MANAGEMENT SCIENCEArticles in Advance, pp. 1–20ISSN 0025-1909 (print) ISSN 1526-5501 (online) http://dx.doi.org/10.1287/mnsc.2016.2515

© 2016 INFORMS

Inventory, Risk Shifting, and Trade Credit

Jiri ChodCarroll School of Management, Boston College, Chestnut Hill, Massachusetts 02467, chodj@bc.edu

This paper has two objectives. First, we show how debt financing distorts a retailer’s inventory decision whenthe retailer orders multiple items that differ in cost, revenue, or demand parameters. Taking advantage

of limited liability, a debt-financed retailer favors items with a low salvage value, those with a high profitmargin, and those that represent a large proportion of the total inventory investment. Second, we argue thatthis distortion is mitigated when the financing is provided by the supplier who can observe the actual orderquantities before determining the credit terms. Borrowing goods rather than borrowing cash limits the retailer’sability to deviate from the first-best inventory decision. On the flip side, few suppliers can access capital at thesame low cost as banks. We study a combination of bank and supplier financing that allows the retailer to getthe best of both worlds.

Keywords : trade credit; supplier financing; debt; agency; newsvendor model; inventoryHistory : Received February 16, 2015; accepted March 29, 2016, by Serguei Netessine, operations management.

Published online in Articles in Advance August 22, 2016.

1. IntroductionDebt financing creates an agency conflict between thecreditor and the borrower generally known as “riskshifting” or “asset substitution.” Due to limited lia-bility, a firm maximizing the value of equity has anincentive to increase risk after issuing debt. Becausethe lender anticipates the firm’s risk-shifting behav-ior and prices the debt accordingly, the equity hold-ers ultimately bear the agency cost of debt through ahigher cost of borrowing. Whereas the seminal workof Black and Scholes (1973), Jensen and Meckling(1976), and Myers (1977) gives us an understand-ing of risk-shifting incentives, this paper articulateswhat contributes to the riskiness of an inventoryitem. Namely, it shows how debt financing distorts aretailer’s inventory decision when the retailer ordersmultiple items that differ in cost, revenue, or demandparameters. In addition, we argue that this distortionis mitigated when the financing is provided by thesupplier who can observe the actual order quantitiesbefore determining the credit terms.

In reality, most firms use a combination of bankand supplier financing. The advantage of banks pro-viding credit stems primarily from their access tocapital markets and ability to diversify credit risk.A supplier, on the other hand, may have an advan-tage in being able to understand the buyer’s busi-ness, enforce repayment, or salvage the repossessedinventory upon the buyer’s default. Trade credit canbe also used to circumvent anti-price discriminationlaws, reduce precautionary money holdings, or miti-gate moral hazard. Rather than attempting to capture

all potential pros and cons of the two types of financ-ing, this paper simply describes one particular benefitof trade credit, which should play a role in a firm’schoice of the optimal financing strategy, but has notyet been identified in the literature.

We model a retailer with limited wealth who ordersinventory of two products while facing uncertaindemand. We first consider bank financing that theretailer obtains prior to purchasing inventory andwithout any covenants regarding order quantities.Assuming that the retailer maximizes the value ofequity, his order quantities are distorted by the lim-ited liability effect and deviate from the first-bestlevels that maximize total firm value. This scenariohinges on the assumption that the bank choosescredit terms before the retailer chooses inventory. Thiswould be the case with a standard or revolving lineof credit, which the borrower can tap at its discre-tion. The agency problem will not arise if the bankties financing to a particular transaction, e.g., througha letter of credit.

Under trade credit financing, the retailer’s incen-tives depend on the number of suppliers. When pur-chasing both products from the same supplier whoobserves the entire inventory order before determin-ing the credit terms, the retailer has the incentive toorder the first-best quantities. When each product isordered from a different supplier, trade credit cannoteliminate the retailer’s incentive to seek risk at theexpense of the first supplier when ordering from thesecond supplier.

Because banks have access to cheaper capital thansuppliers, trade credit does not necessarily dominate

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bank financing even when the retailer sources multi-ple items from a single supplier. Whether the retailerprefers bank financing or trade credit depends on themagnitude of the agency problem and the differencebetween the cost of capital of the bank and that of thesupplier. We also consider a mixed financing strategythat dominates pure trade credit financing by lever-aging the bank’s access to cheap capital and the sup-plier’s monitoring advantage.

An important assumption of our model is that thesupply market is competitive, i.e., there are manypotential suppliers offering similar products, none ofwhom are able to influence prices or credit terms.1

Free entry ensures that at equilibrium suppliers makezero economic profit. This has two important impli-cations. First, wholesale prices are equal to the unitproduction costs, which we assume to be constant.Second, trade credit is fairly priced, i.e., suppliersoffer credit terms at which they expect to break even.2

Consider a powerful retailer who proposes to apotential supplier a take-it-or-leave-it contract thatspecifies order quantities, up-front payment, anddelayed payment. In the presence of many compet-ing suppliers, the retailer will always choose contractterms at which the supplier expects to break even,and the supplier will always accept. Such a contractis equivalent to the supplier selling at the unit pro-duction costs and providing fairly priced credit. Anexample is a discount department store that carriesgeneric-brand clothing, cosmetics, electronics, or foodproducts, and sources multiple stock keeping unitsfrom any given supplier. Wal-Mart, for instance, isknown to rely heavily on trade credit financing, bor-rowing more from its considerably smaller suppliersthan in bank and bond markets despite its outstand-ing credit rating (Murfin and Njoroge 2015).

Our model provides the following managerialimplications. First, we demonstrate how the opti-mal order quantities of a multiproduct newsven-dor should be adjusted in the presence of debt.In particular, we show that relative to the classicalall-equity newsvendor, a shareholder-controlled butdebt-financed newsvendor should favor (i) items witha low salvage value, (ii) those with a high profit mar-gin, and (iii) those that represent a large portion ofthe total inventory investment.

1 The existence of many competing suppliers does not mean that aparticular retailer sources from more than one.2 This is a fairly common assumption in the trade credit literature(e.g., Biais and Gollier 1997, Wilner 2000, Burkart and Ellingsen 2004).This also means that trade credit interest reflects the actual credit risk,which seems to be supported by empirical data. Namely, Klapper et al.(2012) use a data set of almost 30,000 trade credit contracts of 56 largebuyers to show that early payment discounts (trade credit interest)tend to be offered to riskier buyers.

A basic tenet of the corporate finance theory isthat debt induces risk seeking. For example, Jensenand Meckling (1976, pp. 335–336) argue that a debt-financed firm is better off with a higher profit vari-ance, ceteris paribus. Interestingly, we show thatbetween two otherwise identical products, a debt-financed newsvendor may favor the product with lessvolatile demand. This occurs when the product withless volatile demand represents a larger portion of theinventory investment, and thus, a greater bankruptcyrisk.

Because the second-best risk-seeking strategy isoptimal after financing has been obtained, managershave a fiduciary duty to pursue it. To achieve the firstbest, they would have to commit to their inventorydecision before obtaining financing, for example, byrelying on trade credit and consolidating the numberof suppliers. According to our model, a firm bene-fits from trade credit financing most when it sourcesmultiple differentiated items from a single supplier,and when the bankruptcy risk and the limited lia-bility effect are significant. Finally, we show that thefirm may be best off by using a mixed financing strat-egy involving some bank credit, complemented withadditional supplier financing.

Relation to the LiteratureThe agency theory dates back to Black and Scholes(1973), who observed that in the presence of debt,the value of equity increases in the volatility of thereturn on the firm’s assets. Jensen and Meckling(1976) introduced the notion of “agency costs” thatstem from shareholders’ opportunity to extract wealthfrom debt holders by increasing risk. Myers (1977)showed that in addition to risk shifting, the existenceof debt can also lead to underinvestment. Myers andMajluf (1984) described the agency conflict that arisesbetween old and new shareholders when managersacting in the interest of the former know more aboutthe firm value than potential investors.

Despite the prominent role agency theory hasplayed in the finance literature, its operational impli-cations have received relatively little attention. Oneof the few exceptions is Chod and Zhou (2014),who consider a two-product firm that chooses theoptimal capital structure and, subsequently, investsin product-flexible and product-dedicated produc-tion capacity. They show that while leverage gener-ally leads to underinvestment and less flexibility, theagency problem disappears when the firm invests ina single fully flexible resource and, thus, cannot favorone type of capacity over another. This result has aparallel in the context of our model: A retailer thatsells a single undifferentiated product cannot engagein risk shifting by favoring the “riskier items” overthe “safer ones.” The incremental contribution of our

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Chod: Inventory, Risk Shifting, and Trade CreditManagement Science, Articles in Advance, pp. 1–20, © 2016 INFORMS 3

paper vis-à-vis Chod and Zhou (2014) is twofold.First, we examine how debt affects stocking levelsof products that differ in cost, revenue, and demandparameters, whereas Chod and Zhou study how debtaffects capacity to produce symmetrical, undifferenti-ated products. Second, we argue that trade credit mit-igates the agency cost associated with other forms ofdebt, whereas Chod and Zhou do not consider tradecredit at all.3

Another recent paper that studies operational re-percussions of financial leverage is by Iancu et al.(2016), who consider a lending contract between aretailer and a bank that is collateralized by inven-tory. The agency conflict arises when the retailer, afterobserving initial sales, must decide whether to liq-uidate inventory or continue to operate. Iancu et al.(2016) demonstrate that this conflict can be eliminatedby a suitably designed financial covenant, whichdepends on the borrower’s operational characteris-tics such as demand distribution, inventory depreci-ation, or profit margin. Our work is also related tothat of Van Mieghem (2007) and Chod et al. (2010),who examine the role of risk in multi-item newsven-dor networks. Whereas this literature assumes riskaversion, we endogenize risk attitude by taking intoaccount the role of external financing and limited lia-bility. Boyabatli and Toktay (2011) examine the effectof external financing on optimal technology choice,but no agency frictions arise in their model.

The question of why most firms provide tradecredit to their customers while receiving credit fromtheir suppliers in the presence of specialized finan-cial intermediaries, has puzzled financial economistsfor several decades. Among the existing explanations,our work is most related to those based on a moni-toring advantage of the supplier and those based onmoral hazard. The former argue that in the course oftheir business relationship the supplier gains superiorinformation about the buyer’s credit worthiness (see,e.g., Smith 1987, Brennan et al. 1988, Biais and Gollier1997, Jain 2001). The latter focus mostly on the moralhazard faced by the buyer. When product quality isnot immediately observable, deferred payment pro-vides the supplier with an incentive to exert effort andserves as a guarantee or a signal of product quality(see, e.g., Lee and Stowe 1993, Long et al. 1993, Babichand Tang 2012, Kim and Shin 2012).

The literature that links trade credit to the moralhazard faced by the lender, is much more limited.Cunat (2007) points out that the supplier of a uniqueinput has an advantage over a bank in controlling the

3 Another difference from Chod and Zhou (2014) is our assumptionthat equity is given. This reflects the fact that a firm is unlikely toraise new equity and reoptimize its capital structure each time itorders inventory.

buyer’s behavior. According to Burkart and Ellingsen(2004), the advantage of lending goods over lendingcash is that goods are more difficult for an oppor-tunistic borrower to divert for private benefits. Sim-ilar to Burkart and Ellingsen, we link the benefitof trade credit to an agency conflict between theborrower and the lender. However, whereas Burkartand Ellingsen consider cash diversion by an oppor-tunistic entrepreneur, we focus on asset substitutiondue to limited liability. Therefore, unlike Burkart andEllingsen, we must consider multiple products andbankruptcy risk.

Other theories attempt to explain the prevalenceof trade credit based on price discrimination (e.g.,Brennan et al. 1988, Mian and Smith 1992), transac-tion cost (Ferris 1981, Emery 1987), imperfect compe-tition (Emery 1984, Wilner 2000), and the supplier’sadvantage in salvaging the repossessed merchandisein case the buyer defaults (Frank and Maksimovic2004, Mian and Smith 1992). Petersen and Rajan(1997) tested the various theories empirically conclud-ing that trade credit is likely to serve multiple pur-poses. More recently, Giannetti et al. (2011) reportedempirical evidence favoring the theories based onborrower’s opportunism and suppliers’ informationaladvantage.

The operations literature has studied the effect oftrade credit on optimal inventory control (e.g., Haleyand Higgins 1973, Jaggi et al. 2008, Gupta and Wang2009, Luo and Shang 2013), as well as on supplychain performance (Chaharsooghi and Heydari 2010,Lee and Rhee 2010). Kouvelis and Zhao (2012) modelthe strategic interaction between a retailer and itssupplier, both of whom are financially constrainedand face bankruptcy risk. They characterize the opti-mal trade credit contract from the supplier’s perspec-tive, and they show that the retailer always prefersthis contract to bank financing and that trade creditimproves supply chain efficiency.

Yang and Birge (2013) consider the interaction be-tween a supplier and a retailer who finances inven-tory using the optimal mix of cash, trade credit, andshort-term debt. They demonstrate that trade creditenhances supply chain efficiency by serving as a risk-sharing mechanism. In particular, trade credit shiftssome of the inventory risk from the retailer to the sup-plier, which induces the retailer to increase its orderquantity. Dong et al. (2016) show that a manufacturercan use trade credit to induce a retailer to carry moreof the manufacturer’s products.

By identifying a new rationale for consolidating thenumber of suppliers, our paper also relates to the lit-erature on optimal supplier selection. This literaturegenerally ignores financing decisions with the notableexception of Babich et al. (2012), who study the opti-mal financing strategy and supplier selection, assum-ing that suppliers are unreliable and offer trade credit.

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The authors show that bank loans and trade creditare complementary, and that firms should use moresuppliers when bank financing is not available.

Seifert et al. (2013) provide a comprehensive surveyof the trade credit literature spanning all disciplines.As they point out, most existing trade credit mod-els consider a single product. We show that consider-ing multiple products, or “assets,” makes a differencebecause of the role that trade credit can play in alle-viating the asset substitution problem.

2. ModelIn the following, Ɛ denotes expectation, bold font de-notes vectors, and prime denotes transpose. We con-sider a retail firm that orders an inventory of twoproducts under demand uncertainty. Let Q= 4Q11Q25

′

denote the order quantities, and let c= 4c11 c25′ denote

the unit product costs, or wholesale prices. Assumingthat the upstream industry is competitive and suppli-ers incur constant marginal costs of production, wetake the wholesale prices as given.

Once demand uncertainty is resolved, the firmgenerates revenue 4D1Q5, where D = 4D11D25

′

represents demand shocks corresponding to the twoproducts. The random vector D is allowed to followan arbitrary continuous distribution over 2

+, and we

denote its mean vector and covariance matrix as =

41125′ and

è=

(

21 12

12 22

)

1

respectively. The revenue function 4D1Q5 is as-sumed to be continuous and nondecreasing in allarguments, and jointly concave in Q. Finally, weassume that i ≡ ¡/¡Qi, i = 112, exists almost every-where. These assumptions are satisfied, among others,in the following two instances.

Newsvendor Model. Suppose that the random vec-tor D represents actual product demands, the firmsells as much inventory as possible at given retailprices p = 4p11 p25

′, and it salvages the rest at givensalvage values s = 4s11 s25

′. In other words, the firm’srevenue is given by

4D1Q5=2∑

i=1

4pimin4Qi1Di5+si4Qi−min4Qi1Di5550 (1)

To avoid trivial scenarios, we assume si < ci < pi,i = 112.

Linear Demand Model. Suppose that the firm faceslinear demand curves that are subject to additiveuncertainty. Namely, the retail market-clearing priceof product i is given by pi4D, x5 = Di − xi − x3−i,where D is the vector of stochastic demand curveintercepts, x= 4x11x25

′ is the vector of quantities sold,

and ∈ 60115 is a measure of product substitutability,for i = 112. Suppose further that the firm chooses theretail prices or, equivalently, how much inventory tosell, after observing the realized demand shocks. Theunsold inventory is again salvaged at fixed salvagevalues s. In this case, the firm’s revenue is given by

4D1Q5 = maxx∈2

+

4x′p4D1x5+ 4Q− x5′s5

s.t. xi ≤Qi1 i = 1120 (2)

Regardless of the functional form of , it is usefulto define the following four events: Let ì0 be the setof demand shock realizations for which the firm endsup with excess inventory of both products; let ì1 (ì2)be the event in which the firm sells the entire inven-tory of product 1 (2), but ends up with excess inven-tory of product 2 (1); and let ì3 be the event in whichthe firm sells the entire inventory of both products. Itis also useful to define ìij ≡ ìi ∪ìj , i1 j ∈ 801112139.The formal characterization of these events, as wellas the functional form of corresponding to each ofthem, are given in (34) for the newsvendor model andin (35) for the linear demand model.

Let W be the firm’s remaining wealth after all non-inventory costs have been incurred. Assuming thatthis wealth does not suffice to finance the entireinventory investment, i.e., W < c′Q, we consider twoforms of external financing: bank loan and tradecredit.

2.1. Bank FinancingSuppose that before buying the inventory, the firmborrows L dollars from a bank, promising to repayL+r dollars once the selling season is over. Assumingthat debt covenants prohibit the firm from paying outdividends before repaying the loan, the firm has noreason to borrow more than what it intends to spendon inventory, i.e.,

L+W = c′Q0 (3)

Since it is unusual for debt covenants to prescribe aspecific operational policy, we assume that it is withinthe firm’s discretion to allocate the available capitalL+W between the inventory of the two products.

Depending on the realized demand, the firm mayor may not be able to repay the debt in full. Thus, wedistinguish two events:

(a) If 4D1Q5 ≥ L + r , the firm’s revenue is suffi-cient to repay the debt in full. The value of debt isL+ r , whereas the value of equity is 4D1Q5−L− r0

(b) If 4D1Q5 < L+ r , the firm is unable to repaythe debt in full, declares bankruptcy, and is takenover by the lender. Assuming a fixed bankruptcy cost

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Figure 1 Partitioning of the Demand Shock State Space for the Newsvendor and Linear Demand Models, Where the Shaded Areas Correspond to theBankruptcy Event ìb

Ω2b Ω2a Ω3a

Ω0b

Ω0a

Ω1b

Ω1a

D2D2

D1

2Q2 + s2

2 L + r – Q22 – sQ + s1

s1

s2

2Q1 + s1 L + r – s2Q2 + Q1Q1

L + r – Q21 – sQ + s22

(b) Linear demand model

Ω2b Ω2a Ω3a

Ω0b

Ω0a

Ω1b

Ω1a

D1

(a) Newsvendor model

L + r – s1Q1 – p2Q2

p1 – s1

L + r – p1Q1 – s2Q2

p2 – s2

Q1

Q2

Q2L + r – s1Q1

Q2

+

denoted as b, the payoff to the lender is 4D1Q5− b.4

The equity becomes worthless.We denote these two events as ìa and ìb, respec-

tively, so that subscript b is mnemonic for bankruptcy.If we further define ìij ≡ìi ∩ìj for i ∈ 801112139 andj ∈ 8a1 b9, we can distinguish seven events, which areillustrated in Figures 1(a) and 1(b) for the newsvendorand linear demand models, respectively.5

Following the literature (e.g., Biais and Gollier 1997,Wilner 2000, Burkart and Ellingsen 2004), we assumethat debt is fairly priced, i.e., the bank expects to makezero profit. If D ∈ ìa, the bank is repaid the princi-pal plus interest, L + r . If D ∈ ìb, the bank receivesthe firm’s revenue minus the bankruptcy cost, − b.Therefore, fair pricing requires that

41 +B5L= Pr4ìa54L+ r5+ Pr4ìb5Ɛ4 − b ìb51 (4)

where B denotes the bank’s cost of capital, i.e., theinterest rate at which the bank obtains funds.

We also assume that the firm is fully controlledby shareholders who maximize the expected terminalvalue of equity, which we denote by V . Because thevalue of equity is −L− r if D ∈ìa, and it is zero ifD ∈ìb, we have

V 4Q1L1 r5= Pr4ìa5Ɛ4 −L− r ìa50 (5)

The firm chooses the amount of borrowing andinventory in two stages. In the second stage, the credit

4 We assume that upon default, the creditor initiates the bankruptcyprocess before observing the retailer’s liquidation value and, thus,initiates bankruptcy even if b > .5 The exact shape of the bankruptcy region in Figures 1(a) and 1(b)depends on model parameters, as we discuss later.

terms 4L1 r5 are given and the firm chooses the opti-mal inventory levels. Using the superscript BF todenote the optimal solution under bank financing, thesecond-stage problem can be written as

QBF 4L1 r5= arg maxQ≥0

V 4Q L1 r5 subject to (3). (6)

In the first stage, the firm chooses the optimal loanamount, taking into account the interest determinedby the bank according to (4). In this stage, both thefirm and the bank anticipate the firm’s second-stageinventory choice given by (6). Formally, we have

LBF= arg max

L≥0V 4L1 rBF 4L51QBF 4L551 (7)

where rBF 4L5 and QBF 4L5 satisfy (4) together with (6).To simplify the notation, we let QBF ≡QBF 4LBF 5, rBF ≡

rBF 4LBF 5, and V BF ≡ V 4QBF 1LBF 1 rBF 5. The next lemmacharacterizes the optimal loan and inventory levelsassuming they are all positive.

Lemma 1. For any given loan amount L, the optimalinventory QBF 4L5 and the corresponding interest rBF 4L5must satisfy

Ɛ41 ìa5

c1=

Ɛ42 ìa5

c21 (8)

together with (3) and (4). The optimal loan amount LBF

must satisfyƐ41 ìa5

c1= 1 +

drBF

dL0 (9)

The optimality condition (8) provides the ruleaccording to which the firm allocates its availablecapital between the inventory of the two products.

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Chod: Inventory, Risk Shifting, and Trade Credit6 Management Science, Articles in Advance, pp. 1–20, © 2016 INFORMS

It ensures that the last dollar invested in the inventoryof each product has the same expected payoff. Whenevaluating this payoff, the firm takes into accountonly nonbankruptcy states because the revenue real-ized in bankruptcy states accrues to the bank, notthe shareholders. The optimality condition (9) equatesthe expected revenue generated for the shareholdersby the last dollar borrowed and the correspondingincrease in the loan principal and interest that theshareholders expect to repay.

To understand exactly how debt distorts the firm’sinventory decision, we need to define the “first best.”Note that our definition of the first-best strategyhas to do with the agency conflict between theshareholders and the debt holders, rather than withthe misalignment of incentives between the retailerand the supplier.

2.2. First BestIn this section, we consider an alternative scenario inwhich the firm chooses the amount of borrowing andinventory that jointly maximize total firm value, i.e.,the value of equity plus the value of debt. Otherwise,we keep all of our assumptions unchanged. Namely,we continue to assume that the firm’s wealth is lim-ited, debt is fairly priced, and the lender’s cost of cap-ital is B. Formally, we define the first-best strategy as

Q∗1L∗1 r∗= arg max

Q1L1 r≥06Ɛ4Q5− c′Q−BL− Pr4ìb5b7

subject to (3) and (4), (10)

and we let V ∗ = V 4Q∗1L∗1 r∗50The reason why a bank-financed firm does not gen-

erally achieve the first best is that after obtaining theloan, it has the incentive to choose inventory thatmaximizes the value of equity rather than the totalfirm value. The bank can anticipate this and pricesthe loan accordingly. The resulting loss of total firmvalue, known as the agency cost of debt, V ∗ −V BF , isthus ultimately borne by the shareholders through ahigher cost of borrowing. The next lemma character-izes the solution to (10) when the optimal inventorylevels and loan amount are positive.

Lemma 2. The first-best inventory Q∗ must satisfy

Ɛ1 = 41 +B5c1 +dPr4ìb5

dQ1b and

Ɛ2 = 41 +B5c2 +dPr4ìb5

dQ2b1

(11)

where L and r embedded in ìb are given by (3) and (4).

The optimality conditions (11) equate the expectedmarginal revenue and expected marginal cost ofthe two products, respectively. In general, the first-best inventory levels depend on the bankruptcy cost

because the amount of inventory affects the prob-ability of bankruptcy. In the special case of zerobankruptcy cost, the optimality conditions (11) canbe combined into the following rule, according towhich the firm allocates total capital between the twoproducts:

Ɛ1

c1=

Ɛ2

c20 (12)

Recall the analogous rule that we established inLemma 1 for the bank financing scenario:

Ɛ41 ìa5

c1=

Ɛ42 ìa5

c20 (13)

The difference between the two allocation rules, (12)and (13), is that a bank-financed firm takes intoaccount only the revenue that accrues to shareholders,i.e., the revenue generated in nonbankruptcy statesìa. In the next section, we examine how exactly thisaffects the stocking levels.

3. Risk Shifting and InventoryIn this section, we characterize the effect of debt onoptimal inventory for several special cases, and verifythe robustness of our analytical findings in a series ofnumerical experiments. It is well known that in thepresence of debt, equity is equivalent to a call optionon the firm’s assets, and its value increases in volatil-ity of the return on these assets (Black and Scholes1973). One would, therefore, expect a debt-financedfirm to favor the “riskier” of the two products. Thequestion is what makes an inventory item risky. Weaddress this issue in the next proposition assumingthat the revenue function is given by the newsven-dor or linear demand model.

Proposition 1. Suppose that b = 0 and is given bythe newsvendor model (1) or by the linear demand model(2) with = 0. Suppose further that

4i5 Pr4ì2b4QBF 55 > 01 4ii5 Pr4ì1b4QBF 55= 01

and 4iii5 s1/c1 ≤ s2/c20(14)

Under bank financing, the firm overinvests in product 1,or it underinvests in product 2, i.e.,

QBF1 >Q∗

1 or QBF2 <Q∗

20 (15)

The three conditions in (14) ensure that product 1is in some sense riskier than product 2. Accordingto (i), low sales of product 1 can lead to bankruptcyeven if product 2 is sold out. According to (ii), lowsales of product 2 cannot result in bankruptcy as longas product 1 is sold out. This situation is illustratedin Figure 2 for both revenue models. Finally, condi-tion (iii) ensures that product 1 has a lower relativesalvage value. In the remainder of this section, weexamine how risk shifting depends on various modelparameters, focusing on the newsvendor model.

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Chod: Inventory, Risk Shifting, and Trade CreditManagement Science, Articles in Advance, pp. 1–20, © 2016 INFORMS 7

Figure 2 Partition of the Demand State Space Under Conditions (i) and (ii) in (14)

D2

D1

2Q1 + s1

D2

D1Q1

Q2

L + r – sQp1 – s1

2Q2 + s2

s1

s2

L + r – s1Q1 – p2Q2p1 – s1

2 L + r – Q22 – sQ + s1

Ω0a

Ω0a

Ω2bΩ2a

Ω2a

Ω3a

Ω3a

Ω0b

Ω0b

Ω1a

Ω1a

Ω2b

(a) Newsvendor model (b) Linear demand model

2 L + r – sQ + s1

Q2L + r – s1Q1

Q2

+

Notes. The shaded areas represent the bankruptcy regions when conditions (i) and (ii) in (14) are satisfied. Product 1 is riskier because its low sales can leadto bankruptcy even if product 2 is sold out.

3.1. Newsvendor ModelWhen the revenue function is given by (1), condi-tions (i) and (ii) in (14) can be written as

4p2 − c25QBF2 − 4c1 − s15Q

BF1 < rBF −W1 (16)

and 4p1 − c15QBF1 − 4c2 − s25Q

BF2 ≥ rBF −W1 (17)

respectively. Thus, we would expect a bank-financedfirm to favor product 1 as a result of one or severalof the following factors:

1. product 1 has a low salvage value; product 2 hasa high salvage value

2. product 1 has a high profit margin; product 2 hasa low profit margin

3. product 1 represents a large proportion of totalinventory, e.g., because of high mean demand.

In the following analysis, we examine how thedistortion of the inventory decision depends onindividual product parameters. When the productsare symmetrical (p1 = p2, c1 = c2, s1 = s2, 1 = 2, and1 = 2), the firm does not have an incentive to favoreither of them, thus, it follows the first-best strategy,i.e., QBF

1 = QBF2 = Q∗

1 = Q∗2 . We start by considering

this symmetrical case, and examine how the optimalinventory levels deviate from the first best, as one ofthe model parameters changes by a sufficiently smallamount.

We complement this analysis by a series of numeri-cal experiments in which we also begin with the fullysymmetrical “base case,” and examine the impact ofincreasing asymmetry in various model parameterson the inventory decision, cost of borrowing, and the

agency cost of debt. In these experiments, we assumethat demand follows a bivariate lognormal distribu-tion, and we use the following base-case parametervalues: ci = 1, pi = 102, si = 0, i = 100, and i = 75 fori = 112; W = 0, B = 0, b = 5, and ∈ 8−00510100759.

We consider a relatively low profit margin and arelatively high coefficient of variation of demand tocapture situations in which the bankruptcy risk, andtherefore, the agency cost of debt, are significant. Withour base-case parameter values, the probability ofbankruptcy ranges between 8%–11%, depending ondemand correlation. Setting W = 0 means that thefirm’s internal capital covers all noninventory costs,whereas the inventory is financed entirely by debt.When W > 0, our results are qualitatively the same,but of a smaller magnitude. Our choice of bankruptcycost, b = 5, means that at the base-case parametervalues, the bankruptcy cost represents, dependingon demand correlation, 6.8%–8.2% of the expectedfirm value at default Ɛ4 ìb5. This is in line withempirical data. For example, Bris et al. (2006) esti-mate the default cost to vary between 0%–20% ofthe firm value at default. Although we ran all ofour numerical experiments for three different val-ues of demand correlation, we only show results forthe correlation at which the agency cost is most sig-nificant, and which we indicate in the figure cap-tions. The numerical study is based on simulating100,000 bivariate demand scenarios, and the optimiza-tion is done directly using “Global Search” algorithmin Matlab.

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Figure 3 Effect of Increasing Salvage Value, s2, On Optimal Inventory Levels, Interest Rate, Total Inventory Investment (Borrowing), and theRelative Agency Cost of Debt for = 0075

00

0.3 0.6 0.9

20

40

60

80

100

120

s2

(a) Inventory levels

Q2*

Q1*

Q2BF

Q1BF

0

20

40

60

80

100

120

140

160

0 0.3 0.6 0.9s2

(c) Total inventory investment (borrowing)

L*

LBF

0

1

2

3

0 0.3 0.6 0.9s2

(%)

(b) Interest rate

(%)rBF

r*

L*

LBF

%

0

1

2

3

4

5

6

0 0.3 0.6 0.9s2

(%)

(d) Agency cost of debt

V* – VBF%

VBF

Salvage Value. The next proposition characterizesthe effect of salvage value assuming that ì12b4Q∗5= ,i.e., the firm cannot go bankrupt when at least one oftwo products is sold out.

Proposition 2. Assume that b = 0, all product param-eters are symmetrical, and ì12b4Q∗5 = . Now supposethat the salvage value of product 2 increases by . Underbank financing, the proportion of product 2 in the inven-tory mix increases less than it does under the first-beststrategy, i.e.,

[

d

ds2

QBF2

QBF1 +QBF

2

]

s2=s1

≤

[

d

ds2

Q∗2

Q∗1 +Q∗

2

]

s2=s1

0 (18)

A debt-financed firm ignores the salvage revenuegenerated in bankruptcy states. As a result, the prod-uct with a higher salvage value represents a smaller

portion of the firm’s total inventory than in the first-best scenario. This effect is illustrated in Figure 3.When s2 = s1 = 0, the products are symmetrical andthe firm orders the same quantity of each. As s2

increases, product 2 becomes more attractive and itsoptimal inventory level increases (Figure 3(a)). Thehigher salvage value of product 2 also reduces creditrisk and, hence, the interest rate, r/L (Figure 3(b)).The lower cost of capital, together with the highersalvage value of one of the products, results in alarger total inventory investment and the correspond-ing loan, c′Q = L (Figure 3(c)). The lower cost ofcapital means that the firm increases the order quan-tities of both products, but the increase is more sig-nificant for product 2, whose salvage value has goneup (Figure 3a).

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Figure 4 Effect of Increasing Retail Price, p2, on Optimal Inventory Levels and the Relative Agency Cost of Debt for = −005

p2

0

20

40

60

Q2*

Q1*

Q2BF

Q1BF

(a) Inventory levels

1.15 1.20 1.25 1.30 1.35 1.400

2

4

6

8

p2

(b) Agency cost of debt

%

1.15 1.20 1.25 1.30 1.35 1.40

(%)V* – VBF

VBF

Although all of these effects are similar under bankfinancing and the first-best scenario, they have dif-ferent magnitudes. Consistent with Proposition 2, abank-financed firm underinvests in the high-salvage-value product, QBF

2 < Q∗2 , and overinvests in the

low-salvage-value product, QBF1 > Q∗

1 (Figure 3(a)).Because the bank anticipates this distortion of thefirm’s inventory decision, it charges a higher inter-est rate compared to the first best (Figure 3(b)), towhich the firm responds by borrowing and invest-ing less (Figure 3(c)). Finally, Figure 3(d) shows theagency cost of debt as a percentage of shareholdervalue, 44V ∗ −V BF 5/V BF 5100%. As the salvage value ofproduct 2 increases, so does the difference in the risk-iness of the two products. As a result, risk shiftingbecomes more significant and the agency cost of debtincreases.

Unit Revenue 4Retail Price5. The next propositioncharacterizes the effect of asymmetry in unit rev-enues, assuming that demands are independent andfollow distributions with an increasing failure rate(IFR), such as the normal or the uniform distribution.

Proposition 3. Assume that b = 0, all product param-eters are symmetrical, and demands are independent andfollow IFR distributions. Now suppose that the unit rev-enue of product 2 increases by . Under bank financing,the proportion of product 2 in the inventory mix increasesmore than it does under the first-best strategy, i.e.,

[

d

dp2

QBF2

QBF1 +QBF

2

]

p2=p1

≥

[

d

dp2

Q∗2

Q∗1 +Q∗

2

]

p2=p1

0 (19)

The product with a higher unit revenue representsa larger inventory investment and a larger risk ofexcess inventory. As a result, sales of this product aremore critical to whether the firm goes bankrupt or

not. In this sense, the high-margin product is riskier.Therefore, it is favored by a debt-financed firm. Thiseffect is illustrated in Figure 4(a). When p2 = p1 = 102,the order quantities of the two products are equaland are not distorted by the agency problem. Whenp2 6= p1, a bank-financed firm overinvests in the high-margin, high-inventory product, and underinvests inthe low-margin, low-inventory product, as predictedby Proposition 3. The larger the asymmetry in profitmargins, the larger the distortion of the inventorydecision, and the larger the agency cost of debt (Fig-ure 4(b)). Our numerical experiments also indicatethat the agency problem increases the cost of bor-rowing, rBF /LBF ≥ r∗/L∗, and decreases the inventoryinvestment, c′QBF ≤ c′Q∗.

Unit Cost 4Wholesale Price5. The effect of the unitcost is similar to that of the unit revenue in the sensethat a debt-financed firm favors, once again, the high-margin product.

Proposition 4. Assume that b = 0, s1 = s2 = 0, allproduct parameters are symmetrical, and demands are inde-pendent and follow IFR distributions. Now suppose thatthe cost of product 2 increases by . Under bank financing,the proportion of product 2 in the inventory mix decreasesmore than it does under the first-best strategy, i.e.,

[

d

dc2

QBF2

QBF1 +QBF

2

]

c2=c1

≤

[

d

dc2

Q∗2

Q∗1 +Q∗

2

]

c2=c1

0 (20)

A debt-financed firm favors the low-cost prod-uct because it represents a larger portion of totalinventory, and its sales are the main driver of thebankruptcy risk.6 To examine the effect of asymmetryin demand parameters, we resort directly to numeri-cal results.

6 The reason why Proposition 4 assumes s1 = s2 = 0 is that, in gen-eral, the secondary effect of a lower unit cost is a higher relative

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Figure 5 Effect of Increasing Mean Demand, 2, on Optimal Inventory Levels and the Relative Agency Cost of Debt for = −005

0

2

4

6

8

125100752

(b) Agency cost of debt

%

0

10

20

30

40

50

60

70

12510075

(a) Inventory levels

Q2BF

Q2*

Q1*

Q1BF

2

(%)V* – VBF

VBF

Mean Demand. The effect of asymmetry in meanproduct demands is shown in Figure 5. At 2 = 1 =

100, the products are fully symmetrical and thereis no agency cost of debt. When 2 6= 1, the low-demand product has less inventory, and thus, is lessrisky, in the sense that its low sales are unlikelyto cause bankruptcy as long as the other productsells out. Therefore, consistent with Proposition 1, abank-financed firm overinvests in the high-demand,high-inventory product, and underinvests in the low-demand, low-inventory product, as shown in Fig-ure 5(a). Figure 5(b) illustrates how this distortion ofthe inventory decision impacts firm value.

Demand Volatility. As shown in Figure 6(a), abank-financed, risk-seeking firm overinvests in thelow-volatility product and underinvests in the high-volatility product. However surprising this may seem,it is consistent with Proposition 1 and conditions(16)–(17), according to which a debt-financed firmfavors the item that represents a large portion oftotal inventory. When profit margins are relativelylow, as is the case in our numerical experiments, thehigh-volatility product represents a smaller portion oftotal inventory. Therefore, its low sales are unlikely tocause bankruptcy as long as the low-volatility, high-inventory product sells out. Thus, the low-volatility,high-inventory product turns out to be riskier. Assuch, it is favored by a bank-financed retailer.

In newsvendor-like models and under Gaussiandemand, the optimal inventory level depends onthe product of the demand standard deviation andthe z-value. Thus, the fact that higher demand

salvage value, s1/c1, which reduces this product’s riskiness. It isstraightforward to show that Proposition 4 continues to hold if,instead of imposing s1 = s2 = 0, we vary s2 together with c2, keepingthe relative salvage value, s2/c2, constant.

volatility leads to lower inventory is true only fornegative z-values, which correspond to low profitmargins. Therefore, we expected to observe the oppo-site effect for high-margin products whose optimalinventory increases in demand volatility. However,in our numerical experiments based on lognormaldemand distribution, optimal inventory does notincrease in demand volatility unless the newsven-dor ratio 4pi − ci5/pi ¦ 008, i.e., pi ¦ 5. With such highprofit margins, the firm cannot go bankrupt with oneof the products being sold out (ì12b = ), in whichcase asymmetry in demand volatility alone does notresult in any risk shifting. In other words, we did notfind any parameter combination for which we wouldobserve results contrary to those shown in Figure 6.

As we have seen, when any parameter of oneinventory item changes, the optimal order quantitiesof both items change in response. This is becauseboth order quantities depend on the cost of borrow-ing, which in turn depends on characteristics of bothproducts. As product 2 becomes more attractive (say,its salvage value, s2, or unit revenue, p2, increases),the marginal cost of borrowing decreases, which pro-vides the firm with an incentive to increase both orderquantities. At the same time, the firm has an incen-tive to reallocate some of its inventory budget awayfrom product 1 toward product 2. As a consequence ofthese two opposing effects, the optimal order quantityof product 1 can increase or decrease.7 For example,QBF

1 increases in s2 (Figure 3(a)), but mostly decreasesin p2 (Figure 4(a)). When the products are substitutesor complements from the consumers’ point of view,their optimal order quantities are further linked onthe revenue side through cross-price elasticity. This isthe case under the linear demand model.

7 These two effects are analogous to the income and substitutioneffects in the standard consumer choice model.

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Figure 6 Effect of Increasing Demand Volatility, 2, on Optimal Inventory Levels and the Relative Agency Cost of Debt for = −005

(a) Inventory levels

0

20

40

60

1007550

Q2*

Q1*

Q2BF

Q1BF

2

(b) Agency cost of debt

0

0.3

0.6

0.9

1.2

1007550

2

%

(%)V* – VBF

VBF

3.2. Linear Demand ModelThe electronic companion (available as supple-mental material at http://dx.doi.org/10.1287/mnsc.2016.2515) provides a similar numerical analysis forthe linear demand model (2). It examines the effectof asymmetry in product-specific parameters, ci, i,and i, under different levels of product substitutabil-ity . The results are consistent with the newsvendormodel in the following sense. As we have seen in Fig-ures 4(a), 5(a), and 6(a), when the two products differin profit margin, mean demand, or demand volatil-ity, a bank-financed newsvendor overinvests in theproduct that represents a larger portion of the totalinventory, and underinvests in the other product, i.e.,

QBFi ≥Q∗

i ≥Q∗

j ≥QBFj 0

We observe exactly the same phenomenon underthe linear demand model. Moreover, in all of ournumerical experiments, the relative agency cost,4V ∗ −V BF 5/V BF , increases with product substitutabil-ity . As the products become closer substitutes, theirprofit margins shrink, bankruptcy risk increases, andso does the risk-shifting problem.

4. Trade Credit4.1. Single SupplierSuppose, once again, that the firm orders inventoryQ at wholesale prices c1 while its wealth W is lessthan the total inventory cost c′Q. Suppose further thatboth items are ordered from the same supplier whoprovides the entire residual financing, L= c′Q−W , inthe form of trade credit. The interest is again denotedas r , so the firm pays the supplier W up front and

L+ r at the end of the selling season.8 Consistent withour assumption of a competitive upstream indus-try and the finance literature (e.g., Biais and Gollier1997, Wilner 2000, Burkart and Ellingsen 2004), weassume that, similar to bank credit, trade credit isfairly priced.

Trade credit differs from bank credit in two ways.First, following Burkart and Ellingsen (2004), weassume that the supplier faces a higher cost of capi-tal than do banks. Denoting the interest rate at whichthe supplier obtains funds as S , fair pricing of tradecredit requires

41 +S5L= Pr4ìa54L+ r5+ Pr4ìb5Ɛ4 − b ìb50 (21)

Second, we assume that unlike a bank, the suppliersets the interest rate after observing the actual orderquantities. Using superscript TC for the optimal solu-tion under trade credit financing, the firm’s decisionproblem can be written as

QTC=arg max

Q≥0V 4Q1LTC4Q51 rTC4Q551 (22)

where the value of equity V is given by (5), andthe credit terms, LTC4Q5 and rTC4Q5, are given by(3) and (21). Let LTC ≡ LTC4QTC5, rTC ≡ rTC4QTC5,and V TC ≡ V 4QTC1LTC1 rTC5 be the optimal amount

8 This contract corresponds to a common form of trade credit termsknown as “two-part terms,” which specify the early payment dis-counted price, the discount period, the payment due date, and theeffective interest rate (Klapper et al. 2012). In our model, these cor-respond to c, the delivery time, the end of the selling season, andr/L, respectively. It is fairly common for firms to delay paymentsto their suppliers beyond the terms stipulated contractually (e.g.,Giannetti et al. 2011). Although the greater leniency of suppliersrelative to banks may be an important consideration in practice, itis outside the scope of our static model.

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of trade credit, and the corresponding interest andequity value, respectively. The next proposition for-malizes the advantage of using trade credit.

Proposition 5. Under trade credit financing, the firmchooses the first-best inventory levels, for which it receivesthe first-best credit terms. Formally, 4QTC1LTC1 rTC5 solves(10) with B replaced by S .

There is no agency cost associated with trade creditbecause the trade credit interest is contingent on theinventory decision. The retailer knows that taking toomuch risk when ordering inventory would automat-ically result in less favorable credit terms. The orderquantities that maximize the value of equity undersupplier financing are those that maximize total firmvalue.

At the same time, the supplier faces a higher cost ofcapital than a bank, S >B. This means that for anygiven order quantities, the supplier requires a higherinterest rate from the retailer than does a bank. Obvi-ously, if there is no agency cost associated with bankfinancing, such as when the two products are com-pletely symmetrical or there is no demand risk, theretailer prefers the cheaper bank credit. In the moreinteresting case in which the agency cost is positive,V BF < V ∗, there is a trade-off between using tradecredit and bank financing, which we characterize inthe next corollary to Proposition 5.

Corollary 1. Suppose that V BF < V ∗. The retailerprefers trade credit over bank financing, i.e., V TC > V BF ,as long as the difference between S and B is sufficientlysmall. When this difference is sufficiently large, the retailerprefers bank financing over trade credit, i.e., V TC <V BF .

Because it eliminates risk-shifting incentives, sup-plier financing can dominate bank financing despitethe supplier’s cost of capital being higher than thatof the bank, as long as the difference is not too large.A natural question arises whether a retailer can com-bine the benefit of trade credit with that of bankfinancing by using a mixed financing strategy.

4.2. Mixed FinancingSuppose that the retailer first obtains a fairly pricedbank loan, and let LB and rB denote its principal andinterest, respectively. As is common is practice (e.g.,Longhofer and Santos 2000 and references therein),we assume that this bank loan is given seniority withrespect to all subsequent creditors. The retailer thenorders inventory, using trade credit for the remainingamount LS = c′Q−LB −W . The supplier provides thegoods and the financing, and charges fair interest rS .At the end of the selling season, one of two events,denoted again as ìa and ìb, takes place.

(a) If 4D1Q5≥ LB +LS + rB + rS , both creditors arefully repaid, and the value of equity is V = − LB −

LS − rB − rS .(b) If 4D1Q5 < LB + LS + rB + rS , the retailer

defaults, the bank receives min4 − b1LB + rB5, thesupplier receives max401−b−LB −rB5, and the valueof equity is zero.

The expected value of equity is therefore V =

Pr4ìa5Ɛ4 − LB − LS − rB − rS ìa5. Using superscriptMF for the optimal mixed financing strategy, theretailer’s second-stage problem is

QMF 4LB1 rB5= arg maxQ≥0

V 4Q1LS1 rS LB1 rB51 (23)

where LS = c′Q− LB −W , and rS is given by the fol-lowing fair pricing condition:

41 +S5LS = Pr4ìa54LS + rS5

+ Pr4ìb5Ɛ4max401 − b−LB − rB5 ìb50 (24)

The retailer’s first-stage problem is

LMFB = arg max

LB≥0V 4Q1LS1 rS1LB1 rB51 (25)

where Q is given by (23), LS = c′Q−LB −W , rS is givenby (24), and rB is given by the following fair pricingcondition:

41 +B5LB = Pr4ìa54LB + rB5+ Pr4ìb5

· Ɛ4min4 − b1LB + rB5 ìb50 (26)

Let VMF ≡ V 4QMF 1LMFB 1LMF

S 1 rMFB 1 rMF

S 5. The next pro-position confirms our intuition about the advantageof mixed financing.

Proposition 6. The retailer prefers mixed financingover trade credit financing, i.e., VMF ≥ V TC . If is givenby the newsvendor model (1) or by the linear demandmodel (2), s> 0, W > 0, and b = 0, the preference is strictand

VMF−V TC

≥ 4S −B5W min8s1/c11 s2/c29

1 +B

0 (27)

The first part of the proposition is not surprising.The retailer cannot be worse off given the optionto borrow any amount from a bank before goingto the supplier. The lower bound on the benefit ofmixed financing (27) is based on a particular mixedfinancing strategy, which is not necessarily optimalbut is feasible and strictly dominates trade creditfinancing as long as inventory has some salvagevalue, the retailer has some wealth, and there is nobankruptcy cost. Suppose that the retailer borrows4W min8s1/c11 s2/c295/41 + B5 from a bank, which isan amount that is guaranteed to be repaid in full,

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including the fair interest, regardless of demand real-ization. Because this loan is not risky, it does not dis-tort the retailer’s incentives when he orders inventoryand requests additional credit from the supplier. Theretailer thus orders the same first-best quantities as inthe trade credit scenario, while enjoying a lower inter-est rate applied by the bank to the secured portion oftotal debt. In practice, a firm can use a similar mixedfinancing strategy even if s= 0, W = 0, and b > 0, aslong as it has some assets to collateralize the bankloan.9

The benefit of trade credit hinges on the assumptionthat both products are ordered from the same sup-plier. In the next section, we examine what happensif we depart from this assumption.

4.3. Two SuppliersSuppose that the retailer buys each product from adifferent supplier and relies entirely on trade creditfinancing. We consider the following sequence ofevents. The retailer first orders Q1 units of product 1from supplier 1 at unit cost c1. Supplier 1 providesthe goods as well as full trade credit financing L1 =

c1Q1, and charges fair interest r1. Subsequently, a sim-ilar transaction takes place between the retailer andsupplier 2.

Let ìa ≡ 8D2 4D1Q5 ≥ L1 + r1 + L2 + r29 be theevent in which the retailer repays both suppliers infull, and let ìb ≡2

+\ìa be the complementary event,

in which the retailer goes bankrupt. We assume thatupon bankruptcy, the suppliers split the liquidationvalue of the retailer proportionally to the face valueof their claims.10 Assuming that both suppliers incurthe same cost of capital S , fair pricing of trade creditrequires

41 +S5Li = Pr4ìa54Li + ri5+ Pr4ìb5Ɛ4 − b ìb5

·Li + ri

L1 + r1 +L2 + r21 for i = 1120 (28)

Interestingly, conditions (28) imply that both suppli-ers charge the same interest rate, i.e., r1/L1 = r2/L2,regardless of the amounts of credit provided, productcharacteristics, or the lending sequence.

The retailer maximizes the value of equity V =

Pr4ìa5Ɛ4 − L1 − r1 − L2 − r2 ìa5 in two stages. Inthe second stage, the retailer chooses how much toorder from the second supplier, taking the inventory

9 If the firm can provide enough collateral to secure the entirefinancing, there is no credit risk, and pure bank financing is alwaysoptimal.10 This assumption is consistent with U.S. bankruptcy law, accord-ing to which in the absence of collateral, trade creditors are nor-mally treated as general unsecured creditors, who obtain pro ratashares of the debtor’s remaining assets, in accordance with the sizeof their claims.

of product 1 and the corresponding trade credit con-tract as given, i.e.,

QTC2 4Q11L11r15=argmax

Q2≥0V 4Q21L21r2 Q11L11r151 (29)

where L2 = Q2c2 and r2 satisfies (28) for i = 2. In thefirst stage, the retailer chooses how much to orderfrom the first supplier, i.e.,

QTC1 = arg max

Q1≥0V 4Q1L1 r51 (30)

where Li = Qici and ri satisfies (28) for i = 112; andwhere Q2 =QTC

2 4Q11L11 r15 is the anticipated solutionto the second-stage problem (29).

In the next lemma, we compare the optimal quan-tity ordered from the second supplier under tradecredit financing with its first-best counterpart.

Lemma 3. The first-best inventory of product 2, Q∗2 ,

and the optimal inventory of product 2 chosen under tradecredit financing, QTC

2 , must satisfy the following condi-tions, respectively:

Pr4ìa5Ɛ42 ìa5+

(

Pr4ìb5Ɛ42 ìb5− bdPr4ìb5

dQ2

)

= 41 +S5c21 (31)

and

Pr4ìa5Ɛ42 ìa5+

(

Pr4ìb5Ɛ42 ìb5− bdPr4ìb5

dQ2

)

·L2 + r2

L1 + r1 +L2 + r2+ Pr4ìb5Ɛ4 − b ìb5

·d

dQ2

(

L2 + r2

L1 + r1 +L2 + r2

)

= 41 +S5c20 (32)

The first-best condition (31) is analogous to the oneprovided in Lemma 2, but it is written so that wecan distinguish the payoffs in nonbankruptcy andbankruptcy states. The optimality condition (32) dif-fers from its first-best counterpart in two ways. First,the retailer takes into account only a portion of thepayoff realized in bankruptcy states, namely, the por-tion that is allocated to the second supplier, 4L2 + r25/4L1 +r1 +L2 +r25. This gives the retailer an incentive tounderinvest. Second, the retailer takes into account theeffect of the order quantity on this allocation, whichis captured by the term 4d/dQ2544L2 + r25/4L1 + r1 +

L2 + r255. The larger the order from the second sup-plier, the larger this supplier’s claim on the retailer’sliquidation value. This provides an incentive to over-invest. It is difficult to analytically determine whichof the above two effects dominates, except for the fol-lowing special case.

Proposition 7. Suppose b = 0, ¡i/¡Qj = 0 for i 6= j1

and 24D1QTC5 = 0 for any D ∈ ìb4QTC5. The retaileroverorders from the second supplier in the sense thatQTC

2 >Q∗2 .

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The assumptions b = 0 and ¡i/¡Qj = 0 (no cross-price effect) are made for tractability. The conditionthat 2 = 0 for any D ∈ ìb (the last unit of prod-uct 2 has no value in bankruptcy states) eliminates theunderinvestment incentive. The retailer has an incen-tive to overorder product 2 because it increases theclaim of the second supplier on the retailer’s assetsin case of bankruptcy. In general, trade credit alignsincentives between the retailer and the supplier whoprovides this particular credit. However, it cannoteliminate the retailer’s incentive to seek risk at theexpense of the suppliers who provided trade creditearlier.

In sum, supplier financing can eliminate the assetsubstitution problem only if the assets are purchasedfrom the same supplier. Interestingly, there are alsocircumstances under which trade credit provided bymultiple suppliers introduces risk-shifting incentivesthat would not exist under bank financing. Recall thatbank financing does not lead to any agency conflictwhen the products have symmetrical cost, revenue,and demand parameters. As is apparent from condi-tions (31) and (32), parameter symmetry, or equal risk-iness of the two products, does not eliminate agencyfrictions when financing is provided by two suppli-ers. This is because sequential financing is, in itself,a form of asymmetry, which leaves the retailer withrisk-shifting opportunities.

There are limited circumstances under which differ-ent rules for liquidating the firm’s assets may apply,e.g., a supplier may have the “reclamation right” torepossess the goods sold (see, e.g., Ayer et al. 2004).This does not change the fact that there are risk-shifting incentives present when sourcing from twoor more suppliers. When there are two suppliers, theagency problem can be eliminated only if the creditfrom the first supplier is fully secured with inventoryor any other collateral.

5. Model Implications and Limitations5.1. Managerial ImplicationsThe comparative statics in Section 3 provide guide-lines for managing an inventory portfolio underdebt financing. Relative to a classical equity-financednewsvendor, a shareholder-controlled but debt-finan-ced newsvendor should favor items (i) with a lowsalvage value, (ii) with a high profit margin, and(iii) those that represent a large portion of total inven-tory because of high or relatively predictable demand.

This strategy is optimal after debt has been issuedand managers have a fiduciary duty to pursue it.Yet, it creates an agency cost that is ultimately borneby the shareholders through the higher cost of exter-nal financing. Risk-seeking behavior and the resultingagency cost can be avoided when inventory and credit

terms are determined simultaneously, as is often thecase when financing is provided by a single supplierin the form of trade credit. This has several impli-cations. First, a firm benefits from supplier financingmost (i) when sourcing multiple items from a sin-gle supplier, and (ii) when bankruptcy risk and thelimited liability effect are significant. Second, whenbuying multiple items on trade credit, consolidat-ing the number of suppliers is likely to alleviate theagency problem associated with sourcing from multi-ple suppliers.

This is not to suggest that trade credit always dom-inates bank financing, even when sourcing multipleitems from a single supplier. Because banks tend toface lower cost of capital than suppliers, bank creditcan ultimately be cheaper than trade credit. We studya mixed financing strategy that allows the firm toenjoy the best of both worlds. It involves obtainingbank credit up to an amount that can be secured ornearly secured by the firm’s assets, and using tradecredit for the remaining financing.

5.2. Empirical ImplicationsIn our model, there is an agency cost of debt only ifthe retailer offers multiple products; and the greaterthe asymmetry in product parameters, the greater thiscost (see Figures 3(b), 4(b), 5(b), and 6(b)). Therefore,we would expect firms that offer high product varietyto rely less on debt financing.11

We have also observed that a debt-financed retailertends to favor products that already represent a largeportion of total inventory (see Propositions 3 and 4and Figures 4(a), 5(a), and 6(a)). By extrapolation,we would expect a debt-financed retailer to prefer aless diversified product portfolio. This further corrob-orates our conjecture regarding the negative relation-ship between debt and product variety. We formalizeit as our first empirical prediction.

Prediction 1. A firm’s reliance on debt financing,excluding trade credit, is negatively related to itsproduct mix variety.

Our remaining predictions are related to the use oftrade credit. In the absence of demand uncertainty, theretailer has no risk-shifting opportunities and, accord-ing to our model, no reason to use trade credit. Inthe presence of demand uncertainty, bank financingcreates agency frictions, which favors the use of tradecredit. Our numerical experiments indicate that whendemand volatilities are increased proportionally, therelative agency cost, 4V ∗ −V BF 5/V BF , increases under

11 Although our focus has been on bank credit, publicly traded debtleads to similar, if not greater, agency frictions.

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both the newsvendor and linear demand model.12

Therefore, we expect higher overall demand risk toresult in greater use of supplier financing.

Prediction 2. Higher demand risk leads to greaterreliance on trade credit financing.

The challenge in testing this prediction is thatdemand volatility is difficult to measure directly, andits natural proxy—sales volatility—is endogenous inour model. (Sales volatility depends on inventory lev-els, which in turn depend on the financing method.)However, there is some evidence that the use of tradecredit increases with various measures of exogenousrisk. For example, Elliehausen and Wolken (1993) findthat probability of using trade credit and the amountof trade credit outstanding are positively related tohigher financial and business risks, where the formeris measured by the debt-to-equity ratio, and the lat-ter is a function of the firm’s corporate governanceand age. In addition, Nilsen (2002) documents thatthe importance of trade credit relative to bank financ-ing increases during an economic downturn. This isconsistent with our theory that ties the advantage ofusing trade credit to bankruptcy risk, which for mostfirms increases during recessions. Finally, Huyghe-baert and Van de Gucht (2007) find evidence thatrisk-shifting incentives measured by industry-wideasset liquidity multiplied by historical bankruptcyrate, have a negative impact on bank debt and a posi-tive impact on leasing and trade credit as proportionsof total debt.

Admittedly, the aforementioned empirical evidenceis consistent with several existing trade credit theo-ries, and thus, does not prove that our explanationis at work.13 The distinguishing feature of our theoryis that the benefit of trade credit hinges on multipleitems being procured from a given supplier. Whenmanaging a single product, the retailer cannot favorone item over another, and therefore, no agency prob-lem arises in inventory management. This leads to thefollowing prediction.

Prediction 3. A firm is more likely to use tradecredit when sourcing multiple items from a givensupplier.

Finally, we have seen that trade credit eliminatesthe agency problem in the single supplier scenario

12 Interestingly, increasing demand volatility of only one of theproducts can reduce the relative agency cost. Suppose the productsare identical in all parameters except that 2 < 1. As 2 increasesand approaches 1, the products become more similar and theagency cost diminishes, vanishing completely when 1 = 2. Thissituation is captured in Figure 6(b), but the effect appears to be toosmall to have any practical significance.13 In particular, it is consistent with explanations based on infor-mation asymmetry regarding the buyer’s default risk (Smith 1987,Biais and Gollier 1997).

(Proposition 5), but not in the multiple supplier sce-nario (Proposition 7). We formalize this observationin our last prediction.

Prediction 4. A firm is less likely to use tradecredit when sourcing from multiple suppliers.

Testing Predictions 3 and 4 would be a good wayto validate our theory as a distinct motivation for theuse of supplier financing.

5.3. Model LimitationsOur model relies on two important assumptions, fixedwholesale prices and fairly priced trade credit, both ofwhich can be justified by perfect competition amongsuppliers. In the following, we discuss robustness ofour results under alternative market structures.

Price-Setting Supplier. Suppose that any type ofcredit is fairly priced, but the supplier strategicallychooses wholesale prices as the first mover. Whenchoosing inventory and financing, the retailer facesthe same problem as in our model. With bank financ-ing, the best-response order quantities, QBF 4c5, mustsatisfy the optimality conditions given in Lemma 1,and they are obviously distorted by risk shifting. Withsupplier financing, the best-response order quantities,QTC4c5, maximize total firm value, and they are givenby Lemma 2. However, even though supplier financ-ing eliminates the agency problem associated withbank financing, it does not produce the first-best out-come because the retailer’s inventory decision is stilldistorted by double marginalization (see Lariviere andPorteus 2001). To instate the first-best outcome in thissituation, trade credit terms would have to be part ofa larger supply chain coordinating contract.

Monopolistic Lender. Now suppose that the whole-sale prices are given exogenously, but the credit mar-ket is monopolistic. Suppose further that the lender,whether it is the supplier or a bank, sets the inter-est rate z as the first mover. The retailer responds byordering Q4z5, borrowing 4c′Q − W5, and promisingto repay 41 + z54c′Q − W5. Regardless of who pro-vides financing, the retailer chooses inventory to max-imize the value of equity V = Pr4ìa5Ɛ4 − 41 + z5 ·

4c′Q − W5 ìa5. The best-response inventory vec-tor, Q4z5, must satisfy the first-order conditions,¡V /¡Qi = 0, i = 112, which together imply

Ɛ41 ìa5

c1=

Ɛ42 ìa5

c20 (33)

This is the same condition that characterizes the opti-mal inventory under bank financing in Lemma 1 withr/L replaced by z. In other words, when the lenderhas the power to set the interest rate as the first mover,there is no difference between trade credit and bankfinancing. Trade credit can induce the first-best inven-tory decision only if the supplier sets the credit termsafter observing the order quantities, fairly reflectingthe actual credit risk.

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Other Limitations. Our model assumes that the re-tailer maximizes shareholder value. If a retailer is con-trolled jointly by shareholders and debt holders, andtheir objective is to maximize total firm value, thebenefit of trade credit described in this paper disap-pears. We also assume that it is the retailer who makesthe inventory decision. That may not be the caseunder “vendor managed inventory,” when the sup-plier manages stocking levels. Finally, our model doesnot capture all of the potential differences betweentrade credit and bank financing, most of which havebeen described in the literature. Rather, its purpose isto demonstrate one additional benefit of trade creditthat the literature has not yet identified.

Supplemental MaterialSupplemental material to this paper is available at http://dx.doi.org/10.1287/mnsc.2016.2515.

AcknowledgmentsThe author is grateful to the whole review team fortheir excellent comments, which improved the paperconsiderably.

AppendixWe use f 4x1y5 and fi4x5, i = 112, to denote the joint andmarginal p.d.f. of D, respectively. When assuming symmet-rical demand distribution, we replace fi4x5 with f 4x5. Wealso define x+ ≡ max4x105.

Newsvendor Model. If is given by (1), it can be also writ-ten as

=

p′D+ s′4Q−D5

if D ∈ì0 = 8D2 Di ≤Qi1 i = 11291

p1Q1 + p2D2 + s24Q2 −D25

if D ∈ì1 = 8D2 D1 >Q11D2 ≤Q291

p2Q2 + p1D1 + s14Q1 −D15

if D ∈ì2 = 8D2 D2 >Q21D1 ≤Q191

p′Q if D ∈ì3 = 8D2 Di >Qi1 i = 11290

(34)

Linear Demand Model. If is given by (2) with = 0, itcan be also written as

=

s′Q+

2∑

i=1

(

4Di−si5+

2

)2

if D∈ì0 =8D2 Di ≤2Qi+si1i=11291

s2Q2 +

(

4D2 −s25+

2

)2

+D1Q1 −Q21

if D∈ì1 =8D2 D1>2Q1 +s11D2 ≤2Q2 +s291

s1Q1 +

(

4D1 −s15+

2

)2

+D2Q2 −Q22

if D∈ì2 =8D2 D2>2Q2 +s21D1 ≤2Q1 +s191

D1Q1 −Q21 +D2Q2 −Q2

2

if D∈ì3 =8D2 Di>2Qi+si1i=11290

(35)

Proof of Lemma 1. The optimal QBF 4L1 r5 > 0 solvingproblem (6) must satisfy the first-order condition, which canbe written as (8), together with constraint (3). The optimalLBF > 0 must satisfy the first-order condition dV /dL = 0,where QBF 4L5 and rBF 4L5 are given jointly by (8) togetherwith (3) and (4). We have

dV

dL=

¡V

¡L+

dr

dL

¡V

¡r+

dQ1

dL

¡V

¡Q1+

dQ2

dL

¡V

¡Q2

= −Pr4ìa5− Pr4ìa5dr

dL+

dQ1

dLPr4ìa5Ɛ41 ìa5

+dQ2

dLPr4ìa5Ɛ42 ìa50

Using (8) and (3), the optimality condition dV /dL = 0 canbe written as in (9).

Proof of Lemma 2. The optimal Q∗ > 0 has to sat-isfy the first-order conditions 4d/dQi56Ɛ4Q5− c′Q−BL−

Pr4ìb5b7 = 0, i = 112, where L and r are given by (3)–(4).Taking the total derivatives and setting them equal to zerogives (11).

Proof of Proposition 1. When b = 0, the optimal Q∗

and QBF satisfy the first-order conditions (12) and (13),which can be written as

Ɛ14Q∗5

c1=

Ɛ24Q∗5

c2and (36)

Pr4ìa4QBF 55Ɛ414QBF 5 ìa4QBF 55

c1

=Pr4ìa4QBF 55Ɛ424QBF 5 ìa4QBF 55

c21 (37)

respectively. Conditions (14) imply

Ɛ414QBF 5 ìb4QBF 55

c1<

Ɛ424QBF 5 ìb4QBF 55

c20 (38)

Condition (38), together with (37), implies

Ɛ14QBF 5

c1<

Ɛ24QBF 5

c200 (39)

Combining (36) and (39) implies that

Ɛ14Q∗5 > Ɛ14Q

BF 5 or Ɛ24Q∗5 < Ɛ24Q

BF 50 (40)

Because ¡i/¡Qj = 0, inequalities (40) can be written as

Ɛ14Q∗

15 > Ɛ14QBF1 5 or Ɛ24Q

∗

25 < Ɛ24QBF2 50 (41)

Because Ɛi4Qi5 is decreasing in Qi, inequalities (41)imply (15).

Proof of Proposition 2. The desired inequality (18) canbe written as

4dQBF2 /ds254Q

BF1 +QBF

2 5−QBF2 4dQBF

1 /ds2 + dQBF2 /ds25

4QBF1 +QBF

2 52

≤4dQ∗

2/ds254Q∗1 +Q∗

25−Q∗24dQ

∗1/ds2 + dQ∗

2/ds25

4Q∗1 +Q∗

252

∣

∣

∣

∣

s2=s1

0

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When all parameters are symmetrical, QBF1 =QBF

2 =Q∗1 =Q∗

2 ,and the above inequality simplifies into

[

dQBF1

ds2−

dQBF2

ds2

]

s2=s1

≥

[

dQ∗1

ds2−

dQ∗2

ds2

]

s2=s1

0 (42)

Recall that when b = 0, the optimal Q∗ and QBF satisfy thefirst-order conditions (12) and (13), which can be written,respectively, as

FOC∗≡ Pr4ì135

p1

c1+Pr4ì025

s1

c1

−Pr4ì235p2

c2−Pr4ì015

s2

c2=0 (43)

and FOCBF≡ Pr4ì13a5

p1

c1+Pr4ì02a5

s1

c1

−Pr4ì23a5p2

c2−Pr4ì01a5

s2

c2=00 (44)

Taking the total differential of (43) and (44) with respectto s2, we get, respectively,

dFOC∗

ds2=

¡FOC∗

¡s2+

2∑

i=1

dQ∗i

ds2

¡FOC∗

¡Qi

+¡FOC∗

¡L

dL

ds2+

¡FOC∗

¡r

dr

ds2= 01 (45)

anddFOCBF

ds2=

¡FOCBF

¡s2+

2∑

i=1

dQBFi

ds2

¡FOCBF

¡Qi

+¡FOCBF

¡L

dL

ds2+

¡FOCBF

¡r

dr

ds2= 00 (46)

Taking the partials of (43) and (44), and using the parametersymmetry, we can write

¡FOC∗

¡Q1

∣

∣

∣

∣

s2=s1

= −¡FOC∗

¡Q2

∣

∣

∣

∣

s2=s1

=

(

p2

c2−

s2

c2

)(

¡Pr4ì15

¡Q1−

¡Pr4ì25

¡Q1

)

1

¡FOC∗

¡L

∣

∣

∣

∣

s2=s1

=¡FOC∗

¡r

∣

∣

∣

∣

s2=s1

= 01

¡FOCBF

¡Q1

∣

∣

∣

∣

s2=s1

= −¡FOCBF

¡Q2

∣

∣

∣

∣

s2=s1

=

(

p2

c2−

s2

c2

)(

¡Pr4ì1a5

¡Q1−

¡Pr4ì2a5

¡Q1

)

1 and

¡FOCBF

¡L

∣

∣

∣

∣

s2=s1

=¡FOCBF

¡r

∣

∣

∣

∣

s2=s1

= 00

(47)Plugging these expressions into (45) and (46), and applyingsome algebra, we obtain

dQ∗1

ds2−dQ∗

2

ds2

∣

∣

∣

∣

s2=s1

= −¡FOC∗

¡s2

(

p2

c2−

s2

c2

)−1

·

(

¡Pr4ì15

¡Q1−¡Pr4ì25

¡Q1

)−1

(48)

anddQBF

1

ds2−dQBF

2

ds2

∣

∣

∣

∣

s2=s1

= −¡FOCBF

¡s2

(

p2

c2−

s2

c2

)−1

·

(

¡Pr4ì1a5

¡Q1−¡Pr4ì2a5

¡Q1

)−1

0 (49)

Using (48) and (49), the desired inequality (42) can be re-written as

¡FOCBF

¡s2

(

¡Pr4ì1a5

¡Q1−

¡Pr4ì2a5

¡Q1

)−1

≤¡FOC∗

¡s2

(

¡Pr4ì15

¡Q1−

¡Pr4ì25

¡Q1

)−1 ∣∣

∣

∣

s2=s1

0 (50)

Using the assumption Pr4ì12b5 = 0, and taking advantageof the parameter symmetry, we can obtain the followingexpressions:

¡FOC∗

¡s2

∣

∣

∣

∣

s2=s1

= −Pr4ì0151c21

¡FOCBF

¡s2

∣

∣

∣

∣

s2=s1

= −Pr4ì01a51c21 and

¡Pr4ì15

¡Q1−¡Pr4ì25

¡Q1

∣

∣

∣

∣

s2=s1

=¡Pr4ì1a5

¡Q1−¡Pr4ì2a5

¡Q1

∣

∣

∣

∣

s2=s1

= −

∫

0f 4Q11y5dy0

(51)

Plugging these expressions into the desired inequality (50),this inequality simplifies into Pr4ì01a5 ≤ Pr4ì015, which isevidently true.

Proof of Proposition 3. Using the same procedure thatwe used to convert (18) into (50) in the proof of Proposi-tion 2, we can convert the desired inequality (19) into

¡FOCBF

¡p2

(

¡Pr4ì1a5

¡Q1−

¡Pr4ì2a5

¡Q1

)−1

≥¡FOC∗

¡p2

(

¡Pr4ì15

¡Q1−

¡Pr4ì25

¡Q1

)−1 ∣∣

∣

∣

p2=p1

0 (52)

Taking the partials of (43) and (44) with respect to p2, andusing the parameter symmetry, we obtain

¡FOC∗

¡p2

∣

∣

∣

∣

p2=p1

= −Pr4ì2351c21 and

¡FOCBF

¡p2

∣

∣

∣

∣

p2=p1

= −Pr4ì23a51c2

+

(

p1

c1−

s1

c1

)(

¡Pr4ì1a5

¡p2−

¡Pr4ì2a5

¡p2

)

0 (53)

Taking the partials of

Pr4ì1a5 =

∫ Q2

4L+r−p1Q1−s2Q25/4p2−s25

∫

Q1

f 4x1y5dx dy1

Pr4ì2a5 =

∫

Q2

∫ Q1

4L+r−p2Q2−s1Q15/4p1−s15f 4x1y5dx dy1

Pr4ì15 =

∫ Q2

0

∫

Q1

f 4x1y5dx dy1 and

Pr4ì25 =

∫

Q2

∫ Q1

0f 4x1y5dx dy1

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and using some algebra, we obtain

¡Pr4ì1a5

¡p2=

L+r−p1Q1 −s2Q2

4p2 −s252

·

∫

Q1

f

(

x1L+r−p1Q1 −s2Q2

p2 −s2

)

dx1

¡Pr4ì2a5

¡p2=

Q2

p1 −s1

∫

Q2

f

(

L+r−p2Q2 −s1Q1

p1 −s11y

)

dy1

¡Pr4ì1a5

¡Q1=

p1

p2 −s2

∫

Q1

f

(

x1L+r−p1Q1 −s2Q2

p2 −s2

)

dx

−

∫ Q2

4L+r−p1Q1−s2Q25/4p2−s25f 4Q11y5dy1

¡Pr4ì2a5

¡Q1=

s1

p1 −s1

∫

Q2

f

(

L+r−p2Q2 −s1Q1

p1 −s11y

)

dy

+

∫

Q2

f 4Q11y5dy1

¡Pr4ì15

¡Q1= −

∫ Q2

0f 4Q11y5dy1 and

¡Pr4ì25

¡Q1=

∫

Q2

f 4Q11y5dy0

(54)

Substituting (53) and (54) into the desired inequality (52)and using some algebra, this inequality simplifies into

(

Pr4ì23a5−L+ r − p1Q1 − p2Q2

p2 − s2

·

∫

Q1

f

(

x1L+ r − p1Q1 − s2Q2

p2 − s2

)

dx

)

·

(

∫

4L+r−p1Q1−s2Q25/4p2−s25f 4Q11y5dy

−

∫

Q1

f

(

x1L+ r − p1Q1 − s2Q2

p2 − s2

)

dx

)−1

≥Pr4ì235

∫

0 f 4Q11y5dy1 (55)

where 4L1 r1Q5 = 4L∗1 r∗1Q∗5 = 4LBF 1 rBF 1QBF 5. Using theassumptions of symmetry and demand independence,inequality (55) further simplifies into

(

1 −L+ r − p1Q1 − p2Q2

p2 − s2h

(

L+ r − p1Q1 − s2Q2

p2 − s2

))

·

(

h4Q15−h

(

L+ r − p1Q1 − s2Q2

p2 − s2

))−1

≥1

h4Q151 (56)

where h4x5 = f 4x5/Pr4Di > x5 is the demand failure rate.Using the assumption that h4x5 is increasing, and the factthat Q1 > 4L + r − p1Q1 − s2Q25/4p2 − s25, inequality (56) isequivalent to

L+ r − p1Q1 − p2Q2

p2 − s2≤

1h4Q15

1 (57)

where 4L1 r1Q5 = 4L∗1 r∗1Q∗5 = 4LBF 1 rBF 1QBF 5. We knowthat inequality (57) is satisfied because its left-hand side isnegative while its right-hand side is positive.

Proof of Proposition 4. Using the same procedure thatwe used to convert (18) into (50) in the proof of Proposi-tion 2, we can convert the desired inequality (20) into

¡FOCBF

¡c2

(

¡Pr4ì1a5

¡Q1−

¡Pr4ì2a5

¡Q1

)−1

≤¡FOC∗

¡c2

(

¡Pr4ì15

¡Q1−

¡Pr4ì25

¡Q1

)−1 ∣∣

∣

∣

c2=c1

0 (58)

Taking the partials of (43) and (44) with respect to c2, andusing the assumption that s1 = s2 = 0, we obtain

¡FOCBF

¡c2= Pr4ì23a5

p2

c22

1 and

¡FOC∗

¡c2= Pr4ì235

p2

c22

1 respectively.

(59)

Substituting (59) and (54) into the desired inequality (58),and taking the advantage of parameter symmetry anddemand independence, we can rewrite (58) as

4Pr4ì23a55 ·

(

f 4Q15Pr(

D1 >L+ r − p1Q1

p2

)

− f

(

L+ r − p1Q1

p2

)

Pr4D1 >Q15

)−1

≥Pr4ì235

f 4Q151 (60)

where 4L1 r1Q5 = 4L∗1 r∗1Q∗5 = 4LBF 1 rBF 1QBF 5. Using theassumption that h4x5 is increasing and the fact that Q1 >4L+ r − p1Q15/p2, we know that the desired inequality (60)must hold if

Pr4ì23a5≥ Pr4ì235Pr(

D1 >L+ r − p1Q1

p2

)

0 (61)

With independent demands, inequality (61) is clearly satis-fied as an equality.

Proof of Proposition 5. Because LTC4Q5 and rTC4Q5 aredetermined by (3) and (21), problem (22) can be rewritten as

QTC1LTC1 rTC = arg maxQ1L1 r≥0

V 4Q1L1 r5

subject to (3) and (21). (62)

Using the definition of V in (5) and constraints (3) and (21),we can rewrite the objective in (62) as

V = Pr4ìa5Ɛ4 −L− r ìa5

= Ɛ − c′Q−sL− Pr4ìb5b+W0 (63)

Thus, we observe that formulation (62) is equivalent to thefirst-best formulation (10) with B replaced by s .

Proof of Corollary 1. It follows from Proposition 5that when S = B , V TC = V ∗ > V BF . By continuity, V TC >V BF for any S ∈ 4B1B +5. Because i is finite, for a suf-ficiently high S , LTC = 0, and the trade credit scenario isequivalent to the bank financing scenario with an additionalconstraint L= 0. Because we are assuming LBF > 01 we haveV TC <V BF .

Proof of Proposition 6. The inequality VMF ≥ V TC fol-lows from the fact that the trade credit scenario is equiv-alent to the mixed financing scenario with an additional

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Chod: Inventory, Risk Shifting, and Trade CreditManagement Science, Articles in Advance, pp. 1–20, © 2016 INFORMS 19

constraint LB = 00 Now suppose that the premise of the sec-ond part of the proposition holds, and consider the mixedfinancing scenario with LB = 4W min8s1/c11 s2/c295/41 +B5,and rB = BLB0 With probability one, we have ≥ s′Q ≥

min8s1/c11 s2/c29c′Q ≥ min8s1/c11 s2/c29W = LB + rB . There-fore, the bank credit is riskless and fairly priced. In thiscase, the retailer’s second-stage problem (23) simplifies intoQMF 4LB5= arg maxQ≥0 V 4Q LB5, where

V 4Q1LB5= Ɛ − 41 +S54c′Q−W5+ 4S −B5LB0

In the trade credit scenario as presented in (22), we haveQTC = arg maxQ≥0 V 4Q5, where

V 4Q5= Ɛ − 41 +S54c′Q−W50 (64)

Therefore, QMF 4LB5 = QTC1 and the value of equity undermixed financing equals V TC +4S −B54W min8s1/c11 s2/c295/41 +B5. The value of equity under mixed financing and theoptimal LB cannot be lower than that, and inequality (27)follows.

Proof of Lemma 3. The optimality condition (31) for Q∗2

is equivalent to the second condition in (11) except that thelender’s cost of capital is S . The optimal QTC

2 > 0 solvingproblem (29) must satisfy the first-order condition

dV

dQ2=

¡V

¡Q2+

dr2

dQ2

¡V

¡r2= 00 (65)

Taking the partial derivatives, we get

¡V

¡Q2= Pr4ìa5Ɛ42 ìa5− Pr4ìa5c2 and

¡V

¡r2= −Pr4ìa50

(66)

Implicitly differentiating condition (28) for i = 21 and apply-ing some algebra gives

dr2

dQ2=

1Pr4ìa5

[

Pr4ìb5c2 +Sc2 −L2 + r2

L1 + r1 +L2 + r2

·

(

Pr4ìb5Ɛ42 ìb5− bdPr4ìb5

dQ2

)

−Pr4ìb5Ɛ4 − b ìb5d

dQ2

(

L2 + r2

L1 + r1 +L2 + r2

)]

0 (67)

Plugging (66) and (67) into (65), we get the optimality con-dition (32).

Proof of Proposition 7. When b = 0 and Ɛ42 ìb5 = 0at QTC , the optimality conditions for QTC

2 and Q∗2 , (31) and

(32), can be written as

Ɛ2 = 41 +S5c21 and (68)

Ɛ2 + Pr4ìb5Ɛ4 ìb5d

dQ2

(

L2 + r2

L1 + r1 +L2 + r2

)

= 41 +S5c21 (69)

respectively. Taking the total derivative, we get

d

dQ2

(

L2 +r2

L2 +r2 +L1 +r1

)

=L1 +r1

4L2 +r2 +L1 +r152

(

c2 +dr2

dQ2

)

0 (70)

Implicitly differentiating condition (28) for i = 2 yields

dr2

dQ2= c2

(

Pr4ìb5−Pr4ìb5Ɛ4 ìb54L1 +r15

4L2 +r2 +L1 +r152+S

)

·

(

Pr4ìa5+Pr4ìb5Ɛ4 ìb54L1 +r15

4L2 +r2 +L1 +r152

)−1

0 (71)

Using (70), (71), and (28), the optimality condition (69) canbe written as

Ɛ2 = 41+S5c2 −41+S5c2 Pr4ìb5Ɛ4 ìb5

41+S5L1 +Pr4ìa54L2 +r25

L1 +r1

L2 +r2 +L1 +r10

Thus, we have Ɛ24Q∗5 = 41 + S5c21 whereas Ɛ24QTC5 <41+S5c2, which means Ɛ24QTC5 < Ɛ24Q∗5. Using the factthat ¡i/¡Qj = 0 for i 6= j1 the last inequality is equivalent toƐ24Q

TC2 5 < Ɛ24Q

∗250 Because Ɛ2 is decreasing in Q2, we

have QTC2 >Q∗

2 .

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