bs, under the PMSA, given the loan amount in the trade-credit, Br. The supplier can give the full buyback rebate to the

4 Even if the supplier makes an insufficient operating profit from this

particular transaction, it repays its debt to the lender with cash from the reserved

working capital for overall operations, from its prior investment in liquid assets, or

from an emergency loan using the firm’s other assets as collateral.

retailer only if the rebate is less than the leftover, Br�Bs, which ispositive due to wZc in a buyback contract. Otherwise, the retaileracquires only Br�Bs as a rebate. Thus, the collateral is Br�min [full

rebate, Br�Bs], which is greater than (or equal to) Bs.On the other hand, the collateral can be smaller than the loan

amount, Bs, in the second group, called ‘‘Contract 2.’’ Revenue-sharing belongs in this category. The collateral includes the loanamount in the trade-credit, Br, and the supplier’s revenue share,(1�Z) py, where Z is revenue-sharing parameter 0oZo1(Cachon and Lariviere, 2005). Since Br (=wq�kr) can be oBs

(=cq�k) because of wrc in revenue-sharing, the collateral,Br+(1�Z) py, is smaller than the supplier’s loan amount, Bs, ifyo(Bs�Br)/(1�Z)p.

3.1. Contract 1

The contracts in the first group enable the lender to recover thefull loan amount, Bs, from the supplier’s default, unless the trade-credit is in default. However, if the retailer also defaults, thefinancial institution may not fully recover the loan amount.Specifically, the financial institution recovers the loan amounteither if the supplier acquires the retailer’s full payment from thedefault, or if the supplier’s collateral on the trade-credit, py, isgreater than loan amount Bs. On the other hand, the financialinstitution recovers only py if yoBs/p. Therefore, if the supplierdefaults, the financial institution expects to recover

EðRÞ ¼ ð1�frÞBsþfr½

Z Bs=p

0pyf ðyÞdyþBs½1�FðBs=pÞ��

¼ Bs�frp

Z Bs=p

0FðyÞdy

Thus, the financial institution expects default loss Bs�EðRÞ ¼

frpR Bs=p

0 FðyÞdy. Subsequently,

DfsðBsÞ ¼ ðBs�EðRÞÞ=Bs ¼ ðfr=BsÞp

Z Bs=p

0FðyÞdy; ð3Þ

and, then, interest rate Rfs is determined as

RfsðBsÞ ¼ ðyþfsD

fsðBsÞÞ=ð1�fsD

fsðBsÞÞ: ð4Þ

Let f=fr�fs denote the joint probability that both the supplier

and retailer go bankrupt. We can easily show that fsDfsðBsÞ ¼

ðf=BsÞpR Bs=p

0 FðyÞdy. This implies that the supplier fails to fulfill

the financial obligation only if both the supplier and retailerdefault. In other words, trade-credit has a ‘‘pooling’’ effect ofdefault risks in the supply chain. Despite the supplier’s defaultrisk, trade-credit enables a financial institution to lower itsdefault loss through this pooling effect.

3.2. Contract 2

In a revenue-sharing contract, the lender secures Br+(1�Z)py

as collateral from the supplier’s default. Thus, when the trade-credit does not default, the financial institution recovers the fullloan amount if BsrBr+(1�Z)py, i.e., yZ (Bs�Br)/(1�Z)p. Other-wise, the financial institution recovers only the collateral,Br+(1�Z) py.

When the trade-credit is also in default, the supplier recoversmin[py, Br+(1�Z)py], where py is the supplier’s collateral on thetrade-credit. If py is oBr+(1�Z)py, i.e., yrBr/Zp, the financialinstitution recovers only py if yrBs/p. Otherwise, the financialinstitution recovers the full loan amount, Bs, from the supplier’sdefault. On the other hand, if py is 4Br+(1�Z)py, the financialinstitution recovers only Br+(1�Z)py if Br+(1�Z)pyrBs, i.e., yr

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(Bs�Br)/(1�Z)p. Otherwise, the financial institution recovers thefull loan amount, Bs.

Note that the above partitions along actual sales y may notcoexist at various values of Br and ZBs. For example, if Br=ZBs,(Bs�Br)/(1�Z)p=Br/Zp=Bs/p, which rules out the case of Br/Zpoyo(Bs�Br)/(1�Z)p. Thus, we restrict the values of Br andZBs for the derivation of the expected default loss. Since ourinvestigation focuses on supply chain coordination, we considerthe case in which the supply chain can be coordinated in revenue-sharing. The supply chain gains the joint profit, PT

t ðqÞ in Eq. (2),while the retailer makes profit

PTr ðqÞ ¼ ZSðqÞp�Brð1þRs

rÞ�krð1þyÞ;

where Rsr is the trade-credit interest rate. We can easily show that

Br=ZBs and Rsr ¼ Rf

s achieve full coordination in the supply chain asthese conditions make the retailer’s profit an affine transforma-tion of the joint profit, i.e., PT

r ðqÞ ¼ ZPTt ðqÞþ½Zks�ð1�ZÞkr �ð1þyÞ.

Hence, we consider the case of Br=ZBs. At Br=ZBs, the financialinstitution recovers

ZBsþð1�ZÞpy if yoBs=p

Bs if yZBs=p

)if the trade�credit does not default;

py if yoBs=p

Bs if yZBs=p

)if the trade�credit defaults:

These recovery amounts yield the following fraction of defaultloss,

DfsðBs;ZÞ ¼ ½ð1�ZþZfrÞ=Bs�p

Z Bs=p

0FðyÞdy;

which is always greater than the fraction in Eq. (3). This showsthat the interest rate under the contracts in the first group islower than the rate with revenue-sharing. Intuitively, under thecontracts in the first group, the financial institution alwaysrecovers the full loan amount, Bs, from the supplier, unless thetrade-credit is in default, because the collateral is greater than thesupplier’s loan amount. In revenue-sharing, however, the collat-eral, ZBs+(1�Z)py, can be smaller than the loan amount. There-fore, the financial institution charges a higher interest rate, whichleads to a larger inventory financing cost and the resulting lowersupply chain profit.

4. Integrated supply chain with direct financing

Suppose that the retailer borrows Br directly from a financialinstitution. As in the case of trade-credit, the interest rate variesover the four coordination contracts. Consider (i) all-unit quantitydiscount, (ii) two-part tariff, and (iii) revenue-sharing, first. Theretailer pays a lump sum fee at the beginning of the sale period intwo-part tariff, or at the end in revenue-sharing. Thus, the retailerborrows Br ¼wq�kr at the beginning of the period, wherew ¼wðpÞ in quantity discount, w ¼wþg=q in two-part tariff,and w ¼w in revenue-sharing. The lender recovers min[Br, py]in the case of the retailer’s credit default, where py is the collateralon the loan. Hence, the three contracts lead to Df

rðBrÞ ¼

ð1=BrÞpR Br=p

0 FðyÞdy.In a buyback contract, the retailer collects the buyback rebate,

b(q�y), from the supplier if yoq. As the rebate is part of thecollateral that the lender secures, the financial institution re-covers

Br if the retailer does not default;

pyþbðq�yÞ if yo ðBr�bqÞ=ðp�bÞ

Br if yZ ðBr�bqÞ=ðp�bÞ

)if the retailer defaults:

Thus, given the above recovery amounts, the buyback contractyields

DfrðBr ; bÞ ¼ ð1=BrÞðp�bÞ

Z ðBr�bqÞ=ðp�bÞ

0FðyÞdy:

Lemma 1. Interest rate Rfr strictly decreases as buyback rate b

increases in a buyback contract.

Proof. See Appendix A.

Lemma 1 shows that the financial institution charges a lowerinterest rate as the supplier shares more demand uncertainty withthe retailer by granting a higher buyback rate on leftover products.

Consequently, the financial institution sets the followinginterest rate on the loan to the retailer,

RfrðBr ; xÞ ¼

yþfrDfrðBr ; xÞ

1�frDfrðBr ; xÞ

;where DfrðBr ; xÞ

¼1

Br

� �ðp�bxÞ

Z ðBr�bxqÞ=ðp�bxÞ

0FðyÞdy;

where x is a dummy variable: x=1 for a buyback contract.Otherwise, x=0. Let PD

t ðq; xÞ denote the joint profit with theretailer’s direct financing at the end of the sale period,

PDt ðq; xÞ ¼ pSðqÞ�Br ½1þRf

rðBr ; xÞ��ðkr�wqþcqÞð1þyÞ: ð5Þ

The supplier incurs the procurement cost, cq, whereas the retailerpays the invoice, wq (=Br+kr), to the supplier at the beginning ofthe sale period. After the sale, the retailer remits the loan withinterest, Br½1þRf

rðBr ; xÞ�, to the financial institution, and also incursthe expense, kr (1+y), which is the time value of its internalcapital kr.

The previous section shows that the contracts in the firstgroup, i.e., quantity discount, buybacks, and two-part tariff, yielda higher profit than revenue-sharing in the case of trade-creditfinancing. Hence, let P

T

t ðqÞ be the joint profit in the supply chainwith trade-credit under the contracts in the first group. Thefollowing proposition compares the supply chain’s profits withthe two financing methods: trade-credit and direct financing.

Proposition 1. At any qZ0, PT

t ðqÞZPDt ðq; xÞ under

(a)

all-unit quantity discount and two-part tariff, (b) buyback contract if brBr=q �min½1�ðBsðp�bÞ=BrpÞ;1�fs�,

otherwise if L(Bs, Br)o0, where LðBs;BrÞ ¼ Bs=Br�frDfr

ð1�fsDfsÞ=fsD

fsð1�frDf

rÞ

(c)

revenue-sharing if ksZ(c�w)q, otherwise if L(Bs, Br)o0.

Proof. See Appendix B.

In quantity discount and two-part tariff, the supply chain isalways better off with trade-credit, not only because of a smallerloan amount, but because of a lower interest rate as well. Duringthe past decades, numerous theories, mainly from financeperspectives, have been proposed to explain why a seller borrowsfunds from a financial institution and then grants credit to thebuyer: To mention a few, the financing advantage theory (Biaisand Gollier, 1997); price discrimination theory (Brennan et al.,1988); financial intermediary theory (Lang and Nakamura, 1995);asymmetric information theory (Smith, 1987), etc. See Petersenand Rajan (1997) for details. Proposition 1(a) puts forth anotherviewpoint from supply chain management that trade-creditmaximizes the joint supply chain profit through the poolingeffects of (i) default risks (i.e., fofr) and (ii) internal capitals(i.e., k4kr).

Lemma 1 shows that, if the supplier grants a higher rebate ratein a buyback contract, the resulting low interest rate in direct

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financing may lead to a lower inventory financing cost than thefinancing costs with trade-credit under the contracts in the firstgroup. Proposition 1(b), however, shows that the contracts in thefirst group are more economical with trade-credit for the supplychain, if the supplier has a sufficiently large internal capital, ks

(i.e., small Bs) or has a low default rate, fs. Even when buybackrate b is relatively high, the supply chain is better off with trade-credit because the condition L(Bs, Br)o0 can be easily satisfied inthe real marketplace if the supplier is a credit-worthy borrower.Standard and Poor’s Credit Week (April 15, 1996) shows that theaverage credit default probability of a AAA-firm within one year isfiE0.0000; a AA-firm, fiE0.0000; a A-firm, fi=0.0006; a BBB-firm, fi=0.0018; a BB-firm, fi=0.0106; a B-firm, fi=0.0520; andthe lowest-rated CCC firm, fi=0.1979 (Crouhy et al., 2000). Hence,if the supplier is a firm of A or higher rate, even a retailer of thelowest credit rate results in almost zero joint default probability,fE0, via the risk pooling effect. Subsequently, frDf

r is greaterthan fsD

fs ¼ ðf=BsÞp

R Bs=p0 FðyÞdy� 0, which makes the second term

in L(Bs, Br) 41. Therefore, L(Bs, Br) is always negative because Bs/Bro1 as krok and cow in a buyback contract.

Proposition 1(c) shows that the contracts in the first groupyield a higher profit with trade-credit than revenue-sharing withdirect financing if ks is relatively large. Even when kso(c�w)q,the first term in L(Bs, Br) decreases as ks increases, whereas thesecond term increases as Df

s decreases with the increase in ks.Thus, L(Bs, Br) becomes negative easily, and the contracts in thefirst group produce greater profit with trade-credit than revenue-sharing with the retailer’s direct financing. Furthermore, as shownin the above, L(Bs, Br) can easily be negative if the supplier is acredit-worthy borrower. Recall that, in the case of trade-creditfinancing, the contracts in the first group make greater profitfor the supply chain than revenue-sharing. This finding andProposition 1(c) show that revenue-sharing yields a lower profitfor the supply chain than the contracts in the first group withtrade-credit, regardless of whether the revenue-sharing is grantedwith or without trade-credit.

5. Decentralized supply chain and coordination

We now examine a decentralized supply chain, in which thesupplier and retailer maximize their own profits independently.We derive the coordination scheme that entices the retailer toreplicate the outcomes of the integrated supply chain in makingdecisions. The reasoning is that it is optimal from the supplier’sperspective to coordinate the supply chain for joint profitmaximization (Lariviere, 1999). After coordinating the supplychain, the supplier takes the largest portion of the joint profit byarranging a profit-sharing scheme, which gives the retailer barelyenough profit to participate in the coordination.

Proposition 1 shows that the integrated supply chain achieves

the largest profit PT

t ðqÞ with the contracts in the first group using

trade-credit. Thus, PT

t ðqÞ is a benchmark, with which we compare

a coordination mechanism in a decentralized supply chain. Thefollowing proposition presents the optimal order quantity, qc, that

maximizes joint profit PT

t ðqÞ.

Proposition 2. PT

t ðqÞ is strictly concave in q. Let qc(f) be the order

quantity that maximizes PT

t ðqÞ, and let Bcs ¼ cqc�k. Then, qc(f)

satisfies the first order condition,

1�c

p

� ��

1

pfcRf

sðBcs Þþ½fcð1þyÞp

Z Bcs=p

0yf ðyÞdy=Bc

s ð1�fsDfsÞ

2�g ¼ FðqcÞ

qc(f) decreases as f increases. Thus, BcsðfÞ40 if F�1[1�c(1y)/p]4

k/c and fo ~f, where qcð~fÞ ¼ k=c.

Proof. See Appendix C.

Let qo be the optimal order quantity at f=0, i.e., qo=qc(f=0).Proposition 2 shows qc(f40)oqo. Obviously, if both the supplierand retailer have no risk of credit default, a financial institutioncharges only risk-free interest rate y on the loan, which enablesthe supply chain to order the largest quantity. The supply chaindecreases the order quantity as financial institutions charge ahigher interest rate with an increase in f. If the joint default rateis higher than ~f, at which k¼ cqcð

~fÞ, the supplier does not financethe inventory from a financial institution because the internalcapital in the supply chain is large enough for the procurement.

In a decentralized supply chain, the supplier fully coordinatesthe supply chain by inducing the retailer (i) to finance the invoiceusing the supplier’s trade-credit, and (ii) to order the samequantity, qc, as in the integrated supply chain. We propose threecoordination mechanisms in this paper, using (a) buybacks, (b)quantity discount, and (c) two-part tariff, respectively.

5.1. Buyback rebate

We first consider buybacks as an incentive for supply chaincoordination. At the beginning of the sale period, the supplier setswholesale price w, buyback rate b, and trade-credit interest rateRs

r , which is lower than Rfr in direct financing. Given the supplier’s

offer, the retailer sets order quantity q and takes the trade-creditfor the invoice. The supplier’s financial institution has a higherpriority in recovering value from the supplier’s credit defaultunder the PMSA. Then, the retailer receives the following amountas buyback rebate from the supplier when yoq,

bðq�yÞ when the supplier does not default;

bðq�yÞ if yZ qðb;wÞ

Br�Bs if yo qðb;wÞ

)when the supplier defaults;

where qðb;wÞ ¼ q�ðBr�BsÞ=b. Hence, the retailer expectsb½q�SðqÞ��bfs

R qðb;wÞ0 FðyÞdy as buybacks. The second term is

buyback loss, which accounts for the loss in buyback rebate dueto the supplier’s default. Therefore, the retailer expects thefollowing profit in the decentralized supply chain at the end ofthe sale period:

PTr ðq;b;w;R

srÞ ¼ pSðqÞþb½q�SðqÞ�fs

Z qðb;wÞ

0FðyÞdy��Brð1þRs

rÞ�krð1þyÞ:

ð6Þ

There have been two different approaches in determining thevalue of the coordination parameter in the channel coordinationliterature: The first approach uses first order conditions (e.g.,Pasternack, 1985; Tsay, 2001), whereas the second one manip-ulates the coordination parameter to make the agents’ profits anaffine transformation of the joint supply chain profit (e.g., Cachon,2003; Cachon and Lariviere, 2005). We employ the secondapproach in this paper.

The retailer’s profit in Equation (6) can be written as

PTr ðq; b;w;R

srÞ ¼ z1pSðqÞ�z2cq�Gðq;b;w;Rs

r�kry;

where z1=(p�b)/p, z2=(w�b)/c, and Gðq;b;w;RsrÞ ¼ bfs

R qðb;wÞ0

FðyÞdyþBrRsr . We can make PT

r ðq; b;w;RsrÞ ¼ rP

T

t ðqÞþ

y½rks�ð1�rÞkr� by setting z1=z2=r and Gðq; b;w;RsrÞ ¼ rBsR

fs . As

PTr ðq; b;w;R

srÞ is an affine transformation of P

T

t ðqÞ, the retailer’s

independent decisions on order quantity also lead to the optimalquantity, qc, for the entire supply chain.

We can implement this task with the following two contracts.First, the supplier sets the interest payment in the trade-creditat BrRs�

r ðr;BrÞ ¼ rBsRfs—buyback loss, given any BsoBr, where

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buyback loss is defined previously. The supplier determinesb*(w)=p(w�c)/(p�c) by equating z1 with z2, and derivesr=(p�w)/(p�c). Note that there are multiple coordinationschemes because b*(w) and Rs�

r ðr;BrÞ are functions of w. Eachscheme at a given value of w allocates the maximized joint profit,P

T

t ðqcÞ, to the supplier and retailer. The supplier gains

PTs ðqc ; b

�;w;RfsÞ ¼ P

T

t ðqcÞ�PTr ðqc; b

�;w;Rs�

r Þ

¼ ð1�rÞPT

t ðqcÞ�y½rks�ð1�rÞkr�;

and, then, extracts a larger fraction of the joint profit by raising w.Specifically, at wU=p�rL (p�c) where rL ¼ kry=½pSðqÞ�cq�BsR

fs �,

the supplier obtains the entire supply chain profit, i.e.,PT

s ðqc ; b�;wU ;RfrÞ ¼ P

T

t ðqcÞ. On the other hand, the retailer takesthe entire joint profit, i.e., PT

r ðqc ; b�;wL;Rs�r Þ ¼ P

T

t ðqcÞ, at wL=p�rU

(p�c) where rU ¼ ½pSðqÞ�cq�BsRfs�ksy�=½pSðqÞ�cq�BsR

fs �. There-

fore, the supplier determines wA[wL, wU] to maximize its portionof the joint profit under coordination. Note that wA[c, p] ifkr=ks=0.

The above contract, however, has a drawback in implementa-tion. Let Bc

r ¼wqc�kr and Bcs ¼ cqc�k. Interest rate on trade-credit

should be non-negative, i.e., Rs�r ðr;Bc

r ÞZ0, which asksrBc

sRfs Zbuyback loss at Bc

s . We can easily show that, atb*(w)=p(w�c)/(p�c) and r=(p�w)/(p�c), the non-negativitycondition leads to

w

prD;where D¼

G1þG2

G1þðp=cÞG2; G1 ¼ Rf

sð1�k=cqcÞ; andG2

¼fs

Z fcqc=p�ksðp�cÞ=pðw�cÞg

0FðyÞdy=qc :

This shows that the wholesale price should be less than Dp fornon-negative interest on the trade-credit. Note that D is o1because of p4c. The supplier increases w in order to extract alarger portion of the maximized joint profit. Nevertheless, thesupplier’s wholesale price and the resulting profit are restrictedby Dp. Thus, ceteris paribus, if the ratio p/c is relatively high, D issignificantly lower than 1, which may make the supplier worse offwith coordination.

We propose another contract, which is free from the short-coming in the above. The second scheme employs both buybacksand quantity discount as shown in the following proposition.

Proposition 3. a The supplier’s and retailer’s profits, PTs ðq; b;w;R

fsÞ

and PTr ðq; b;w;R

srÞ, become ð1�rÞP

T

t ðqÞ�y½rks�ð1�rÞkr � and

rPT

t ðqÞþy½rks�ð1�rÞkr �, respectively, where r¼ ðp�wÞ=ðp�cÞ, at

(i)

buyback rate b�ðwÞ ¼ ðw�cÞp=ðp�cÞ, (ii) wholesale price w�ðq; wÞ, satisfying w�q¼ wq�b�

fs

R qðb� ;w�Þ0 FðyÞdy, and wð4Bs=qÞ is the upper bound of whole-

sale price,

(iii) interest rate Rs�

r ðB�r ;rÞ ¼ rðBs=B�r ÞR

fsðBsÞ, where B�r ¼w�q�kr .

5½w�ðq;rÞ�c�q¼ ð1�rÞ½pSðqÞ�cq�40.

Thus, the optimal order quantity, qc, also maximizes both the

supplier’s and retailer’s profits, and there are multiple coordination

schemes as each scheme is a function of w.

3.b. At any given w, there exists a unique w�ðqc ; wÞ that satisfies

w�qc ¼ wqc�b�fs

R qc ðb� ;w�Þ

0 FðyÞdy, where qc ¼ qc�ðB�r�BcsÞ=b�ðwÞ

and w�ðqc ; wÞZc.

3.c. w�ðq; wÞ is strictly decreasing with an increase in q, and

w�ðq¼ 0; wÞ ¼ w.

Proof. See Appendix D.

We have implicitly assumed wZc in deriving the amount thatthe lender recovers from the supplier’s credit default under abuyback contract. Proposition 3(b) confirms w�ðqc ; wÞZc. Thus, afinancial institution sets interest rate at Rf

sðBsÞ in Eq. (4) in the case

of trade-credit financing. Interest rate Rs�r ðB

�r ;rÞ in the trade-credit

provision should be lower than the retailer’s direct borrowingrate, Rf

rðB�r ; b�Þ, at b* and w*, so that the retailer takes the supplier’s

trade-credit. We can easily show Rs�r oRf

r , because B�r Rs�r ¼

½ðp�wÞ=ðp�cÞ�BsRfs as shown in Proposition 3(a)(iii), crw�r

wrp, and BsRfs oB�r Rf

r , which is satisfied if buyback rate b issufficiently low or if LðBs;B�r Þo0, as shown in Proposition 1(b).These two conditions are easily satisfied when the supplier has asufficiently large internal capital or has a low rate of creditdefault.

Note that the interest payment in the trade-credit, B�r Rs�r , is

lower than that the supplier’s financing cost, BsRfs . The difference,

BsRfs�B�r Rs�

r ¼ BsRfsð1�rÞ, can be viewed as ‘‘the supplier’s share’’

(i.e., 1�r) of the joint inventory financing costs in the supplychain. In the case of the retailer’s direct financing, the suppliermakes no contribution to the retailer’s financing cost. Then, theretailer pays a higher financing cost, which makes the retailerorder less than the optimal quantity for the entire supply chain.Trade-credit, on the other hand, makes the supplier subsidize theretailer by paying fraction 1�r of the joint financing costs forjoint profit maximization, which enables the supplier to gain thesame share of the maximized joint profit. Moreover, the supplieroffers a quantity-discount wholesale price, satisfying w�q¼ wq

�buyback loss, in order to compensate the retailer for the loss inbuyback rebate from the supplier’s credit default.

5.2. Quantity discount and two-part tariff

As in the case of buybacks, we can easily design quantitydiscount and two-part tariff for coordination by making theretailer’s profit an affine transformation of the joint profit. Theretailer gains

PTr ðq;wÞ ¼ pSðqÞ�wq�BrRs

r�kry; ð7Þ

where w ¼wðqÞ in quantity discount, and w ¼wþg=q in two-parttariff. We can make PT

r ðq;w�;Rs�

r Þ ¼ rPT

t ðqÞþy½rks�ð1�rÞkr� bysetting w�q¼ ð1�rÞpSðqÞþrcq and Rs�

r ðB�r ;rÞ ¼ rRf

sðBsÞBs=B�r .Hence, the supplier charges wholesale payment w*(q,r)q=(1�r) pS(q)+rcq in quantity discount; or fixed lump sum feeg*(q,w,r)=(1�r) pS(q)+rcq�wq in two-part tariff, for fullcoordination.

w�ðq;rÞ is 4c.5 Thus, financial institutions set interest rate atRf

sðBsÞ as shown in Eq. (4). We derive

@w�ðq;rÞ=@q¼�ð1�rÞpZ q

0yf ðyÞdy=q2r0;

which confirms that w�ðq;rÞ decreases with an increase in q.Using Proposition 1(a), given w�ðq;rÞ4c, we easily showRs�

r ðB�r ;rÞ ¼ ðrBs=B�r ÞR

fsðBsÞr ðBs=B�r ÞR

fsðBsÞrRf

rðB�r Þ. This confirms

that the retailer takes the supplier’s trade-credit to finance theinvoice. Therefore, the supplier fully coordinates the supply chainat w�ðq;rÞ and Rs�

r ðB�r ;rÞ, and extracts the largest portion of the

maximized joint profit by decreasing r.The non-negativity condition of lump sum fee g*(q,w,r)Z0 in

two-part tariff, however, restricts the retailer’s profit share torr[pS(q)�wq]/[pS(q)�cq]. We can make r take any valuebetween 0 and 1 by setting w*q=cq which leads tog*(q,w,r)=(1�r)[pS(q)�cq]. Note that, at the optimal orderquantity qc, g*(qc,w*,r) has a similar structure to the one underthe assumption of free inventory financing (Lariviere, 1999),which demands w=c and g*(qc)r[pS(qc)�cqc]�o, where o is theretailer’s opportunity cost of carrying the product.

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6. Summary and discussion

Trade-credit has been a critical source of working capital for amajority of businesses. Trade-credit sales represent nearly 18.5%of sales for large US firms (Berlin, 2003). 30% of the participatingfirms in a recent survey report that their clients rely on trade-credit for routine financing (Ng et al., 1999). A vast amount ofliterature has accumulated to explain this source of workingcapital, but mainly from finance perspectives. Trade-credit hasalso been an important issue in supply chain management,because it is one of the major transactions between a supplierand its retailers. Surprisingly, trade-credit has not been suffi-ciently investigated by supply chain academics. This researchpresents an alternative perspective on the pervasive use of trade-credit, and shows that trade-credit can be employed as a tool forsupply chain coordination.

We explicitly assume capital-constrained supply chain agents.The agents incur positive costs of the funds for inventory, not onlybecause of (a) the firms’ credit default risks, but also because of(b) the opportunity costs of the funds even if the firms have zerocredit default risk. Under this more realistic assumption, thisresearch shows that a supplier’s trade-credit leads to greater jointsupply chain profit than direct financing, due to the pooling effectof default risks and the effect of smaller loan amount in trade-credit financing.

Specifically, we consider four widely examined coordinationmechanisms for investigation: (i) all-unit quantity discount, (ii)buybacks, (iii) two-part tariff, and (iv) revenue-sharing. Inquantity discount and two-part tariff, the supply chain is alwaysbetter off with trade-credit. The supply chain is also better offwith trade-credit even in a buyback contract, if the supplier has asufficiently large internal capital or is a credit-worthy borrower,such as a firm with a credit rating of A or higher. Trade-creditenables the supply chain to make the same joint profits with thesethree contracts. On the other hand, positive inventory financingcosts make revenue-sharing less profitable than the other threecontacts.

This paper offers three different schemes for coordination in adecentralized supply chain, using buybacks, quantity discount,and two-part tariff, respectively. The supplier shares demanduncertainty with the retailer using buyback rebate or a lowerwholesale price in quantity discount and two-part tariff. Inaddition, the supplier grants trade-credit in order to subsidizethe retailer’s inventory financing costs. Using these multiple-schemes, the supplier makes the retailer order the optimalquantity for the entire supply chain. Given that the supply chainis fully coordinated for the largest joint profit, the supplierextracts a greater share of the joint profit by increasing wholesaleprice or lump sum fee.

This research guides decisions on trade-credit rate when thetrade-credit is employed for supply chain coordination. In order togeneralize the findings and implications, we ought to extend thisresearch to other types of coordination contracts, such as quantityflexibility (Tsay, 1999), target-level sales rebate (Taylor, 2002),holding cost subsidy (Wang and Gerchak, 2001), etc. Anotherextension would be to relax the assumption of financial institu-tions’ PMSA, which is prevalent in the financial market. WithoutPMSA, the lender becomes a ‘‘general’’ unsecured creditor underthe Bankruptcy Code, and may have a lower priority in recoveringvalue from the supplier’s default than the retailer. If so in the caseof buybacks, the retailer claims the rebate before the lenderrecovers its loan. The change in claim priority may yield differentresults. We need further investigation even though we conjecturefundamentally consistent implications.

We should also relax the assumption of risk-neutral agents,and consider channel partners’ different attitudes toward risk

(Lau and Lau, 1999; Webster and Weng, 2000). For instance, themodel can be extended to examine coordination schemes for aretailer with downside risk (Gan et al., 2004). This research shouldalso be extended to the case of multiple retailers who competeeach other in the same market or to the case of multiple retailerswho have differing demand conditions (Cachon and Lariviere,2005).

Appendix A. Proof of Lemma 1

The first derivative of DfrðBr ; bÞ with respect to b is always

negative as shown in

@DfrðBr ; bÞ=@b¼�ð1=BrÞ½

Z ðBr�bqÞ=ðp�bÞ

0FðyÞdyþ½1=ðp�bÞ�F½ðBr�bqÞ=

ðp�bÞ�ðpq�BrÞ�o0:

Thus, Rfr is strictly decreasing as b increases in a buyback contract.

Appendix B. Proof of Proposition 1

PT

t ðqÞ4PDt ðq;bÞ always if Bsð1þRf

sÞþkð1þyÞoBr½1þRf

rðBr ; xÞ��ðwq�kr�cqÞð1þyÞ, which leads to BsðRfs�yÞoBrðR

fr�yÞ.

Proposition 1(a). We rewrite RfsðBsÞ and Rf

rðBr ; xÞ in the form ofR¼ ðyþAÞ=ð1�AÞ. Thus, A¼ ðf=aÞ

R a0 FðyÞdy, where a=Bs/p, in

RfsðBsÞ, and A¼ ðfr=aÞ

R a0 FðyÞdy, where a=Br/p, in Rf

rðBr ; x¼ 0Þ. Wecan easily show that (i) R is strictly increasing with A, (ii) A isstrictly increasing with f and fr, and (iii) @A=@a¼ðfi=a2Þ

R a0 yf ðyÞdy40, where i¼ �; r. Thus, Rf

rðBr ; x¼ 0Þ4RfsðBsÞ

and PT

t ðqÞ4PDt ðq; x¼ 0Þ, because fr4f and Br4Bs in the cases

of all-unit quantity discount and two-part tariff.Proposition 1(b). In a buyback contract, A¼ ðfr=aÞ½ðBr�bqÞ

=Br �R a

0 FðyÞdy, where a=(Br�bq)/(p�b), in RfrðBr ; x¼ 1Þ. Note that

(i) a in RfrðBr ; x¼ 1Þ is greater than a=Bs/p in Rf

sðBsÞ ifbqrBr�Bs(p�b)/p, and (ii) fr[(Br�bq)/Br]4f if bqrBr(1�fs).Thus, Rf

rðBr ; x¼ 1Þ4RfsðBsÞ and P

T

t ðqÞ4PDt ðq; bÞ if bqrmin

[Br�Bs(p�b)/p, Br(1�fs)]. Even if b is sufficiently large,P

T

t ðqÞ4PDt ðq; bÞ when BsðR

fs�yÞoBrðR

fr�yÞ, which is satisfied if

L(Bs, Br)o0.Proposition 1(c). In revenue-sharing, BrZBs if cq�(kr+ks)

rwq�kr. Thus, as shown in 1(a), RfrðBr ; x¼ 0ÞZRf

sðBsÞ andP

T

t ðqÞZPDt ðq;x¼ 0Þ. Otherwise, P

T

t ðqÞ4PDt ðq; bÞ if L(Bs, Br)o0.

Appendix C. Proof of Proposition 2

Let a=Bs/p, b¼R a

0 yf ðyÞdy, and d¼ 1�fsDfs 40. P

T

t ðqÞ isconcave in q because

d2PT

t ðqÞ

dq2¼� pf ðqcÞþ

c2ð1þyÞfd2p

f ðaÞþ 2b2fa3

" #( )o0:

Consider the implicit function Iðq;fÞ ¼ dPT

t ðqÞ=dq¼ 0. By differ-entiating I(q,f)=0 with respect to f, we derivedqc=df¼�½@2P

T

t ðqÞ=@q@f�=½d2PT

t ðqÞ=dq2�, which is always nega-tive because

@2PT

t ðqÞ

@q@f¼�

cð1þyÞd2

FðaÞþ 2bfda2

Z a

0FðyÞdy

� �o0:

Therefore, qc decreases as f increases. Let qo=qc(f=0) satisfyingthe first order condition, p[1�F(q)]�c(1+y)=0. Then,qo=F�1[1�c(1+y)/p], and Bo

s ¼ cqo�k40, i.e., F�1[1�c(1+y)/p]4k/c, implies that there exists fA ½0; ~f�, which yieldsBc

s ðfÞ ¼ cqcðfÞ�k40, where cqcð~fÞ�k¼ 0, because qc decreases

with an increase in f.

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Appendix D. Proof of Proposition 3

Proposition 3(a). Buyback loss, bfs

R qðb;wÞ0 FðyÞdy in

PTr ðq; b;w;R

srÞ, makes it difficult to design a coordination contract.

Thus, let the supplier set wholesale price w�ðq; b; wÞ satisfying

w�þðbfs=qÞR qðb;w�Þ

0 FðyÞdy¼ w, where w is the upper bound of

w�ðq; b; wÞ as shown in the proof of Proposition 3(c). Then, by

substituting w in PTr ðq;b;w;R

srÞ with w�ðq; b; wÞ, we derive

PTr ðq; b;w

�;RsrÞ ¼ ðp�bÞSðqÞþðb�wÞq�B�r Rs

r�kry. When substituting

b and Rsr in PT

r ðq; b;w�;Rs

rÞ with b� and Rs�r , we obtain

PTr ðq; b

�;w�;Rs�r Þ ¼ rP

T

t þy½rks�ð1�rÞkr�. Thus, the supplier’s

profit is PTs ¼ P

T

t�PTr ¼ ð1�rÞP

T

t�y½rks�ð1�rÞkr �. We also derive

w�ðq; wÞ by substituting b in w�ðq; b; wÞ with

b�ðwÞ ¼ ðw�cÞp=ðp�cÞ.Proposition 3(b). As shown in the above, w�ðqc; wÞ satisfies

Lðqc;w�Þ ¼ w, where Lðq;wÞ ¼wþ½pfsðw�cÞ=qðp�cÞ�R q

0 FðyÞdy.

Since @Lðq;wÞ=@w¼ 1�fsFðqÞ40, there exists a unique

w�ðqc ; wÞZc satisfying Lðqc ;w�Þ ¼ w if Lðqc ;w¼ cÞ ¼ c

þ½pfsðw�cÞ=qcðp�cÞ�R qc

0 FðyÞdyrw. Let w¼ 1�ðc=pÞ. Then, the

condition Lðqc;w¼ cÞrw requires ðfs=qcwÞR qc

0 FðyÞdyr1, which

is satisfied if ðqcw=fsÞZR qc

0 FðyÞdy. Proposition 2 shows wZFðqcÞ

at qc. Thus, w=fsZFðqcÞ, which leads to ðqcw=fsÞZR qc

0 FðyÞdy

because qcFðqcÞZR qc

0 FðyÞdy. Hence, Lðqc ;w¼ cÞrw and there

exists a unique w�ðqc ; wÞZc satisfying Lðqc;w�Þ ¼ w.Proposition 3(c). Consider the implicit function, Iðq; b�;w�; wÞ

¼w�þkðqÞ�w ¼ 0, where kðqÞ ¼ ðb�fs=qÞR qðb� ;w�Þ

0 FðyÞdy andb�ðwÞ ¼ p½ðw�cÞ=ðp�cÞ�. By implicitly differentiating Iðq; b�;

w�; wÞ ¼ 0 with respect to q, we derive

@w�ðq; wÞ=@q¼�b�fS½

Z qðb� ;w�Þ

0yf ðyÞdy

þFðqÞks=b�=q2½1�fsFðqðb�;w�ÞÞ�o0:

By L’Hopital’s rule, limq-0kðqÞ ¼ limq-0ðcþb��w�ÞfsFðqðb�;w�ÞÞ

¼ 0. Thus, w�ðq¼ 0; wÞ ¼ w.

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