Hedging Commodity Procurement in a Bilateral Supply Chain

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Hedging Commodity Procurement in a Bilateral SupplyChainDanko Turcic, Panos Kouvelis, Ehsan Bolandifar

To cite this article:Danko Turcic, Panos Kouvelis, Ehsan Bolandifar (2015) Hedging Commodity Procurement in a Bilateral Supply Chain.Manufacturing & Service Operations Management 17(2):221-235. http://dx.doi.org/10.1287/msom.2014.0514

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MANUFACTURING & SERVICEOPERATIONS MANAGEMENT

Vol. 17, No. 2, Spring 2015, pp. 221–235ISSN 1523-4614 (print) � ISSN 1526-5498 (online) http://dx.doi.org/10.1287/msom.2014.0514

© 2015 INFORMS

Hedging Commodity Procurement in aBilateral Supply Chain

Danko Turcic, Panos KouvelisOlin Business School, Washington University in St. Louis, St. Louis, Missouri 63130

{[email protected], [email protected]}

Ehsan BolandifarChinese University of Hong Kong, Shatin, N.T., Hong Kong, [email protected]

This paper explores the merits of hedging stochastic input costs (i.e., reducing the risk of adverse changesin costs) in a decentralized, risk-neutral supply chain. Specifically, we consider a generalized version of

the well-known “selling-to-the-newsvendor” model in which both the upstream and the downstream firmsface stochastic input costs. The firms’ operations are intertwined—i.e., the downstream buyer depends on theupstream supplier for delivery and the supplier depends on the buyer for purchase. We show that if leftunmanaged, the stochastic costs that reverberate through the supply chain can lead to significant financiallosses. The situation could deteriorate to the point of a supply disruption if at least one of the supply chainmembers cannot profitably make its product. To the extent that hedging can ensure continuation in supply,hedging can have value to at least some of the members of the supply chain. We identify conditions underwhich the risk of the supply chain breakdown will cause the supply chain members to hedge their input costs:(i) the downstream buyer’s market power exceeds a critical threshold; or (ii) the upstream firm operates on alarge margin, there is a high baseline demand for downstream firm’s final product, and the downstream firm’smarket power is below a critical threshold. In absence of these conditions there are equilibria in which neitherfirm hedges. To sustain hedging in equilibrium, both firms must hedge and supply chain breakdown must becostly. The equilibrium hedging policy will (in general) be a partial hedging policy. There are also situationswhen firms hedge in equilibrium although hedging reduces their expected payoff.

Keywords : risk management; commodity procurement; supply chain contracting; hedgingHistory : Received: July 25, 2011; accepted: September 27, 2014. Published online in Articles in Advance

February 19, 2015.

1. IntroductionMuch of the extant supply chain and inventory litera-ture takes demand as stochastic and production costsas fixed. The latter assumption, however, is in contrastto many industrial settings in which manufacturingfirms are exposed to price changes of commoditiesthat affect the direct costs of raw materials, compo-nents, subassemblies, and packaging materials as wellas indirect costs from the energy consumed in oper-ations and transportation. According to a survey oflarge U.S. nonfinancial firms (e.g., Bodnar et al. 1995),approximately 40% of the responding firms routinelypurchase options or futures contracts in order tohedge1 price risks such as those mentioned above.Why do these firms hedge?

To provide rationale for hedging, extant theorieshave relied on the existence of taxes (e.g., Smith andStulz 1985), asymmetric information (e.g., DeMarzoand Duffie 1991), hard budget constraints and costly

1 Throughout the paper, to “hedge” means to buy or sell commod-ity futures as a protection against loss or failure because of pricefluctuation. For details, see Van Mieghem (2003, p. 271).

external capital (e.g., Froot et al. 1993, Caldenteyand Haugh 2009), and risk aversion (e.g., Gaur andSeshadri 2005, Chod et al. 2010). All these papers takea firm as the basic “unit of analysis” and, broadlyspeaking, analyze situations in which the firm plansto either purchase or sell output at a later datewhile facing one of the above frictions. When viewedfrom today, the output’s price and consequently thefirm’s transaction proceeds are random. However, bysomehow eliminating the randomness in price—or byhedging—the firm is able to increase its (expected)payoff, which makes hedging beneficial to the firm.

The thesis behind this paper is that when severalfirms are embedded together in a supply chain whilesimultaneously facing the situation we describedabove, there will be an added level of complex-ity, which the existing theories cannot necessarilyaddress. First, in a supply chain environment, firmswill likely find themselves exposed not only to theirown price risks but also to their supply chain part-ner’s price risk. This is illustrated with an exam-ple from the auto industry (Matthews 2011). In 2011,steelmakers increased prices six times, for a total

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Turcic, Kouvelis, and Bolandifar: Hedging Commodity Procurement in a Bilateral Supply Chain222 Manufacturing & Service Operations Management 17(2), pp. 221–235, © 2015 INFORMS

increase of 30%. However, consistent with the extanttheory, many car and appliance manufactures didnot necessarily pay attention to steel prices becausethey did not directly purchase steel from steelmak-ers, only to find themselves in difficulty with someof their financially weaker suppliers, for whom steelrepresents their biggest cost. In a supply chain, priceincreases can therefore result in the supplier experi-encing significant financial losses if it does not havethe resources to manage price risk. The situationcould deteriorate to the point of a supply disruptionif the supplier cannot profitably make products forits buyer. Although firms can try to mitigate the riskof disruption by contractually imposing penalties, aswill be seen, default penalties are often not a credibleguarantee of supply unless they are also supportedwith hedging.

Second, unlike in the stand-alone firm’s setting,supply chain firms cannot necessarily control anyprice risk they want by hedging alone. Consequently,firms will care whether and how their supply chainpartners hedge, and any hedging activity will beimpounded in transfer prices. Finally, the hedgingbehavior of their supply chain partners will affecttheir own hedging behavior, as will be seen.

Following the seminal contribution of Lariviere andPorteus (2001)—who point out that there is an inti-mate relationship between wholesale price, demanduncertainty, and the distribution of market power inthe supply chain—this paper explores the merits ofhedging stochastic input costs to ensure continuityof supply in a decentralized supply chain. Specifi-cally, we consider a generalized version of the well-known “selling-to-the-newsvendor” model (Lariviereand Porteus 2001) in which both the upstream and thedownstream firms face stochastic production costs.(In contrast to our paper, in Lariviere and Porteus2001, production costs are linear with constant coef-ficients.) The intuition that underlies our model ofhedging in a supply chain is as follows. By agreeingto sell their output at (contractually) predeterminedprices before all factors affecting their productivity areknown, both the upstream supplier and the down-stream buyer subject themselves to default risk. Thefact that a supply chain member may default in somestates of the world will be, ex ante, impounded intothe wholesale price as well as into the order quantity.However, if the firms could somehow “guarantee”that they will fulfill the supply contract (no matterwhat happens to their inputs costs), their supplychain counterpart may respond with a more favor-able price or a more favorable order quantity. Eitherfirm would like to guarantee supply contract perfor-mance if the expected profit associated with the guar-antee is greater than the expected profit associatedwith default in some states. In such circumstances,

the firm can commit to the guarantee by purchasingfutures contracts on the underlying input commodity.The futures contracts will pay off precisely at the timewhen the firm’s resources are strained. Hence, futurescontracts have value because they prevent the firmfrom defaulting on a contractual obligation when notdefaulting is (ex ante) important.

We identify the following conditions under whichthe supply chain firms should be expected to hedgein equilibrium: (i) the downstream buyer’s marketpower exceeds a critical threshold; or (ii) the upstreamfirm operates on a large margin, there is a high base-line demand for downstream firm’s final product, andthe downstream firm’s market power is below a criti-cal threshold. In absence of these conditions there areequilibria in which neither firm hedges. To sustainhedging in equilibrium, both firms must hedge andsupply chain breakdown must be costly. The equi-librium hedging policy will (in general) be a partialhedging policy. There are also situations when firmshedge in equilibrium although hedging reduces theirexpected payoff. (A partial hedging policy leaves thehedger exposed to some residual price risk. A fullhedging policy eliminates all price risk.)

To summarize, we provide a rationale for hedging,which is based on a noncooperative behavior in asupply chain. We illustrate this point with an anec-dotal example from a diversified global manufactur-ing company. The company’s Residential SolutionsDivision produces a collection of home appliancesand power tools, which are sold worldwide. Its Com-mercial Division manufactures industrial automationequipment and climate control technology. Manyof these products are raw-material-intense. When itcomes to supply chain management, the firm playsa variety of roles: Sometimes the firm is a sup-plier; other times, the firm plays a role of an end-product assembler. Recently, the company saw anincreased volatility in the prices of metals, whichaffected its margins. To combat this, the firm hedgedits raw material inputs, but only some. When askedwhen the firm hedges, a manager who presented thecase replied that the firm’s hedging decisions partlydepend both on the product and on the firm’s sup-ply chain partners. For example, as a supplier, thefirm fully hedges the raw material it uses to pro-duce garbage disposals, which are sold through amajor U.S. home-improvement store. This is becausethe firm’s bargaining power against this retailer issmall, and it is the retailer who dictates both produc-tion quantities and pricing. As a buyer, however, thefirm tends not to hedge when it knows that it mayface glitches in supply, because then it is exposed torisk of financial losses from its hedging activities. Theexisting risk-management literature does not alwaysprovide clear guidance on hedging in cases like these.

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Turcic, Kouvelis, and Bolandifar: Hedging Commodity Procurement in a Bilateral Supply ChainManufacturing & Service Operations Management 17(2), pp. 221–235, © 2015 INFORMS 223

Either because it requires frictions (e.g., risk aversion),which do not apply to large multinationals (Carteret al. 2006, Jin and Jorion 2006), or because it doesnot allow hedgers to vary their hedging decisionsacross products and channel relationships. Whereasour model is a simple selling-to-the-newsvendor set-ting, this example illustrates that the equilibrium thatwe advance seems to mirror the facts of these cases(see part (i) of Proposition 2).

In what follows, we review related literature,present the model (§§3 and 4), and derive the firms’equilibrium behavior under the unhedged and hedgedcontracts (§§5.1 and 5.2). Then, by comparing theexpected payoffs both firms can achieve under theunhedged and hedged contracts, we make predictionsas to when offers of hedged wholesale contract shouldbe expected in equilibrium (§5.3). Section 6 concludes.

2. Related LiteratureRecent research in the finance literature has greatlyimproved our understanding of why and how indi-vidual firms should hedge. For instance, it has estab-lished that for hedging to be beneficial, the firm’spayoff must be a concave function of the stochastichedge-able exposure metric. In spirit, this result fol-lows from basic convex-analysis theory: It means thatthe firm’s expected payoff, a weighted average of aconcave function, is lower when the level of variabil-ity of the exposure metric is higher. Thus, since hedg-ing reduces variability, then it increases the firm’sexpected payoff.

Focusing on a single-firm setting, Smith and Stulz(1985) demonstrate that managerial risk aversion ormarket frictions such as corporate taxes cause a firm’spayoff to become a concave function of its hedge-able exposure. One of the implications of this paper’sresult is that firms should hedge fully or not at all.Froot et al. (1993) expand the discussion to settingsin which firms are exposed to multiple sources ofuncertainty that are statistically correlated. They showthat in such type of situations, it may become optimalfor a firm to utilize partial hedging. Building uponthis observation, a large body of the finance litera-ture has studied hedging by taking a firm as the basic“unit of analysis” and exogenously imposing the afore-mentioned concavity property. This property has beencommonly justified by assuming that the hedger’spreferences on the set of its payoffs are consistentwith some form of concave utility function. This lit-erature provides insights into the mechanics of hedg-ing using financial instruments such as futures andoptions. For instance, Neuberger (1999) shows how arisk-averse supplier may use short-maturity commod-ity futures contracts to hedge a long-term commoditysupply commitment. A comprehensive summary ofthis research is given in Triantis (2005).

Whereas the finance literature focuses on mitigat-ing pricing risk, the operations management liter-ature predominantly considers demand and supplyrisks. An excellent survey of the early operations lit-erature can be found in Van Mieghem (2003). Morerecent and influential papers in this area includeGaur and Seshadri (2005) and Caldentey and Haugh(2009). Gaur and Seshadri (2005) address the prob-lem of using market instruments to hedge stochas-tic demand. The authors consider a newsvendor whofaces demand that is a function of the price of anunderlying financial asset. They show how firms canuse price information in financial assets to set optimalinventory levels, and they demonstrate that financialhedging of demand risk can lead to higher return oninventory investment. Caldentey and Haugh (2009)show that financial hedging of stochastic demand canincrease supply chain output. Note, however, thatthere are several important differences between thesupply chain considered in Caldentey and Haugh(2009) and the supply chain in this paper. Their sup-ply chain is centralized, the hedge-able risk in theirchain is stochastic demand, and the risk is faced byonly one of the firms in the supply chain. To provide arationale for the hedging behavior, Gaur and Seshadri(2005) rely upon risk aversion; Caldentey and Haugh(2009) require market frictions, which lead to creditrationing.

Other papers in the operations management lit-erature discuss and analyze hedging strategies formanaging risky supply and ask when it makes a dif-ference if firms hedge financially or operationally. Forexample, while intuition may suggest that operationaland financial hedging may be substitutes, surpris-ingly Chod et al. (2010) show that they can be com-plements. Emergency inventory (e.g., Schmitt et al.2010); emergency sourcing (e.g., Yang et al. 2009); anddual sourcing (e.g., Gümüs et al. 2012) are examples ofoperational hedging strategies capable of mitigatingadverse supply shocks. Supplier subsidy (e.g., Babich2010) is an example of a financial strategy designedto reduce the likelihood of supply contract aban-donment. Haksöz and Seshadri (2007), in turn, putfinancial value on the supplier’s ability to abandon asupply contract. We differ from the aforementionedliterature by focusing on the management of inputcost risk (rather than demand risk) and by extend-ing the discussion of why and how risk-neutral firmshedge from the single-firm setting into a decentral-ized supply chain.

3. Preamble: A Model of a VerticallyIntegrated Supply Chain

Consider the problem of a risk-neutral manufacturer(final good assembler), M , who faces the newsven-

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Figure 1 Inputs Required to Produce One Unit of Assembler’s FinalProduct

Commodity 1 S's component M's final product Commodity 2

dor problem in that at time t0 it must choose a pro-duction quantity, q, before it observes a single real-ization of stochastic demand, D, for its final product.Stochastic demand for the final product, D, has a dis-tribution F and density f . The function F has supporton 6d1 d7, i.e., there is always some demand (Cachon2004, p. 225). The selling price of the final product isnormalized to 1.

To assemble q units of the final product, M requiresq units of commodity 2 as well as q components(subassemblies), which are produced by a componentsupplier, S. To produce q components, S requires qunits of commodity 1. (Figure 1 summarizes the pro-duction inputs required to assemble 1 unit of the finalproduct.) For simplicity, unmet demand is lost, unsoldstock is worthless, and per unit production and hold-ing costs are zero.

Because of operational frictions (e.g., capacity con-straints and lead times), the subassembly produc-tion cannot be completed before time t1 > t0. Con-sequently, the final assembly cannot be completedbefore time t2 ≥ t1. The manufacturer’s, M ’s, sales rev-enue is realized at time t3 ≥ t2 with the proviso thatM cannot sell more at time t3 than what it producedfor at time t2 and what the supplier, S, produced forat time t1. Neither firm can produce more than whatboth firms committed to at time t0.

The commodity i = 112 per-unit price, Si, is re-vealed at time ti. Viewed from t0, Si is a randomvariable modeled on a complete probability space4ì1F1�5 equipped with filtration 8Ft91 t ∈ 6t01 t37. S1and S2 have a joint density function h. For simplicity,the discount rate between times t0 and t3 is taken to bezero. From CAPM, this is equivalent to assuming thatthe risk-free rate is zero and that both input commodi-ties are zero beta assets. Relaxing these assumptionscan be done without changing the main message ofthe results presented here.

We begin by observing that, in the absence of anyworking capital constraints, a vertically integrated sup-ply chain in this context would actually take advan-tage of input price volatility and suspend productionwhenever input costs are unfavorable. To see this, forthe case when t0 < t1 = t2 = t3, the time t0 expectedpayoff of the integrated system is

Ɛt0ç4q5= Ɛt0

min8q1D941 − S1 − S25+0 (1)

The right-hand side of Equation (1) says that at time t0,the integrated supply chain holds min8q1D9 of putoptions, each with a time t3 payoff of 41 − S1 − S25

+.

Because of this optionality, the integrated system’sexpected payoff increases both in q and in input costvolatility. Consequently, the system’s optimal produc-tion quantity will be q = d. (When t0 < t1 ≤ t2 ≤ t3, theexpression for Ɛt0

ç4q5 becomes more complicated, butno new important insights are added.)

The vertically integrated supply chain design iscontrasted with a decentralized supply chain designwhere M and S operate independently. Then,

• each firm will exercise its market power by push-ing its price above its (expected) marginal cost, whichwill lead to double-marginalization;

• each firm will face a stochastic input cost, whichwill lead to an incentive to default when defaulting isnot necessarily optimal for the integrated channel.

Under some conditions, the integrated channel pay-off given in (1) can be replicated with the use ofa state-contingent contract. A generalized version ofthe well-known revenue-sharing contract 8wr1�9 is animportant example of such a contract, where wr =

�S1 −41−�5S2 is a wholesale price the assembler paysto the supplier at time t2 and 0 ≤�≤ 1 is the fractionof the supply chain revenue the assembler keeps attime t3; so 41 −�5 is the fraction the supplier gets.

Notice that wr is a function of both S1 and S2, whichare revealed at times t1 and t2. Pricing this way iswhat allows the firms to share their production costsand thereby achieve the integrated channel outcome.As such, wr > 0 if and only if S1 > S241 − �5/�; oth-erwise wr ≤ 0, which means that the supplier paysthe assembler. If t1 = t2 = t3, then the supplier’s andthe assembler’s time t3 payoffs, respectively, are asfollows:

çS4q5 = 41 −�5min8q1D9+ q4wr − S15 andçM 4q5 = �min8q1D9− q4wr + S250

(2)

A special case of this generalized revenue shar-ing contract is the well-known pass-through con-tract, which is obtained by setting � to 1. Underthe pass-through contract, raw material charges aresimply passed through in full onto the downstreamassembler.

Example 1. Let � = 1/2 and suppose that viewedfrom time t0, 4S11 S21D5 is a Bernoulli random vec-tor that takes on values 44/512/5155, 42/512/5155with respective probabilities �844/512/51559 = 1/2,�842/512/51559 = 1/2. Then both the supplier’s andthe assembler’s time t3 payoffs are either 0 or 1/2,depending on the outcome of S1. The total supplychain payoff is 0 or 1, which matches the integratedchannel payoff given by the right-hand side of (1).

Example 1 illustrates a situation where the state-contingent contract addresses both double-marginali-zation and the adverse default incentive. That is,under the terms of the contract, the forward-looking

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supplier defaults exactly when it is optimal to defaultfor the integrated chain and the sum of both firm’sexpected payoffs is the same as the expected payoffof the integrated chain. However, as seen next, thereare situations where the contract fails.

Example 2 (Continuation of Example 1). Every-thing is the same as in Example 1, except thatviewed from time t0, 4S11 S21D5 is a Bernoulli randomvector that takes on values 42/512/5155, 42/510155with respective probabilities �842/512/51559 = 1/2,�842/5101559= 1/2, and �= 2/5.

Now, the state contingent contract of Example 1 nolonger matches the integrated chain payoff for all out-comes of 4S11 S21D5. To see this, suppose 4S11 S21D5=

42/512/5155. With q = 5, the integrated chain payoffwill be min8D1q9− 44/55q = 1 > 0 while the decentral-ized chain’s total payoff will be 0. The latter followsbecause wr = �S1 − 41 −�5S2 = −42/255. Therefore bynot producing, the assembler will collect 42/255q =

2/5 4= −wr q5 at time t2. At the same time, the high-est payoff the assembler can achieve by producingat time t2 is 42/255q = 2/5. The assembler, therefore,has no incentive to produce at time t2. To avoid los-ing 412/255q = 12/5 (= qS1 −qwr ), the forward-lookingsupplier breaks the contract at time t1.

An obvious resolution to the situation described inExample 2 is to impose contractual default penaltiesthat would induce the firms not to shirk when pro-ducing is profitable for the integrated supply chain.Had the assembler faced a default penalty greaterthan 42/55 at time t2, the state-contingent contract inExample 2 would again achieve the integrated chan-nel payoff. A state-contingent contract with penal-ties, however, is still not a guarantee that all supplychain members will default only when it is optimal todefault for the integrated supply chain.

Example 3 (Continuation of Example 2). Every-thing is the same as in Example 2, except that t1 ≤

t2; S has zero assets; and when viewed from time t0,4S11 S21D5 is a Bernoulli random vector that takes onvalues 42/512/5155, 42/514/5155 with respective prob-abilities �842/512/51559= 1/2, �842/514/51559= 1/2.

Under these assumptions, the supplier will require42/55q in cash-on-hand to procure production inputsat time t1 and either 42/255q or 48/255q (depending onthe state of the world at time t2) to pay the assemblerat time t2.

Facing a hard budget constraint, the upstream sup-plier, S, could do one of the following: (a) borrowfrom a bank and incur an interest payment, which isa cost that the integrated supply chain could avoid;(b) borrow from the downstream assembler. To over-come the supplier’s budget constraint, the assemblercould simultaneously procure commodity 1 at time t1,

worth 42/55q, and put over the payment of wrq totime t2. The problem is that option (b) is not incentivecompatible because if 4S11 S21D5 = 42/514/5155, thenthe assembler will earn a payoff of −42/55q. If, on theother hand, 4S11 S21D5 = 42/512/5155, then the sup-plier will earn 2/5. Since 41/2542/55− 41/2542/55q ≤ 0for all q = 1121 0 0 0, then the assembler has no incen-tive to lend to the supplier. At the same time, theintegrated channel can achieve a payoff of 1 withprobability 1/2.

In summary, characteristics of the industry or chan-nel relationship could make implementing of the state-contingent contract either expensive or ineffective.Since a price-only contract (Lariviere and Porteus2001) is a natural alternative to the state-contingentcontract, evaluating performance under such a con-tract is a first step in determining whether the addi-tional costs of a more sophisticated policy are worthbearing.

Similarly to the state-contingent, the traditionalprice-only contract too does not address the adversedefault issue. The idea that we develop in the remain-der of this paper is that hedging together with contrac-tually imposed default penalties are a way to protectthe entire supply chain from failing when one of itsmembers faces high realized input costs while its sup-ply chain partner would like to produce.

4. Model of a DecentralizedSupply Chain

Our assumptions are exactly the same as in §3 withthe provision that the supply chain is decentralized,implying that the supplier, S, is now selling to thenewsvendor, M . Both firms are risk neutral and areprotected by limited liability. Each firm has a reserva-tion payoff Rj ≥ 0 and cj ≥ 0, j ∈ 8M1S9 dollars in cash-on-hand. Firms have no outside wealth. (As explainedin Cachon 2003, p. 234, exogenous reservation pay-offs, Rj , are a standard way to model a firm’s marketpower.)

One can interpret the firms as playing a game overtimes t0 < t1 ≤ t2 ≤ t3. The supplier offers the assem-bler a wholesale contract at time t0 when only thedistribution function of future commodity input costsis known. Because of production lead times, the sup-plier produces the subassembly at time t1 and theassembler performs the final assembly at time t2. Asbefore, an outcome of the supplier’s commodity inputcost, S1 ≥ 0, is revealed at time t1; an outcome of anassembler’s commodity input cost, S2 ≥ 0, is revealedat time t2; the assembler’s sales revenue is realized attime t3. The discount rate between times t0 and t3 istaken to be zero.

As explained in §3, both firms may have separateincentives to default on the supply contract whentheir time ti commodity input costs turn out be too

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high in relation to their predetermined selling prices.We assume that if a firm defaults on the supply con-tract, then it incurs a contractually predetermineddefault penalty Lj , j ∈ 8M1S9. Throughout the paper,we assume that the penalty, Lj , is payable to firm j’ssupply chain partner. (It is, however, rather straight-forward to change this to a model, in which Lj repre-sents an indirect, reputation cost.) If both firms chooseto produce, then q is the highest payoff either firm canachieve, as will be seen. To prevent a situation in whichone firm experiences a financial windfall when its supplychain partner defaults, throughout the analysis we assumethat LS ≤ q and LM ≤ q. (As an example of a supplycontract penalty, see the Plymouth Rubber Companycontract in §A of the online supplement (provided athttp://dx.doi.org/10.1287/msom.2014.0514); there, Lj

is set as a percentage of the wholesale price. In theDavita, Inc. contract, the penalty can be more than100% of the wholesale price.)

One of the goals of this paper is to investigatewhether firms should be expected to hedge their com-modity input costs to avoid default on the underlyingsupply contract. For this reason, we suppose that foreach commodity i = 112 there exists a futures con-tract to purchase a specific amount of commodity iat time ti for a prespecified (futures) price, Fi. As isconvention, the futures price is set so that the valueof the futures contract at inception is zero. The payoffto a futures contract on commodity i = 112 is realizedat time ti, where the payoff is the difference betweenthe futures price, Fi, and the time ti spot price, Si. Weuse the variable nj , j ∈ 8M1S9 to represent firm j’sposition in the futures market: nj > 0 indicates thatfirm j has a “long position” of nj futures contracts,and nj < 0 indicates that it has a “short position.” Ifthe firm j ∈ 8M1S9 buys—or is long in—nj futurescontracts, its time ti payoff is nj4Si − Fi5, which willbe positive when input prices are high and negative wheninput prices are low. In our setting, both firms will behedging by taking long positions. Therefore, hereafter,nj ≥ 0, j ∈ 8M1S9.

The futures contracts are negotiated at a futuresexchange, which acts as an intermediary. One of thekey roles of the exchange is to organize trading sothat futures contract defaults are completely avoided(for additional discussion, see Hull 2014, §§2 and 3).For this reason, the exchange will limit the numberof contracts a firm j ∈ 8M1S9 can buy if it knows thatfirm j will not have sufficient ex post resources to payoff its long futures contract position when the realizedinput prices at time ti, i = 112 are low.

In summary, there are three key assumptions aboutpenalties and futures contracts that underlie ourmodel of supply chain hedging:

Assumption 1. Any default on the underlying supplycontract is costly; neither firm, however, stands to unusu-ally profit from a default of its supply chain partner.

Assumption 2. To avoid pathological cases, the futuresprices Fi, i = 112 are low enough so that by hedging, nei-ther firm effectively locks in a negative (expected) profit.

Assumption 3. The futures market is credible, mean-ing that neither firm j ∈ 8M1S9 defaults on its futurescontract position in all possible futures states of the world(Hull 2014, §§2 and 3).

Later in the paper, we will translate Assump-tions 1–3 into mathematical conditions. It is worthmentioning that all three assumptions are quite stan-dard in the hedging literature, which includes Smithand Stulz (1985) and Froot et al. (1993).

4.1. Game SequenceWhat follows is a sequence of stages in the gamebetween the supplier and the assembler. At time t0:

• Firms choose to take positions Nj ≥ 0, j ∈ 8M1S9in the futures contracts.

• The futures exchange accepts 0 ≤ nj ≤ Nj , j ∈

8M1S9.• Supplier, S, sets a wholesale price, w.• Assembler, M , sets an order quantity, q.This ends time t0. Before time t1 begins, the future

spot price of commodity 1 price is revealed. Attime t1:

• The supplier produces q units for the assem-bler and incurs a production cost qS1. (If the supplierchooses not to produce, then it incurs a penalty LS .)

This ends time t1. Before time t2 begins, the futurespot price of commodity 2 and the supplier’s produc-tion decision at time t1 are revealed. If the supplierproduced at time t1, then at time t2:

• The assembler produces q units of the final prod-uct and incur a production cost q4w + S25. (If theassembler chooses not to produce, then it incurs apenalty LM .)

This ends time t2. Before time t3 begins, the valueof demand D is revealed. If the assembler producedat time t2, then at time t3 the sales revenue min8q1D9is realized.

In terms of the information structure, we assumethat all market participants can observe the actionstaken by all players and can observe all market out-comes, i.e., information is complete. This assumptioncan best be justified with emerging empirical researchin finance, which reveals that companies commonlyincorporate covenant restrictions into supply con-tracts (e.g., see Costello 2011) that help them reduce oreliminate asymmetric information and moral hazard.

5. Analysis of the DecentralizedSupply Chain

As previously mentioned, one of the objectives of thispaper is to identify conditions under which the sup-ply chain firms should be expected to hedge their

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commodity input costs. We will proceed with theanalysis by first assuming that neither firm hedges itsinput costs and show that in this case both firms sub-ject themselves to default risk, since they agree to selltheir output at prices that are fixed before all factorsaffecting their productivity are known. We’ll refer tothis situation as the “unhedged wholesale contract.”Where useful, the subscript “U” will be used to indi-cate association with this subgame.

We then analyze the case in which the supplieravoids defaulting on the supply contract by enteringinto nS > 0 futures contracts to buy commodity 1 attime t1 and the assembler avoids defaulting on thesupply contract by entering into nM > 0 futures con-tracts to buy commodity 2 at time t2. We will refer tothis subgame as the “hedged wholesale contract” anduse the subscript “H” to represent it. In analyzing thissubgame, we determine the number of futures con-tracts, nj1 j ∈ 8M1S9, both firms will be purchasing inequilibrium and consider what happens if Nj > 0 andNk = 0, j1 k ∈ 8M1S9, j 6= k, which is a case in whichone of the firms attempts to hedge while its supplychain partner does not hedge.

As is standard, the model is solved in two stagesusing backward induction: First stage characterizesthe equilibrium behavior of the assembler; secondstage characterizes the equilibrium behavior of thesupplier and identifies the equilibrium in each sub-game. The final equilibrium in which both firmsdecide whether to hedge can be determined by sim-ply comparing the expected profits the supply chainfirms generate with and without hedging.

The cash flows reveal that, in general, each firm j ∈

8M1S9 will choose not to produce if the firm j’sfuture spot price, Si1 i = 112, is either too high inrelation to firm j’s selling price (retail or wholesaleprice) or too low in relation to firm j’s futures price,Fi1 i = 112. We evaluate the issue of default in detailin Lemma 1 and present the results graphically in

Figure 2 (Color online) Supplier’s and Assembler’s EquilibriumStrategies

S produces;M does notproduce

S and M bothproduce

S does not produce;M unable to produce

S produces; M doesnot produce

S do

es n

ot p

rodu

ce;

M u

nabl

e to

pro

duce

uMH

uML

S2

S1

uSL2

uSH2

uSH1

uSL1

(0, 0)

Figure 2. Given arbitrary futures contract positions,NM ≥ 0 and NS ≥ 0, cash flows to both firms are

çM=

NM 4S2 −F25+min8q1D9−qS2 −wq

if uSL1<S1 ≤uS

H11uML <S2 ≤uM

H 1t2<t31

NM 4S2 −F25+min8q1D941−S25−wq

if uSL1<S1 ≤uS

H11uML <S2 ≤uM

H 1t2 = t31

max8NM 4S2 −F25−LM1−cM 9

if uSL2<S1 ≤uS

H21S2>uMH 1t2 ≤ t31

max8NM 4S2 −F25−LM1−cM 9

if uSL1<S1 ≤uS

H11S2 ≤uML 1t2 ≤ t3

max8NM 4S2 −F25+`S1−cM 9

otherwise3

(3a)

çS=

NS4S1 −F15+q4w−S15

if uSL1<S1 ≤uS

H11uML <S2 ≤uM

H 1t1 ≤ t21

NS4S1 −F15+`M −qS1

if uSL2<S1 ≤uS

H21S2>uMH 1t1 ≤ t21

NS4S1 −F15+`M −qS1

if uSL1<S1 ≤uS

H11S2 ≤uML 1t1 ≤ t21

max8NS4S1 −F15−LS1−cS9

otherwise;

(3b)

where`S ≡ min8LS1 4cS +NS4S1 − F155

+9 and`M ≡ min8LM1 4cM +NM 4S2 − F255

+90(3c)

Above, `j , j ∈ 8M1S9 are expressions for penaltiesthat each firm j winds up paying if it fails to produce.In (3c), Nj4Si − Fi5, j ∈ 8M1S9, i = 112 is firm j’s payofffrom the futures contract position and cj +Nj4Si − Fi5is firm j’s total capital at time ti. Taken together, theexpressions for `j reflect the fact that both firms j havelimited liability. The u’s in Equations (3a) and (3b) arespot prices of commodities 1 and 2 at which the sup-plier and the assembler will break the underlying sup-ply contract at times t1 and t2, respectively. Their val-ues are as follows. (Formal proofs of all of the paper’sresults are included in §B of the online supplement.)

Lemma 1. If t0 < t1 ≤ t2 ≤ t3 and 0 ≤ Nj ≤ q, j ∈

8M1S9, then in (3), uSL1 = uS

L2 = 0, uML = 0, and

uSH1 =

min{(

wq + cS −NSF1

q −NS

)+

1w+LS

q11}

if t1 = t21

min{(

cS −NSF1

q −NS

)+

1(

q w �8S2 ≤ uMH 9

+`M �8S2 >uMH 9+LS

)

· 4q5−111}

if t1 < t23

(4a)

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uSH2 =

min{

`M +LS

q1

(

`M + cS −NSF1

q −NS

)+

11}

if t1 = t21

uSH1 if t1 < t23

(4b)

uMH =

min{(

min8q1D9+ cM −NMF2 −wq

min8q1D9−NM

)+

1

(

min8q1D9+LM −wq

min8q1D9

)+

11}

if t2 = t31

min{(

cM −NMF2 −wq

q −NM

)+

1

(

Ɛt0min8q1D9+LM

q−w

)+

1

Ɛt0min8q1D9

q

}

if t2 < t30

(4c)

If t0 < t1 ≤ t2 ≤ t3 and 0 ≤ q <Nj , j ∈ 8M1S9, then in (3):

uSH1 =

max{

uSL11min

{

w+LS

q11}}

if t1 = t21

max{

uSL11min

{

4q w �8S2 ≤ uMH 9

+ `M �8S2 >uMH 9+LS5 · 4q5

−111}}

if t1 < t23

(5a)

uSL1 =

(

NSF1 −wq − cSNS − q

)+

if t1 = t21

(

NSF1 − cSNS − q

)+

if t1 < t23

(5b)

uSH2 =

max{

uSL21

`M +LS

q11}

if t1 = t21

uSH1 if t1 < t23

(5c)

uSL2 =

(

NSF1 − cS − `MNS − q

)+

if t1 = t21

uSL1 if t1 < t23

(5d)

uMH =

max{

uML 1min

{

min8q1D9+LM −wq

min8q1D911}}

if t2 = t31

{

uML 1min

{(

Ɛt0min8q1D9+LM

q−w

)+

1

Ɛt0min8q1D9

q

}}

if t2<t33

(5e)

uML =

(

wq +NMF2 − min8q1D9− cMNM − min8q1D9

)+

if t2 = t31

(

wq +nMF2 − cMNM − q

)+

if t2 < t30

(5f)

A careful inspection of the functions u specified inthe above lemma reveals that each firm j ∈ 8M1S9 willdefault on the supply chain contract if at least one ofthe following holds:

1. Firm j’s expected profit from production isstrictly less than the penalty Lj1 j ∈ 8M1S9 firm jagreed to pay for breaking the underlying supply con-tract. That is, if input prices at time ti turn out to behigh, and the cost of default, Lj , is low, then firm j willchoose not to produce owing to insufficient expectedpayoff.

2. Firm j lacks sufficient resources to produce. Toillustrate, suppose t1 < t2. Then, the supplier, S, willinevitably default at time t1 if cS + NS4S1 − F15 < qS1because, in such a case, the time t1 cost of productioninputs, qS1, exceeds the time t1 available capital, cS +

NS4S1 − F15.Figure 2, which illustrates the functions u graph-

ically confirms that the supplier, S, will rationallychoose to produce if it knows that the assembler willwant to produce and the time t1 spot price of com-modity 1 is between uS

L1 and uSH1. In this case the

supplier will earn a payoff of q4w − S15+NS4S1 − F15.However, the figure also reveals that the supplier willchoose to produce if it knows that the assembler willnot produce and if its time t1 spot price of commod-ity 1 is between uS

L2 and uSH2. In this case, the supplier

will receive a payoff of NS4S1 − F15+ `M − qS1, whichis in excess of the penalty payment `S . The assembler,on the other hand, will only produce if its commodityinput cost is between uM

L and uMH and if the supplier

produces. (This is because the assembler’s productionprocess requires an input from the supplier.)

A closer examination at the functions u given in thelemma also reveals that the exogenously given capitalendowment, cj ≥ 0, j ∈ 8M1S9, causes “parallel shifts”to the functions u, but it does not qualitatively changehow each firm j behaves as its endogenously chosenposition in the futures market, Nj , and the spot priceof commodity i = 112 change (except when cj is verylarge; then the role of the firm j’s futures contractposition, Nj , diminishes). It is for this reason that here-after we analyze the model in the analytically simplest wayby taking cj = 0. Moreover, to allow firms to producewithout having to borrow, we take 0 < t1 = t2 = t3.Then both firms incur revenues and costs simulta-neously, and production is feasible without cash-on-hand and without borrowing. (Strictly speaking, to

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avoid borrowing, all we require is that 4t3 − t15≥ 0 beno more than the standard contractual grace period,which allows payments to be received for a certainperiod of time after a payment due date. This is usu-ally up to 21 days.)

5.1. Analysis of the Unhedged ContractSuppose now that neither firm hedges its commod-ity input cost. The assembler’s and supplier’s time t0payoffs, çM

U and çSU , can be recovered from Equa-

tions (3) and (4) and by setting NM = NS = 0. SincecM = cS = 0, then Equations (3c) imply `M = `S = 0 forall LS ≥ 0, LM ≥ 0, implying that default penalties in theunhedged case are irrelevant.

maxd≤q≤d

{

∫ uSH1

0

∫ d

q

∫ uMH 4q1q5

0q41−y+wU 5f 4x5h4y1z5dydxdz

+

∫ uSH1

0

∫ q

d

∫ uMH 4x1q5

04x−xy−wU q5f 4x5h4y1z5dydxdz

}

0 (6)

Using the necessary optimality conditions withrespect to q, (e.g., Bertsekas 2003, Proposition 1.1.1)and the Leibniz rule, we can obtain the implicitinverse demand curve, wU 4q5, that the supplier faces:

wU 4q5=

∫ uSH10

∫ d

q

∫ uMH 4q1q5

0 41−y5f 4x5h4y1z5dydxdz

�8S1 ≤uSH11S2 ≤uM

H 90 (7)

By substituting the right-hand side of (7) for thewholesale price, wU , into the maximand in (6), wecan obtain an expression for the assembler’s expectedpayoff (in terms of q):

Ɛt0çM

U 4q5

=

∫ uSH1

0

∫ q

d

∫ uMH 4x1q5

0x41 − y5f 4x5h4y1 z5dy dx dz0 (8)

Since the distribution of demand, F , has support on6d1 d7, then Ɛt0

çMU 4q52 6d1 d7→�, given by (8), is a con-

tinuous real-valued mapping, where 6d1 d7 is a non-empty subset of �. There exists a quantity q∗

M such thatƐt0

çMU 4q5 ≤ Ɛt0

çMU 6q∗

M 7 for all q ∈ 6d1 d7 (Weierstrassproposition). It follows that if Ɛt0

çMU 6q∗

M 7 ≤ RM , thenthe assembler will place no order.

Ɛt0çS

U 4q5 = q(

wU 4q5− Ɛt04S1 � S1 ≤ uS

H11 S2 ≤ uMH 5)

·�{

S1 ≤ uSH11 S2 ≤ uM

H

}

1 or, equivalently,

Ɛt0çS

U 4q5 = q∫ uMH 4q1 q5

0

∫ d

q

∫ uSH1

041 − y5f 4x5h4y1 z5dzdx dy

− q∫ uMH 4q1 q5

0

∫ uSH1

0zh4y1z5dzdy0 (9)

Since F has support on 6d1 d7, then Ɛt0çS

U 4q52 6d1d7→�is a continuous real-valued mapping where 6d1 d7 is

a nonempty subset of �. There exists a quan-tity q∗

U such that Ɛt0çS

U 4q5 ≤ Ɛt0çS

U 6q∗U 7 for all q ∈

6d1 d7 (Weierstrass proposition). The supplier’s opti-mal order quantity q∗∗

U is given by arg maxq≥d Ɛt0çS

U 4q5

s.t. Ɛt0çM

U 4q5 ≥ RM and Ɛt0çS

U 4q5 ≥ RS , where RS1RM

are the firms’ reservation payoffs (note that if RM andRS are excessively large, then q∗∗

U may not exist). Itfollows that if the supplier were able to choose anywholesale price, it would choose wU 4q

∗U 5, where wU

solves Equation (7).

5.2. Analysis of the Hedged ContractSuppose now that both firms hedge by entering intonj ≥ 0, j ∈ 8M1S9 futures contracts to purchase com-modity i = 112 at time ti. Each firm j’s time tipayoff from the futures contract position will benj4Si − Fi5, which is positive when the realized futurespot price, Si, is high (i.e., when 0 ≤ Fi < Si ≤ �) andnegative when it is low (i.e., when 0 ≤ Si < Fi < �).Before we begin the Stage 1 analysis of the supplychain contract with hedging, we present a preliminaryresult that deals with firms’ equilibrium long futurescontract positions.

Proposition 1. Suppose Assumption 3 holds and cj =0, j ∈ 8M1S9. The number of futures contracts, nj ≥ 0,that can be supported as a subgame equilibrium (SE) mustsatisfy the following constraints:

q LS

LS + q4wH − F15≤ nS ≤ min

{

q wH

F11LM

F11 q

}

1 (10a)

d LM

d41 − F25+LM −wH q

≤ nM ≤ min{

d− q wH

F21LS4S2 = 05

F21 q

}

1 (10b)

where LS4S2 = 05 denotes the value penalty LS when S2 = 0.(Of course, this is only relevant in cases when LS is afunction of S2.)

Reading from the left, the first upper bound onnj given in (10) is in force when both supply chainmembers choose to produce at time ti, i = 112 andfirm j’s realized input price, Si, at time ti is less thanthe futures price, Fi. The bound ensures that firm j’soperating profit is sufficient to offset any loss fromfirm j’s futures contract position, nj4Si − Fi5 < 0, j ∈

8M1S9, i = 112. Otherwise firm j would be at risk ofdefaulting on its futures contract position because ofinsufficient resources.

The second upper bound on nj given in (10) is inforce when firm j’s realized time ti input price is againlow (i.e., Si < Fi) and its supply chain counterpart,firm k ∈ 8M1S9, j 6= k, chooses not to produce at time tl,l = 112, l 6= i. This will occur exactly when firm k’srealized input cost, Sl, at time tl is high (infinite). In this

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situation firm k will be contractually obligated to usethe payoff from its futures contract position, nk4Sl −

Fl5 > 0, to compensate firm j with a default penalty,Lk ≥ 0. (Note that if firm k breaks the supply contract,the penalty payment, Lk, is firm j’s only revenue.) Asbefore, the upper bound on nj ensures that Lk is suf-ficient to offset any loss from firm j’s futures contractposition, nj4Si − Fi5 < 0. At the same time, the lowerbound on nk given in (10) ensures that firm k’s pay-off from its futures contract position, nk4Sl − Fl5 > 0, isenough to pay at least Lk to firm j .

In summary, Proposition 1 can be viewed as a setof necessary and sufficient conditions under whichneither supply chain member defaults on its futurescontract position in all possible futures states of theworld—as specified in our Assumption 3.

Since the lower bounds on nM and nS are strictlypositive, then Proposition 1 also reveals that, in equi-librium, either both firms j ∈ 8M1S9 simultaneouslyenter into futures contract positions nj > 0, or theydo not hedge at all. This reflects the fact that a situa-tion in which one firm hedges while its supply chainpartner does not hedge, makes the hedger susceptibleto default on the futures contract. This default occursexactly when the realized future spot price of the firmthat did not hedge is high (causing it to default on theunderlying supply contract), and the realized futurespot price of the firm that did hedge is low (caus-ing it to default on the futures contract). Corollary 1summarizes the result.

Corollary 1. If cj = 0, j ∈ 8M1S9, then in equilib-rium, either both firms j simultaneously enter into futurescontract positions nj > 0, or they do not hedge at all.

Finally, to ensure that the ranges for nj , j ∈ 8M1S9given in Proposition 1 are nonempty, we require thefollowing two conditions:

Condition 1. wHq≤LM≤q and min8q1D941−S25+−

wH q ≤ LS ≤ q0

Condition 2. F1 ≤wH and F2 ≤ 1 − 4q wH 5/d.

Condition 1 corresponds to Assumption 1, whichrequires that a default on the underlying supply con-tract be costly; neither firm, however, should unusu-ally profit from a default of its supply chain part-ner. Interestingly, the penalties that satisfy the boundsgiven in Condition 1 not only ensure that neitherfirm j ∈ 8M1S9 defaults on its futures contract posi-tion, but also that each firm j is indifferent as towhether its supply chain partner chooses to produceor not. To see this, consider the case when LM =wH q,and LS = min8q1D941 − S25

+ − wH q and when bothfirms hedge. Then the supplier is guaranteed to earna payoff of q wH 4q5 − min8q S11min8q1D941 − S25

+9 +

nS4S1 −F151 and the assembler is guaranteed to receivea payoff of min8q1D941 − S25

+ − q wH + nM 4S2 − F25,

whatever action its supply chain counterpart takes. Itis worth mentioning that penalties that satisfy Condi-tion 1 are not inconsistent with what one can observein empirical practice. For example, in the Davita, Inc.contract included in §A of the online supplement, thepenalty can be more than 100% of the wholesale price.

Condition 2 corresponds to Assumption 2, whichrequires upper bounds on the futures prices of com-modities 1 and 2. Without these upper bounds, firmscannot make money in expectation, and hedging isessentially infeasible.

Lemma 2. If Conditions 1 and 2 hold, then the rangefor nj , j ∈ 8M1S9 given in (10) is nonempty for all 0 ≤

S1 <�, 0 ≤ S2 <�, d ≤D ≤ d, and cj = 0.

We illustrate Proposition 1 with the following sim-ple example.

Example 4. Suppose that S1 and S2 are independentand log-normally distributed with volatilities �1 =

�2 = 60%. Lead time, i.e., t1 − t0, is six weeks. F1 =

F2 = $0025. Demand, D, is uniformly distributed onthe interval 610011507. Using Condition 1, we assumeLM = wH q and LS = min8q1D941 − S25

+ −wH q, wherethe (equilibrium) wholesale price wH is given by (12),which will be derived later in the paper. Using theresult in Proposition 1, the following table summa-rizes the range for nS and nM that can be supportedas an SE.

Order Range for nS Range for nM

quantity, q required by (10a) required by (10b)

1001003

≤ nS ≤ 100 nM = 100

10521667

59≤ nS ≤ 105 91 ≤ nM ≤ 105

11011738

29≤ nS ≤ 110

2423

≤ nM ≤ 110

11541439

57≤ nS ≤ 115 69 ≤ nM ≤ 115

1206967

≤ nS ≤ 120 56 ≤ nM ≤ 120

Note that when q = 100, then the assembler faces nodemand risk, but the range for nM reduces to a sin-gle point, implying that the assembler can only hedgeby purchasing q futures contracts. The range for nM ,however, opens up as the order quantity, q increases.Therefore for q > 100, the assembler can hedge byadopting different hedging policies that achieve thesame expected payoff.

Figure 3 illustrates Proposition 1 graphically forthe case when q = 120. Figure 3(a) illustrates a hypo-thetical nonequilibrium situation in which neither ofthe firms purchased a sufficient number of futurescontracts and as a consequence there exist S1 × S2 ×

D combinations for which at least one of the firms

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Figure 3 (Color online) Range of S1 and S2 in Which Both Firms Can Sustain Not Defaulting on Their Futures Contract Positions

(a) nM = 55, nS = 85 (b) nM = nS = 110

1 × 106

500,000

0

140

120

100

0

500,000

1 × 106

S2

S1

Demand

2.01.5

1.0S2

S1

0.5

0.0

140

120Demand

100

0.00.5

1.01.5

2.0

Note. Assumes F1 = F2 = $0025, D ∼U610011507, q = 120, LM = wH q, and LS =min8q1D941−S25+ −wH q.

should be expected to default on its futures contractposition (these combinations are represented by thehollow space in Figure 3(a)). In Figure 3(b), bothfirms purchased a correct number of futures con-tracts and can rationally sustain not to default for allS1 × S2 ×D.

5.2.1. Partial vs. Complete Hedging. FollowingSFAS 52 and 80, which are generally accepted ac-counting principles (GAAP) for the accounting treat-ment of hedged transactions (see Mian 1996, p. 425),one could adopt the assumption that a futures tradeis not considered to be a hedge unless the underlyingtransaction is a firm commitment. Under, this assump-tion, both firms would hedge by taking futures con-tract positions nj = q, j ∈ 8M1S9 and thus completelyinsulate their market values from hedge-able risks.

Here, however, we allow that firms adopt the“take-the-money-and-run” strategy under which firmsdefault on the underlying supply contract and selltheir commodity inputs on the open market—wheneverit is profitable to do so. There is some empirical evi-dence for this conduct: Hillier et al. (2008, p. 768), forexample, describe a case where in 2001 an aluminumproducer Alcoa temporarily shut down its smeltersand sold its electricity futures contracts on the openmarket (since it could make more money by sellingelectricity than by selling aluminum).

Allowing firms to adopt the take-the-money-and-run strategy implies that the hedged supply con-tract is no longer a firm commitment and the resultsin Proposition 1 reveal that in equilibrium firmsmay only hedge partially, meaning that 0 < nj < q,j ∈ 8M1S9.

The hedged contract is operationalized as follows:In Stage 1, both firms infer the equilibrium quantity,say q∗∗

H ∈H , and adopt hedging positions nj , j ∈ 8M1S9

that satisfy Proposition 1. The equilibrium quantitythat firms infer can be derived by adapting resultsfrom the existing literature (for example, Lariviereand Porteus 2001). Using Equations (3), the assem-bler’s expected payoff is given by

Ɛt0çM

H 4q1wH 5=

∫ �

0

∫ d

q

∫ 1

0q41 − y5f 4x5h4y1 z5dy dx dz

+

∫ �

0

∫ q

d

∫ 1

041 − y5xf 4x5h4y1 z5dy dx dz

− qwH +nM 4Ɛt0S2 − F25

︸︷︷︸

=0

0 (11)

Using the fact that F2 = Ɛt0S2, the assembler’s optimal

order quantity implicitly defined by

∫ �

0

∫ d

q

∫ 1

041 − y5f 4x5h4y1 z5dy dx dz−wH = 01

from which we can solve for the inverse demand, saywH 4q5, the supplier faces

wH 4q5 = min{

∫ �

0

∫ d

q

∫ 1

041 − y5f 4x5h4y1 z5dy dx dz1

d41 − F25

q

}

0 (12)

We can then use Equations (11) and (12) to yield

Ɛt0çM

H 4q5

= max{

∫ �

0

∫ q

d

∫ 1

0x41 − y5f 4x5h4y1 z5dy dx dz1

∫ �

0

∫ d

q

∫ 1

0q41 − y5f 4x5h4y1 z5dy dx dz

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+

∫ �

0

∫ q

d

∫ 1

041 − y5xf 4x5h4y1 z5dy dx dz

− d41 − F25

}

1 (13)

which is the assembler’s expected payoff in termsof q. Differentiation reveals that the assembler’s pay-off is increasing in q. To see this, the derivatives ofthe first and the second expression on the right-handside of (13) are

∫ �

0

∫ 1

0q41 − y5f 4q5h4y1 z5dy dz≥ 0 and

∫ �

0

∫ d

q

∫ 1

041 − y5 f 4x5h4y1 z5dy dx dz≥ 00

(14)

The supplier’s expected profit is

Ɛt0çS

H 4q5 = q wH 4q5− Ɛt0min

{

q S11min8q1D941 − S25+}

+nS 4Ɛt0S1 − F15

︸︷︷︸

=0

0 (15)

The supplier’s most preferred order quantity, q∗H , is

given by arg maxq≥d Ɛt0çS

H 4q5; the supplier’s optimalorder quantity q∗∗

H is given by arg maxq≥d Ɛt0çS

H 4q5

s.t. Ɛt0çM

H 4q5 ≥ RM and Ɛt0çS

H 4q5 ≥ RS , where RS1RM

are the firms’ reservation payoffs. In special case,the results in Lariviere and Porteus (2001) show thatƐt0

çSH 4q5, given by (15), is unimodal in q.

5.3. Equilibrium Among Unhedged andHedged Contracts

We now allow the firms to enter into the supplychain contract by choosing whether they wish tohedge. Depending on the model parameters, it is pos-sible to have equilibria where (a) neither firm hedges,

Figure 4 (Color online) Expected Payoffs Under Unhedged and Hedged Wholesale Contracts

(a) Set HM

30 35 40 45

0

5

10

15

Payo

ffs

20

q

E ΠHM

E ΠUM

Set HM

qqMq*

M

(b) Set HS

30 35 40 45

0

2

4

6

8

10

12

q

q

Set HS

Payo

ffs

E ΠHS

E ΠUS

qS^q*

H

^

Notes. D ∼U4301605; S1 and S2 are independent and log-normally distributed; time t0 prices of commodities 1 and 2 are $0.3; lead-time= t1 − t0 = 6 weeks;t2 = t1; annual volatility of commodity 1 and 2 spot prices is 60%. Definitions. (i) For the order quantities qS and qM , respectively, we have Ɛt0

çSH 6qS7 =

Ɛt0çS

U 4q∗

U 5 and Ɛt0çM

H 6qM 7 = Ɛt0çM

U 4q∗

M 5. (ii) For the order quantity q = qS , we have Ɛt0çS

H 6qS7 = Ɛt0çS

U 4q∗

U 5; for the quantity q = qM , we have Ɛt0çM

H 6qM 7 =

Ɛt0çM

U 4q∗

M 5. (iii) For the order quantity q = q, we have Ɛt0çS

H 6q7= 0.

or (b) both firms hedge. Because of Proposition 1,case (a) will occur if one of the firms in the modeldecides not to hedge at time t0. If, on the otherhand, there exists a subgame perfect Nash equilib-rium (SPNE) in which

Ɛt0çS

H 4q∗∗

H 5≥ Ɛt0çS

U 4q∗∗

U 51 (16a)

then the supplier, S, will hedge, and as will be seen inLemma 3, part (i), the assembler will hedge as well.Similarly, the supplier will hedge in cases where theunhedged wholesale contract may be infeasible. Thisoccurs when

Ɛt0çM

U 4q∗

M 5≤RM1 and there existsq ∈H for which

RS ≤ Ɛt0çS

H 4q5 and RM ≤ Ɛt0çM

H 4q50 (16b)

The following Lemma 3 is needed to establish thatconditions (16a) and (16b) can hold in equilibrium.Parts (i) and (ii) describe how the risk of supplycontract default affects both the assembler’s expectedpayoff and the wholesale price. Parts (iii) and (iv)give sufficient conditions under which (for both thesupplier and the assembler) there exist expected pay-off levels that may only be achieved via the hedgedwholesale contract.

Lemma 3. Let wU 4q5, Ɛt0çM

U 4q5, Ɛt0çS

U 4q5, wH 4q5,Ɛt0

çMH 4q5, and Ɛt0

çSH 4q5, respectively, be given by (7), (8),

(9), (12), (13), and (15). If cj = 0 and Conditions 1 and 2hold. Then:

(i) 0 ≤ Ɛt0çM

U 4q5≤ Ɛt0çM

H 4q5.(ii) wH 4q5≤wU 4q5 (if S1 and S2 are independent).(iii) The set HM 2= 8q ∈H � Ɛt0

çMU 4q∗

M 5≤ Ɛt0çM

H 4q59 isnonempty. (An example of the set HM can be seen graphi-cally in Figure 4(a).)

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(iv) HS 2= 8q ∈ H � Ɛt0çS

U 4q∗U 5 ≤ Ɛt0

çSH 4q59 is non-

empty if

�{

S1 ≤SH11 S2 ≤ uM

H 4q∗

U 1 q∗

U 5}

+

(

∫ 1

0

zhZ4z5

F 4q∗U 5

dz+

∫ 1

0yhY 4y5dy

)

≤�8S2 ≤ 19

+

(

∫ uSH1

0

∫ uMH 4q∗U 1q∗

U 5

0

(

z

F 4q∗U 5

+ y

)

h4y1z5dy dz

)

1 (17)

where hY 4y5 and hZ4z5 denote marginal densities. (An ex-ample of the set HS can be seen graphically in Figure 4(b).)

For the case when S1 and S2 are independent,part (ii) of the lemma formally establishes that theequilibrium wholesale price is lower when the down-stream assembler hedges. This result reflects the factthat by hedging, the assembler guarantees the sup-ply contract performance and the rational supplierresponds to this guarantee by reducing the whole-sale price.

Parts (iii) and (iv) give sufficient conditions underwhich there exist expected payoff levels that can onlybe achieved via the hedged wholesale contract. As aconsequence of part (i) of the lemma, the set HM willalways be nonempty.

The set HS will be nonempty if d and the supplier’smargin are sufficiently large. The former is consis-tent with a situation in which there is a high baselinedemand for the assembler’s final product. The latteris consistent with a situation in which F1 is sufficientlylow and wH is sufficiently high. Under these con-ditions, reducing the probability that the assemblerdefaults on the supply contract is ex-ante importantto the supplier. Mathematically, this situation reducesto condition (17). We illustrate this in the followingexample, which reveals that the supplier can be betteroff or worse under the hedged contract.

Example 5 (Continuation of Example 4). If F1 =

F2 = $0025, q∗U = 105 and condition (17) reduces to

10446 ≤ 10477. In fact, with RM = RS = 0, the high-est profit the supplier can attain without hedging isƐt0

çSU 4q

∗U 5 = $39089. However, if the supplier and the

assembler, respectively, purchase 21667/59 ≤ nS ≤ 105and 91 ≤ nM ≤ 105 futures contracts, the supplier’sexpected payoff increases from $39089 to $42086.Clearly, hedging is beneficial to the supplier (Notethat the assembler will join the supplier in hedgingbecause of Lemma 3, part (i).) On the other hand,if F1 = F2 = $0045, then q∗

U = 102 and condition (17)reduces to 1038001 � 103488 and the highest profit thesupplier can attain without hedging is Ɛt0

çSU 4q

∗U 5 =

$8022. With hedging, however, the supplier’s expectedpayoff would decrease from $8022 to $4083. The sup-plier is, therefore, better off without hedging.

The primary use of Lemma 3 is in establishing thenext Proposition 2.

Proposition 2. If cj = 0 and Conditions 1 and 2 hold,then there exist SPNEs in which both firms will hedge.Both firms will hedge if (i) RM ≥ Ɛt0

çMU 4q∗

M 5; or (ii) ifRM = 0 and the set HS is nonempty (see part 3 ofLemma 3).

Prediction 1. Offers of the hedged wholesale con-tract should be expected if the downstream firm’smarket power exceeds a critical threshold.

Prediction 1 essentially identifies situations whenhedging is mainly important to the downstreamassembler. A second empirical prediction regardingthe use of a hedged contract can be made usingpart (ii) of Proposition 2. Prediction 2 identifies sit-uations when hedging is mainly important to theupstream supplier.

Prediction 2. Offers of the hedged wholesale con-tract should be expected if the upstream firm oper-ates on a large margin, there is a high baselinedemand for downstream firm’s final product, and thedownstream firm’s market power is below a criticalthreshold.

Table 1 presents some data on how much bet-ter the assembler can do with the hedged contract:If the unhedged equilibrium order quantity is q, thencolumn (L) of Table 1 shows the percentage profitincrease the assembler experiences if it is offered analternative hedged contract that leaves the supplierno worse off than the original unhedged contract.The data shows that the assembler’s expected pay-off can increase from 0.34% to 75%. Column (G) ofTable 1 presents comparable data for the supplier.As might be expected, the value derived from usingthe hedged contract will depend on lead times, spotprice volatilities, and spot price correlation. In com-puting the results in Table 1, we assumed that thespot prices of commodities 1 and 2 were indepen-dent and chose annual volatilities ranging from 30%to 60% (a range comparable to equities; e.g., see Hull2014, §11). Interestingly, in practice, annual volatilityof commodity prices can easily exceed 60%. To illus-trate, silver, an industrial metal used in electrical con-tacts and in catalysis of chemical reactions, was trad-ing at around $18 per ounce in April and May of2010. Less than a year later, silver almost tripled invalue to $49 per ounce during the final week of April2011 (see Zsidisin and Hartley 2012). Our lead times,namely the difference between t1 and t0, were 4 to12 weeks.

We can also circle back some of our predictionsboth to results found in the research existing literatureand to the existing industry practices. In Example 5,we demonstrate that hedging can be unprofitable

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Table1

TheIm

pactof

Switc

hing

totheSu

pplie

r’s(Assem

bler’s)B

estH

edge

dCo

ntractTh

atLe

aves

theAs

sembler

(Sup

plier)No

Worse

Off

Supplier’s

besthedged

contract4q

′ 1wH4q

′ 55

Assembler’sbesthedged

contract4q

′′1w

H4q

′′55

subjecttoƐ t

0çM H4q

′ 5≥Ɛ t

0çM U4q5

subjecttoƐ t

0çS H4q

′′5≥Ɛ t

0çS U4q5

Lead

time

Annual

4Ɛt 0çS H4q

′ 55/

4Ɛt 0çS U4q5/Ɛ t

0çS U4q5

4Ɛt 0çS H4q

′ 5/4Ɛ t

0çS H4q

′ 54Ɛ

t 0çM H4q

′′5/

4Ɛt 0çM U4q5/4Ɛ

t 0çS U4q5

4Ɛt 0çM H4q

′′5/4Ɛ

t 0çS H4q

′′5

(weeks)

volatility

(%)

qwU4q5

q′

wH4q

′ 5(Ɛ

t 0çS U4q55

−154%

5+Ɛ t

0çM U4q5554%5

+Ɛ t

0çM H4q

′ 5554%

5q

′′wH6q

′′7

4Ɛt 0çM U4q5−1554%5

+Ɛ t

0çM U4q5554%5

+Ɛ t

0çM H4q

′′5554%5

(A)

(B)

(C)

(D)

(E)

(F)

(G)

(H)

(I)(J)

(K)

(L)

(M)

(N)

430

32.00

0.63

31.82

0.64

8057

84.37

85.37

33.78

0.57

≥75

15.38

28.07

34.00

0.57

33.97

0.57

0088

70.38

70.50

34.09

0.56

3.50

29.41

30.07

36.00

0.50

36.00

0.50

0034

57.89

57.90

36.00

0.50

0.43

41.93

41.96

38.00

0.43

37.94

0.44

0098

46.49

46.66

38.02

0.43

1.42

53.37

53.64

845

32.00

0.63

31.56

0.65

19057

84.74

86.84

35.30

0.52

≥75

15.01

38.84

34.00

0.57

33.67

0.58

5083

70.52

71.61

34.70

0.54

30.60

29.27

35.04

36.00

0.50

35.72

0.51

2073

57.79

58.37

36.08

0.50

7.18

42.04

43.67

38.00

0.43

37.36

0.45

6058

47.00

48.51

38.06

0.43

10.89

52.86

55.36

1260

32.00

0.63

31.39

0.65

22039

85.22

87.52

35.31

0.52

≥75

14.52

40.82

34.00

0.57

33.25

0.59

9034

71.49

73.20

34.80

0.54

51.96

28.30

37.44

36.00

0.50

35.09

0.53

5071

59.19

60.45

35.86

0.50

16.73

40.64

44.35

38.00

0.43

36.55

0.48

10014

48.80

51.14

37.62

0.45

18.49

51.05

55.21

Note

s.D

∼U4301605;S1andS2areindependenta

ndlog-norm

allydistrib

uted;timet 0

prices

ofcommodities

1and2are$0.3.C

olum

n(A):

Lead-time

=t 1

−t 0.T

heexam

pleassumes

t 0<

t 1=

t 2=

t 3(see

“"Ga

meSequence"in§4.1).

Column(B):

Measuresstandard

deviations

ofpricereturns.Co

lumns

(C)and(D):

Equilibriu

munhedged

wholesalecontract,4q1w

U4q55.C

olum

ns(E)and(F):Su

pplier’s

best

hedged

contract

4q′ 1wH4q

′ 55subjecttoƐ t

0çM H4q

′ 5≥Ɛ t

0çM U4q5.

Columns

(J)a

nd(K):

Assembler’sbesthedged

contract4q

′′1w

H4q

′′55subjecttoƐ t

0çS H4q

′′5≥Ɛ t

0çS U4q5.

for the upstream supplier. However, such a suppliermay still hedge in equilibrium if Ɛt0

çMU 4q∗

M 5 ≤ RM

(i.e., if the supplier’s counterpart has a high reserva-tion payoff). This observation is not necessarily con-sistent with the existing theories of hedging, whichimply that hedging should increase firms’ marketvalues. It does, however, appear to be consistentwith the empirical findings in Jin and Jorion (2006),who report that although many firms hedge, hedg-ing does not necessarily increase their market values.Likewise, it appears to be consistent with anecdo-tal examples from auto parts supply chains (e.g.,Hakim 2003, Matthews 2011), food processing sup-ply chains (e.g., Eckblad 2012), and energy supplychains (e.g., Thakkar 2013), which appeared in popu-lar business press.

6. ConclusionThe production processes of many firms depend onraw materials whose prices can be highly volatile(e.g., Carter et al. 2006). Moreover, many firms whopurchase raw materials also depend on supplierswho purchase raw materials for their own produc-tion. Of course, suppliers experience price volatility,too, and they may request price increases and sur-charges. In some cases, if commodity prices signif-icantly increase, the suppliers may not be able tofulfill contractual requirements, causing a breakdownin the supply chain. It is possible to find examplesof this phenomenon from various industries, includ-ing auto parts, e.g., Hakim (2003), Matthews (2011);food processing, e.g., Eckblad (2012); energy and util-ities, e.g., Thakkar (2013); and heavy manufacturing,e.g., Matthews (2011). Given the prevalence of thisissue, one might guess that the topic of risk manage-ment would command a great deal of attention fromresearchers in finance, and that practitioners wouldtherefore have a well-developed body of wisdom fromwhich to draw in formulating hedging strategies.

Such a guess would, however, be at best only par-tially correct. Finance theory does a good job of in-structing stand-alone firms on the implementation ofhedges. Finance theories, however, are generally silenton the implementation of hedges in situations where afirm operates in a supply chain and its operations areexposed to price fluctuations not only through directpurchases of commodities but also through its suppli-ers and the suppliers of its suppliers. This is preciselythe question that we study in this paper.

Specifically, we ask the following: (1) When shouldsupply chain firms hedge their stochastic input costs?(2) Should they hedge fully or partially? (3) Do theanswers depend on whether the firms’ supply chainpartners hedge? To answer these questions, we con-sider a simple supply chain model—the “selling-to-the-newsvendor” model (Lariviere and Porteus

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2001)—and generalize it by assuming that both theupstream and the downstream firms face stochasticproduction costs. The stochastic costs could representthe raw material costs. We show that the stochasticcosts reverberate through the supply chain and willbe, ex ante, impounded into the wholesale price. Insome cases, if input costs significantly increase, oneof the supply chain members may not be able tofulfill its contractual requirements, causing the entiresupply chain to break down. We identify conditionsunder which the risk of the supply chain breakdownand its impacts on the firms’ operations will causethe supply chain members to hedge their input costs:(i) the downstream buyer’s market power exceeds acritical threshold; or (ii) the upstream firm operateson a large margin, there is a high baseline demandfor downstream firm’s final product, and the down-stream firm’s market power is below a critical thresh-old. In absence of these conditions there are equilibriain which neither firm hedges. To sustain hedging inequilibrium, both firms must hedge and supply chainbreakdown must be costly. The equilibrium hedgingpolicy will (in general) be a partial hedging policy.There are also situations when firms hedge in equilib-rium although hedging reduces their expected payoff.

As an extension, one could consider the case whenfirms’ operations are financed with borrowing andshow that hedging can be profitable even in theabsence of breakdown risk. There, the equilibriumhedging policy would be a full hedging policy. Futureresearch may consider the role coordinating contracts,financing, and hedging play in the effective manage-ment of decentralized supply chains in the presenceof stochastic demands and stochastic input cost.

Supplemental MaterialSupplemental material to this paper is available at http://dx.doi.org/10.1287/msom.2014.0514.

AcknowledgmentsThe authors thank Brian Tomlin, Genaro Gutierrez, and theparticipants of the Interface of Finance, Operations, andRisk Management (iFORM) SIG 2014 Conference for com-ments. The authors also thank the editor, associate editor,and three anonymous reviewers of this journal for theircomments.

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Hedging Commodity Procurement in a Bilateral Supply Chain

Education

Transcript of Hedging Commodity Procurement in a Bilateral Supply Chain