Simple Contracts to Assure Supply Under Noncontractible

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Simple Contracts to Assure Supply Under NoncontractibleCapacity and Asymmetric Cost InformationEhsan Bolandifar, Tianjun Feng, Fuqiang Zhang

To cite this article:Ehsan Bolandifar, Tianjun Feng, Fuqiang Zhang (2017) Simple Contracts to Assure Supply Under Noncontractible Capacity andAsymmetric Cost Information. Manufacturing & Service Operations Management

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MANUFACTURING & SERVICE OPERATIONS MANAGEMENTArticles in Advance, pp. 1–15

http://pubsonline.informs.org/journal/msom/ ISSN 1523-4614 (print), ISSN 1526-5498 (online)

Simple Contracts to Assure Supply Under NoncontractibleCapacity and Asymmetric Cost InformationEhsan Bolandifar,a Tianjun Feng,b Fuqiang Zhangc

aCUHK Business School, Chinese University of Hong Kong, Shatin, N.T., Hong Kong; b School of Management, Fudan University, Shanghai,China 200433; cOlin Business School, Washington University in St. Louis, St. Louis, Missouri 63130Contact: [email protected] (EB); [email protected] (TF); [email protected] (FZ)

Received: October 28, 2015Revised: July 18, 2016; October 30, 2016Accepted: November 22, 2016Published Online in Articles in Advance:August 11, 2017


Copyright: © 2017 INFORMS

Abstract. This paper studies a global sourcing problem where a buyer sources a prod-uct from a supplier to satisfy uncertain market demand. With the increasing length andcomplexity of today’s global supply chains, the buyer may face two issues when design-ing the sourcing contract: adverse selection (i.e., the supplier’s cost structure is privateinformation) and noncontractible capacity (i.e., the supplier’s capacity investment is notcontractible). We show that noncontractible capacity does not necessarily lead to a lowerprofit for the buyer, but it may require a more complex contract format to achieve theoptimal (second-best) profit. Interestingly, we find that a single, linear contract (or a two-part tariff) could be optimal for the buyer under certain conditions. Even when such acontract is suboptimal, it can deliver close-to-optimal profit for the buyer for a wide rangeof situations. These findings indicate that the value of using a complex menu of contractsis negligible in such a supply chain setting. A simple two-part tariff is an attractive optionfor buyers whose goal is to ensure supply while facing both cost uncertainty and con-tractibility issues. The paper also provides a new explanation for the prevalence of suchsimple contracts in practice.

Funding: This research was supported by the National Natural Science Foundation of China [Grants71371057, 71628203, 71232008, and 71520107002].

Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2017.0628.

Keywords: supply contracts • sourcing • information asymmetry • noncontractibility • linear contracts • two-part tariff

1. IntroductionIncreasing competition in the global market has forcedmany firms to outsource their production processesto focus on core competencies such as product designand marketing. While this practice may help firmsreduce costs and improve competitive advantage, italso gives rise to new challenges on how to managetheir supply chains. Consider a buyer who sources aninput (e.g., a product or a component) from a sup-plier. The buyer is unsure about market demand and,due to procurement time lags, the supplier needs tomake capacity-related decisions before the demand isrealized. The buyer wants the supplier to be flexibleenough to deal with any demand fluctuations, whereasthe supplier needs to keep the risk of overages to aminimum in case the demand for his output ceases.How to achieve sufficient supply to satisfy uncertaindemand is a central issue for such supply chain set-tings. Failing to do so may cause substantial losses inprofits andmarket shares for both individual firms andthe supply chain. Anecdotal evidence abounds in bothbusiness media and academic literature. For example,Sony Ericsson’s third-quarter sales fell short of ana-lysts’ expectations in 2010 because of component short-ages, including printed circuit boards (Parker 2010).

In 2012 Qualcomm executives reported that a capacityshortage at an overseas foundry supplier might pre-vent the company frommeeting strongmarket demand(McGrath 2012). More recently, Tesla Motors revealedthat severe Model X supplier parts shortages mightdelay delivery of the new electric cars to customers(Stevens 2016). In view of these lessons, practitionerssurely understand the importance of securing supplyand have taken this into consideration when design-ing sourcing strategies. Sun Microsystems claims thatavailability and flexible supply will be critical whenmaking their sourcing decisions (Carbone 2005). Sim-ilarly, Apple tends to push suppliers to expand theirmanufacturing facility to meet the rush demand for itsnew products (Dou and Luk 2014).

Two practical issues may arise in a buyer’s sup-ply contracting problem. First, firms in a decentral-ized supply chain are independent organizations andthus often have disparate information. In particular,the supplier’s cost structure is highly confidential infor-mation because it conveys significant bargaining powerwhen negotiating a contract. Thus, when devising asourcing strategy, the buyer often does not have per-fect information about the supplier’s cost structure,which gives rise to the so-called adverse selection issue.

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Bolandifar, Feng, and Zhang: Simple Contracts to Assure Supply2 Manufacturing & Service Operations Management, Articles in Advance, pp. 1–15, ©2017 INFORMS

Second, the suppliers’ actions may not be verifiableor enforceable; see, for example, the discussions inCachon and Lariviere (2001) and Kaya and Özer (2009)on the supplier’s compliance on capacity and qualityagreements, respectively. In many supply chain rela-tionships, the buyer’s desired capacity level may notbe optimal for the supplier who builds the capacity. Insuch cases, the contracted capacity may not be enforce-able for a couple of reasons: First, capacity is a complexdecision that involves many factors, including the sup-plier’s managerial effort (thus, when a supplier fails todeliver the promised quantity, it might be hard to ver-ify whether this is due to underinvestment in effort orreasons beyond the supplier’s control). Second, if thecost of enforcement (e.g., the cost of capacity verifica-tion and the cost of a lawsuit) is prohibitively high, con-tract terms that penalize a dishonest suppliermight notbe credible. Due to lack of enforcement, the suppliermay purposely deviate from the agreed capacity, mak-ing capacity investment a noncontractible decision. Wemay call this the capacity contractibility (or moral haz-ard) issue.Empirical findings suggest that both information

asymmetry and moral hazard are important determi-nants of contract design in the buyer-supplier rela-tionship (Costello 2013). The trend of globalizationhas made these issues more prominent because ofincreased length and complexity of supply chains.Research has shown that long distance between par-ties may increase information asymmetry (Noordewieret al. 1990 and Hortacsu et al. 2009). Thus, it is notuncommon that in global supply chains, buyers do nothave precise information about the cost structure oftheir suppliers located overseas. Moreover, it is knownthat regulatory institutions and law enforcement areweak in emerging economies, which may give rise tocontractibility issues due to higher contract enforce-ment costs (Antras 2015). The World Bank (2008) pro-vides a comparison of the judicial system’s efficiencyin resolving commercial disputes in different countriesor regions. The report shows that developing coun-tries are often associated with less efficient judicialsystems (e.g., longer wait time, higher litigation fees,and more complex procedures), which present a majorobstacle for these countries to attract foreign businessesand investments (see also Ahlquist and Prakash 2010).According to a Deloitte survey, 70% of executives wereextremely or very concerned about suppliers’ com-pliance risks in emerging markets (Sinha 2014). Forinstance, in global supply chains of products such asapparel, reneging on a contract could be a significantconcern (Narayanan and Raman 2000). According toDickinson (2016), when outsourcing to Chinese sup-pliers, firms need to be cautious about potential shortdeliveries on agreed-upon purchase orders. It is notsurprising that many global supply chains are prone to

the above problems because firms are located in differ-ent countries and contract agreements are written longbefore final delivery.

In this paper, we study how firms can deal withinformation asymmetry and contractibility issues thatmay both be present in supply contracting. These chal-lenges, made increasingly prominent by the trend ofglobalization, may have significant implications forfirms’ contracting strategies. The purpose of this paperis to understand how firms should design sourcingcontracts to assure supply in the presence of these chal-lenges. We propose a stylized model where a newsven-dor buyer sources a product from a supplier. The buyerdoes not know the exact capacity cost of the supplierbut has an unbiased belief about its distribution. Sincecapacity investment is not contractible, we assume thatboth parties need to contract based on the actual deliv-ery quantity rather than the supplier’s capacity. Thisgives rise to the first question we investigate: How doesthis constraint due to lack of contractibility affect thecontract efficiency? While most of the contracting liter-ature is dedicated to designing the mechanism to max-imize the principal’s profit, there is a stark discrepancybetween the complexity of the theoretically optimalcontracts and the simplicity of many real-world con-tracts (Caillaud et al. 1992). As a result, many studieshave tried to reconcile this discrepancy by studying theperformance of suboptimal but simpler contracts. Fol-lowing this literature, we investigate the second ques-tion: Are there simple contracts that perform well forour sourcing problem?

The main contribution of the paper is twofold. First,we find that with properly designed incentives, thelack of contractibility does not result in efficiency lossfor the buyer. We show that contracting on the deliveryquantity can induce the optimal (second-best) profit.Further, the existence of an optimal menu of linear con-tracts depends critically on the cost uncertainty facedby the buyer. For ease of exposition, later we will intro-duce the modified reverse hazard rate (MRHR) for thesupplier’s cost distribution. When there is a decreasingMRHR, the buyer can achieve the second-best solu-tion by offering a menu of linear contracts. Each con-tract in the menu is essentially a two-part tariff, wherethe buyer pays the supplier a wholesale price for eachproduct delivered and charges the supplier a lump-sumpayment. However, if theMRHR is strictly increas-ing for any cost range, then the second-best solutioncan no longer be implemented using linear contracts;instead, the buyer can use a menu of quadratic con-tracts to achieve the second-best solution.

Second, we show that the optimal menu of contractsreduces to a simple two-part tariff when there is a con-stant MRHR, i.e., a single, linear contract could be opti-mal (second-best) for the buyer. This condition holds,for example, when the supplier’s cost follows a uniform

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Bolandifar, Feng, and Zhang: Simple Contracts to Assure SupplyManufacturing & Service Operations Management, Articles in Advance, pp. 1–15, ©2017 INFORMS 3

distribution, which corresponds to situations wherethe buyer has very limited cost information. When it isoptimal for the buyer to exclude some high cost types,the two-part tariff will further simplify to a wholesaleprice. This is essentially the pull contract that has beenwidely observed in practice, where the supplier under-takes the capacity risk and charges the buyer a whole-sale price for each unit delivered. In addition, it hasbeen shown that even when the simple two-part tar-iff is not optimal, it still yields near-optimal profit in awide range of situations. This implies that the value ofusing a complex menu is negligible in our problem set-ting. From the supply chain’s viewpoint, the two-parttariff is also preferred over the optimal menu of con-tracts because it generally yields higher supply chainefficiency. We provide an explanation for why a sim-ple contract performs well from both the firm’s and thesupply chain’s perspectives.In summary, we show that in a sourcing problem

where the buyer is exposed to demand risk, noncon-tractible capacity, and asymmetric capacity cost infor-mation, a menu of linear or quadratic contracts canachieve the second-best solution for the buyer. How-ever, the value of using a complex menu is insignif-icant, which suggests that a more effective way toimprove buyer profit is to reduce the informationasymmetry rather than designing sophisticated con-tracts. However, if reducing the uncertainty in suppliercost is not a viable strategy, then we recommend thetwo-part tariff to managers—it has a simple format,yields nearly optimal profit, and does not requireenforcement on the supplier’s capacity investment.This finding corroborates the prevalence of such sim-ple contracts in practice.

The rest of the paper is organized as follows. Sec-tion 2 reviews the literature, and Section 3 introducesthe model. Section 4 first presents a benchmark sce-nario with contractible capacity and then studies thescenario with noncontractible capacity. Section 5 stud-ies simple contracts and their performances. This paperconcludes with Section 6. All proofs are given in theonline appendix.

2. Literature ReviewThere is a growing body of literature that studies sup-ply chain management under asymmetric cost infor-mation. A detailed review of earlier studies can befound in Chen (2007). Among these studies, the closestwork to our study is Ha (2001). Ha (2001) considers asimilar supply chain setting except that the retailer isa price-setting newsvendor with private cost informa-tion. It has been shown that a menu of contracts with acutoff level is optimal for the supplier, and the optimalorder quantity is lower under asymmetric informationthan under complete information. Other representativestudies that involve retailer private cost information

include Corbett (2001), Corbett et al. (2004), and Lutzeand Özer (2008). In our model, a newsvendor procuresfrom a supplier whose capacity cost is private informa-tion, which is different from these existing studies.

This paper is closely related to the literature oncapacity procurement in supply chains. Cachon andLariviere (2001) are among the first to study supplycontracting under asymmetric demand information.In particular, the manufacturer has superior demandforecast information and needs to secure sufficientcapacity at the supplier. They coin the terms “forcedcompliance” and “voluntary compliance” to distin-guish between different capacity enforcement scenar-ios. Voluntary compliance essentially means that thesupplier’s capacity is not contractible as in our model.Tomlin (2003) and Özer andWei (2006) are two notablefollow-up studies along this important line of research.By allowing an intermediate compliance scheme (i.e.,the buyer has some ability to identify supplier shirk-ing), Tomlin (2003) finds that in contrast to Cachonand Lariviere (2001), firm commitment and optionscontracts can be useful in increasing supplier capac-ity under full information (symmetric demand infor-mation). In addition, he shows that under certainconditions, the manufacturer’s optimal contract is aquantity-premium price-only schedule, under whichthe two parties “share the gain” rather than “sharethe pain.” Özer and Wei (2006) propose two contracts(a capacity reservation contract and an advance pur-chase contract) to induce credible forecast informationsharing. They also discuss how the supplier’s compli-ance scheme affects the outsourcing firm’s choice ofcontract. Our paper contributes to this line of researchby incorporating private supplier cost informationand studying the interaction between asymmetric costinformation and moral hazard.

Recently, the adverse selection and moral hazard is-sues in outsourcing have received some research atten-tion. For example, Kaya andÖzer (2009) study the qual-ity risk in an outsourcing setting, where input qualityis noncontractible and the supplier has private infor-mation about the cost of quality. Another feature is thatthe outsourcing firm may or may not be able to com-mit up-front to a price in the end market. They showthat the effect of noncontractible quality could be moresignificant than that of private cost information undercertain conditions. They also find that price commit-ment mitigates the adverse effect of quality risk. Kimand Netessine (2013) examine supply chain parties’incentives to collaborate on cost reduction under infor-mation asymmetry and moral hazard. Among a num-ber of contracting strategies considered, they show thata strategy called expected margin commitment caneffectively promote supply chain collaboration on costreduction. Our paper differs from these studies in both

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model setting and main results. First, we study a sup-ply chain setting in which the supplier makes a capac-ity investment decision (to improve product availabil-ity under demand uncertainty), rather than an effortdecision (to improve cost or quality). Second, our paperdemonstrates that a single, linear contract could be opti-mal for the buyer under certain conditions; even whenit is suboptimal, the contract can yield close-to-optimalperformance.A stream of papers in the economics literature study

principal-agent models with noncontractible agentactions. Laffont and Tirole (1986) consider an agentwho is endowed with private cost information and canexert cost-reduction effort. Assuming the cost is uni-formly distributed, they show that a menu of linearcontracts based on the observed output is optimal forthe principal. McAfee and McMillan (1986) extend thisline of research by studying a general distribution forthe agent’s type. While McAfee and McMillan (1986)claim that in a broad set of circumstances, optimal con-tracts can be linear in observed output levels, Rogerson(1987) shows that this result is not true in general.Therefore, it seems that the optimality of linear con-tracts can be viewed as a very special property only forlimited settings. Whereas linear contracts are of greatinterest in the economics literature (see Bose et al. 2011and references therein), much less attention has beenpaid to this issue in the operations management liter-ature. Chen (2005) is among the first to study the per-formance of linear contracts in an operations setting.He examines the so-called Gonik’s scheme in a sales-force incentive mechanism design problem: the sales-people are endowed with private information aboutthe market and can exert effort to sell in the market.He characterizes the optimal menu of linear contractsand finds that it dominates the Gonik’s scheme usingnumerical experiments. In another similar work, Chenet al. (2016) compare the forecast-based contracts withamenu of linear contracts to show that with an endoge-nous information-acquisition effort, the menu of linearcontracts may be dominated by the forecast-based con-tracts. In this paper, we consider a capacity procure-ment setting and characterize the conditions where amenu of linear contracts arises as an optimal mecha-nism for the buyer.

Finally, the importance of simplicity in procurementcontract design has long been recognized in the lit-erature. Hart and Holmstrom (1987) point out thatthere is a contrast between the complexity of theoreti-cally optimal contracts and the simplicity of many real-world arrangements. A number of economics studieshave been devoted to explaining such a discrepancybetween theory and practice. Please refer to Chu andSappington (2015) for an extensive overview of thisliterature in economics. Recently, there have been a

growing number of studies in operations manage-ment that investigate the performance of simple pro-curement mechanisms, including Cachon and Zhang(2006), Kayış et al. (2013), Zhang (2010), and Duenyaset al. (2013). Our paper is in a similar spirit by searchingfor simple procurement mechanisms in supply chainsettings. However, both the model and insights in ourpaper are quite different from these existing studies.

3. Model DescriptionWe consider a two-stage supply chain in which a buyertrades with a supplier. Hereafter we use “she” to referto the buyer and “he” the supplier. The buyer can beviewed as a newsvendor firm. She sources a perish-able product from the supplier and then sells it to cus-tomers in a single selling season. There is an exoge-nously given market price p, and market demand isuncertain. Let D denote the random demand with dis-tribution function Φ and density function φ. For easeof exposition, we assume that the salvage value for anyexcess capacity is zero. Also, there is no penalty cost forunsatisfied demand during the selling season. Includ-ing a positive salvage value or penalty cost will notchange the qualitative results from our model.

Due to long acquisition lead times, the supplierneeds to make capacity investments before the sell-ing season starts. The capacity may refer to all the re-sources (e.g., raw materials, production facility, andlabor) that are needed to produce the product for thebuyer. We assume the capacity can be instantly con-verted into the product at zero cost. Alternatively, wemay assume the supplier builds up inventory to satisfythe buyer’s order, which is quite common in the sup-ply contracting literature. Thus, the concepts of capac-ity and inventory are interchangeable in this paper.There is a unit capacity cost c, which is the supplier’sprivate information. Due to various uncertain factors(e.g., technology, production yield, and raw materialprice), the buyer does not know the exact cost c, butshe has an unbiased belief about its distribution. With-out loss of generality, let [0, c] (c > 0) be the support ofc, and let F and f denote its distribution and densityfunctions, respectively. We assume F has a decreasingreversed hazard rate (RHR), i.e., f (x)/F(x) is decreasingin x. This is equivalent to the log-concavity assump-tion widely adopted in mechanism design problems.We use the decreasing RHR assumption rather than thelog-concavity assumption because we will introduce amodified reversed hazard rate (MRHR) later in this paper.Most commonly used distributions have a decreas-ing RHR, including uniform, normal, exponential, andlogistic distributions (see Bagnoli and Bergstrom 2005for details).

As discussed in the introduction, firms’ actions arenot always contractible, which may happen when the

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firms’ actions are not verifiable or enforcing the con-tract terms is prohibitively costly. As a result, therecould be noncontractibility issues in the supply chain(e.g., the global supply chain for apparel productsdescribed in Narayanan and Raman 2000). This canbe seen as a moral hazard problem since after sign-ing a contract the supplier might have an incentive todistort his effort (i.e., his capacity investment) whencontracting terms are not enforceable. In this case, itis no longer viable for firms to devise a contract asa function of capacity investment from the supplier.For example, the buyer cannot specify a target capacityinvestment for the supplier and then penalize the sup-plier if he fails to meet the target. This is because thesupplier can always claim that the shortage is not dueto underinvestment in capacity but for reasons beyondhis control. Noncontractibility of firms’ actions resultsin contracting on output level. In this paper, we studyhow firms can optimally contract on the output level todeal with any noncontractibility issue.We assume the buyer takes the initiative to offer a

take-it-or-leave-it contract to the supplier. Both supplychain firms are risk neutral, and the objective is to max-imize their own expected profit. In general, the sup-plier may have a reservation profit, which is definedas the value of the best outside option. A higher reser-vation profit corresponds to a more advantageous bar-gaining position for the supplier. We normalize thesupplier’s reservation profit to zero because adding apositive reservation profit does not generate much newinsight. All parameters are common knowledge exceptthe supplier’s capacity cost, c. The sequence of eventsis as follows: (1) the buyer offers a menu of contracts tothe supplier; (2) the supplier chooses a contract fromthe menu and then invests in capacity accordingly;(3) market demand is confirmed and the buyer ordersfrom the supplier to satisfy demand. For simplicity, weassume therewill be no renegotiation after the supplierreveals his cost information.Define Φ � 1−Φ. Let E denote the expectation oper-

ation and K be the supplier’s capacity level. Thenthe supply chain’s expected profit can be written asπsc(K) � pED[min(K,D)] − cK (we use subscript sc forsupply chain), and the optimal capacity thatmaximizesπsc(K) is given by the critical fractile solution K∗(c) �Φ−1((p− c)/p) (we use superscript ∗ for the supply chainoptimal solution). It is clear that if the buyer knows thesupplier’s cost, then she will choose the supply chainoptimal capacity and extract all the surplus. This is theso-called first-best solution in the literature. Next, westudy the buyer’s contract design problem when she isnot perfectly informed about the supplier’s cost; in thiscase, the buyer’s optimal contract is commonly referredto as the second-best solution. Given the existence ofinformation asymmetry, we will use the second-best

solution as the benchmark to evaluate other subopti-mal contracts. Thus, in the rest of the paper, by “opti-mal” we refer to the “second-best” outcome. For exam-ple, the buyer’s optimal profit is defined as her profitin the second-best solution.

4. Analysis of Optimal ContractsThis section presents the optimal contracts that max-imize the buyer’s profit when capacity is noncon-tractible. As a reference point, first we characterizesecond-best solution when capacity is contractible.Solving the buyer’s optimal contract is essentially anoptimal mechanism design problem. According to therevelation principle (see Bolton and Dewatripont 2005,Laffont and Mattrimort 2002), we can focus on thedirect and truth-inducing mechanisms in this section.

4.1. Adverse Selection with Contractible CapacityFirst we study the buyer’s contract design problemwhen supplier’s capacity is contractible, i.e., the sup-plier will faithfully invest in contracted capacity sothere is no moral hazard. Due to the presence of infor-mation asymmetry, the buyer needs to offer a menuof contracts to the supplier. By choosing a contractfrom the menu, the supplier essentially announces hiscapacity cost. Consider the followingmenu of contracts{K(x),T(x)}: If the supplier announces his capacitycost to be x (by choosing the corresponding contract),then the supplier will build a total capacity K(x) andreceive a lump-sumpayment T(x) from the buyer. Nextwe derive the optimal functions {Ko(x),To(x)} thatmaximize the buyer’s expected profit (we use super-script o for the second-best solution).

Suppose the supplier’s true cost is c but he an-nounces x; then the supplier’s profit function is (we usesubscript s for supplier) as follows:

πs(x; c)� T(x) − cK(x). (1)

In the profit function, x is a decision variable while cis a parameter. The buyer’s profit can be written as (weuse subscript b for buyer) follows:

πb(x)� pED[min(K(x),D)] −T(x). (2)

There are two constraints imposed on the mechanismdesign problem. The first constraint is the incentive-compatibility constraint (IC), which guarantees sup-plier’s truth-telling:

c � arg maxx{T(x) − cK(x)} for all c ∈ [0, c]. (3)

The second is the individual rationality constraint (IR),which guarantees nonnegative profit for the supplier:

T(c) − cK(c) ≥ 0 for all c ∈ [0, c]. (4)

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The right-hand side of the inequality is zero becausewe have assumed that the supplier has a zero reserva-tion profit. Now the buyer’s optimal mechanism can besolved from the following optimization problem:

max{K( · ),T( · )}

∫ c

0(pED[min(K(x),D)] −T(x)) f (x) dx

s.t. (3), (4),(5)

where πb(x) is given in (2). Let co be the solution to

p − (x + F(x)/ f (x))� 0 (6)

if a solution exists within [0, c]; otherwise set co � c.Then we have the following proposition.

Proposition 1. If capacity is contractible, the followingmenu of contracts {Ko(x),To(x)} is optimal for the buyer:For x > co , there is Ko(x) � 0 and To(x) � 0, i.e., the buyerprocures nothing from the supplier; otherwise there is

Ko(x)�Φ−1(

p − (x + F(x)/ f (x))p

), (7)

To(x)�xKo(x)+∫ c

xKo(u) du. (8)

The buyer’s optimal mechanism in Proposition 1represents a threshold policy. When co < c, the buyerwould refuse to do businesswith the supplier if his costis greater than the threshold value, co . Note that co onlydepends on the price p and the distribution function F.Similar threshold policies appear elsewhere in the lit-erature on contracting under asymmetric information;examples include Ha (2001), Corbett et al. (2004), andLutze and Özer (2008). Due to the decreasing reversedhazard rate property of F, an immediate result is thatKo(x) is decreasing in x, i.e., a higher supplier cost leadsto a lower capacity in the optimal mechanism. It is clearthat Ko(x) < K∗(x) for F(x)/ f (x) > 0, i.e., the optimalmechanism does not coordinate the supply chain. Thisdistortion is caused by the buyer’s intention to max-imize her own profit under asymmetric information.The lump-sum payment To(x) in the optimal mecha-nism consists of two parts: the first part (xKo(x)) is tocompensate the supplier’s capacity investment, whilethe second (

∫ c

xKo(u) du) is the supplier’s information

rent (or expected profit).

4.2. Adverse Selection with NoncontractibleCapacity

We proceed to study the buyer’s contracting prob-lem when the supplier’s capacity is noncontractible,i.e., there is moral hazard in capacity investment. Inparticular, the contract cannot force the supplier tobuild sufficient capacity, nor can it penalize the sup-plier for deviating from a target capacity level. In suchsituations, the buyer may offer a contract based on

the supplier’s delivered quantity rather than capacitylevel (see Pierce 2012). Consider a menu of payments{T(x , z)} the buyer offers to the supplier, where x is theannounced capacity cost (or, equivalently, the menuselected by the supplier), z � min(K,D) is the actualquantity delivered by the supplier, and T is a generalfunction of x and z. Under this contract, the supplierchooses the cost-to-announce x and capacity decisionK to maximize his profit function:

πs(x ,K; c)� ED[T(x ,min(K,D))] − cK.

The first-order condition for the supplier’s optimalcapacity choice given cost c is

∂πs(x ,K; c)∂K

� (1−Φ(K))∂T(x ,K)∂K

− c � 0.

Under truth-telling, the above first-order conditionbecomes

(1−Φ(K))∂T(x ,K)∂K

− x � 0 for all x. (9)

The buyer’s profit function given supplier cost x canbe written as

πb(x)� pED(min(K,D)) −ED[T(x ,min(K,D))]. (10)

The IC and IR constraints are the same as in (3) and (4).Then the buyer’s optimal mechanism design problemis as follows:

max{T(·,·)}

∫ c

0(pED(min(K,D))

−ED[T(x ,min(K,D))]) f (x) dxs.t. arg max

xπs(x ,K(x , c))� c ,

πs(c ,K(c , c)) ≥ 0,

(11)

for all c ∈ [0, c], where K(x , c) satisfies (1 − Φ(K)) ·(∂T(x ,K)/∂K) − c � 0.

The analysis of the optimal mechanism when firms’actions are not contractible is more involved becausethe payment schedule T(x , z) is a function of bothx and z. Before presenting the optimal solution, wedefine the modified reversed hazard rate (MRHR) to be

H(x)�x f (x)F(x) . (12)

We call it MRHR because it is defined as the RHRmul-tiplied by x. It has been assumed there is a decreas-ing RHR (i.e., f (x)/F(x) is decreasing in x); however,this does not guarantee the MRHR is also decreas-ing. In fact, the MRHR may be either increasing ordecreasing in x. Later wewill see that themonotonicityproperty of the MRHR plays an important role in theoptimal mechanism. An interesting comparison canbe made with the increasing failure rate (FR) and the

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increasing generalized failure rate (GFR) properties stud-ied in Lariviere and Porteus (2001) and Lariviere (2006).There are two major differences here: First, here wefocus on the reversed hazard rate (rather than the haz-ard rate or failure rate); second, MRHR is not a gener-alized version of RHR because a decreasing RHR doesnot necessarily imply a decreasingMRHR. As prepara-tion, the following lemma characterizes the propertiesof the buyer’s optimal mechanism.

Lemma 1. Under noncontractible capacity, the followingconditions must hold in a truth-telling optimal mech-anism:(i) The payment schedule T(x , z) satisfies

∂T2(x ,K)/∂x∂K(1−Φ(K))(∂2T(x ,K)/∂K2) − (∂T(x ,K)/∂K)φ(K) ≥ 0.

(13)(ii) It induces the capacity level given in (7), i.e., Ko(x)�

Φ−1((p − (x + F(x)/ f (x)))/p).The inequality in Lemma 1(i) is the second-order

condition for truth-telling, or incentive compatibility(see the proof of Lemma 1). This condition will laterhelp explain how H(x) affects the format of the opti-mal mechanism. Lemma 1(ii) states that the (induced)optimal capacity when it is noncontractible should beidentical to that of contractible capacity. This is intu-itive because Ko(x) represents the most efficient capac-ity investment under adverse selection according toProposition 1. Given this condition, we only need tofocus on the payment schedule that can induce thecapacity Ko(x). Next we present the buyer’s optimalmechanism under decreasing MRHR and increasingMRHR, respectively. We first consider the case whereF has a decreasing MRHR. The buyer’s optimal mech-anism is given by the following proposition. AlthoughT(x , z) can take any general functional format, theproposition shows that a menu of linear contracts canachieve the second-best solution for the buyer.

Proposition 2. Suppose F has a decreasing MRHR(H′(x) ≤ 0) for x ∈ [0, c]. The following menu of contracts{To(x , z) � wo(x)z + To(x)} is optimal for the buyer: forx > co , there is To(x , z)� 0, i.e., the buyer procures nothingfrom the supplier; for x ≤ co , there is

wo(x)�pH(x)

1+H(x) , (14)

To(x)�xKo(x) −wo(x)ED[min(Ko(x),D)]

+

∫ c

xKo(u) du , (15)

where Ko(x)�Φ−1((p−(x + F(x)/ f (x)))/p) is the capacityinduced by To(x , z).An important observation from Proposition 2 is that

when H′(x) ≤ 0, the buyer’s second-best solution can

be implemented using a menu of linear contracts. Thatis, for each contract in the menu, the buyer offers awholesale price wo(x) to induce the desired capacityKo(x), and then uses a side payment To(x) to splitthe profit between the two parties. This format repre-sents the commonly observed two-part tariff arrange-ment; see Vettas (2011) for practical examples of sucha contract format in retail, wholesale, and technologylicensing markets. This contract has a linear format, sowe will use linear contract and two-part tariff inter-changeably in the rest of the paper. The two-part tar-iff contractual format has been widely studied in theliterature. It is well known that such a contract cancoordinate a decentralized distribution channel (see,e.g., Moorthy 1987). Although our optimal contract{To(x , z) � wo(x)z + To(x)} has the same format asthe traditional two-part tariff contract, its underlyingrationale is different. In a coordinating two-part tar-iff contract, the unit price must be equal to the sup-plier’s marginal cost to avoid double marginalization.However, here wo must be greater than the supplier’smarginal cost in our setting due to the distortion in theoptimal capacity (i.e., F(x)/ f (x)), which guarantees thesupplier’s truth-telling.

We continue to study the buyer’s optimal mecha-nism when the MRHR can be strictly increasing, i.e.,H′(x) > 0, for some x.

Proposition 3. Suppose F has a strictly increasing MRHR(H′(x)> 0) for some x in [0, c]. Then, if capacity is noncon-tractible, the second-best solution cannot be achieved usinga menu of linear contracts. However, there is a menu ofquadratic contracts {To(x , z)�wo

1 z2+wo2(x)z+ To(x)} that

can achieve the second-best solution for the buyer.

Proposition 3 indicates that under strictly increas-ing MRHR, the buyer can still achieve the second-bestprofit under noncontractible capacity; however, a morecomplicated contract format (e.g., a quadratic paymentschedule) is required for implementation. The expla-nation of this result is as follows. Suppose there is alinear payment function T(x , z) � w(x)z + T(x). Thenthe condition in Lemma 1(i) becomes

w′(x)−w(x)φ(K) ≥ 0,

or simply w′(x) ≤ 0. That is, the second-order condi-tion for truth-telling reduces to w′(x) ≤ 0. Under such alinear payment scheme, the supplier faces a newsven-dor problem and his optimal capacity satisfies Φ(K) �(w(x) − x)/w(x). Together with Lemma 1(ii), we get

w(x)�px

x + F(x)/ f (x) �pH(x)

1+H(x) .

Clearly, w′(x) ≤ 0 if and only if H′(x) ≤ 0. To betterunderstand the decreasing MRHR condition, we may

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compare the capacity functions K∗(x)�Φ−1((p−x)/p)and Ko(x)�Φ−1((p−(x+F(x)/ f (x)))/p) under the first-best and second-best solutions, respectively. The termF(x)/ f (x)denotes the capacity distortion in the second-best solution (optimal mechanism) to induce truth-telling from the supplier. Under contractible capac-ity, a decreasing RHR (i.e., f (x)/F(x)) is sufficient forthe existence of a separating equilibrium, as shown byProposition 1. In contrast, under noncontractible capac-ity, a decreasing MRHR (i.e., x f (x)/F(x)) is needed toguarantee a separating equilibrium under linear con-tracts. This is a stronger condition because a decreasingx f (x)/F(x) requires f (x)/F(x) not only to be decreas-ing, but also to be decreasing fast enough. In fact, itimplies that the distortion in capacity has to be sig-nificant enough (i.e., F(x)/ f (x) increases fast enough).Note that a large enough capacity distortion helps dis-courage the supplier from misreporting his true cost.Thus, for the menu of wholesale prices to induce theright capacity and satisfy the truth-telling condition atthe same time, we need a decreasing MRHR. Given theimportance of the property of MRHR, we have exam-ined many commonly used distribution functions. Itcan be shown that exponential, gamma, beta, chi, andchi-squared distributions have decreasing MRHR, andfor normal, logistic, and extreme value distributions,the condition H′(x)≤0 is violated for at most one inter-val on the support; thedetails are relegated to the onlineappendix.Propositions 2 and 3 indicate that the noncontract-

ibility of capacity does not necessarily result in effi-ciency loss for the buyer. Similar findings have beenreported in different procurement settings (e.g., Laffontand Tirole 1986, Rogerson 1987, and Kaya and Özer2009). In particular, Kaya and Özer (2009) study anoutsourcing problem with noncontractible quality andasymmetric cost information. They find that when thebuyer can commit to a sales price before observingquality, noncontractibility does not reduce the out-sourcer’s profit. We demonstrate that the similar resultholds in an outsourcing setting with noncontractiblecapacity.

Although the buyer can still achieve the second-bestprofit, the quadratic payment schedule is complex andless intuitive. The simplicity of the linear format hasbeen lauded in the incentive literature, andmuch atten-tion has been devoted to characterizing environmentswhere optimal contracts can be implemented throughlinear contracts (see, e.g., Bose et al. 2011 and referencestherein). What will happen if the buyer is restricted toa linear contract format? We also investigate this ques-tion and find that under the more practical linear for-mat, there will be bunching in the optimal mechanismbecause the buyer may be unable to screen all the sup-plier types. The next proposition formalizes this result.It can be shown that H′(x) > 0 holds for at most one

interval for many commonly used distributions; there-fore, for illustration purpose we will focus on this casein the following proposition.

Proposition 4. Suppose F has a strictly increasing MRHR(H′(x) > 0) for a single interval in [0, c]. Then, the linearcontract {T(x , z) � w(x)z + T(x)} will lead to a bunch-ing interval [c1 , c2], within which the contract reduces toa constant payment {To(z) � wo z + To}. In particular,we can solve {c1 , c2 ,wo} from the following simultaneousequations:

wo�

pH(c1)1+H(c1)

, (16)

wo�

pH(c2)1+H(c2)

, (17)

0 �

∫ c2

c1

(x

φ(K(x))(wo)2

)· [(pΦ(K(x)) − x) f (x) − F(x)] dx , (18)

where K(x) � Φ−1((wo − x)/wo). If bunching starts at thelower bound of the support, then we replace (16) by c1 � 0;if bunching ends at the upper bound of the support, thenwe replace (17) by c2 � c. The transfer payment To on thebunching interval is given by

To� c2K(c2) −woED[min(K(c2),D)]+

∫ c

c2

K(u) du. (19)

The optimal menu {To(x , z) � wo(x)z + To(x)} for x <[c1 , c2] is the same as in Proposition 2.

When H′(x) > 0, the linear payment schedule willviolate the incentive compatibility constraint. Thus, thebuyer needs to offer constant contract terms to cer-tain supplier types; the inability to screen the suppliertypes (i.e., pooling) leads to a lower profit for the buyercompared to the case of full separating. This bringsabout another question: What is the trade-off betweensimplicity and lost profit under simple contracts withstrictly increasing MRHR? In Section 5 we will shedsome light on this question by examining performanceof simple contracts in general settings.

5. Simple ContractsThe optimal (second-best) contracts characterized inSection 4 are still complicated and thus may not bestraightforward to implement in practice: First, theyinvolve contract menus that are difficult to derive andexplain to managers; second, they require the supplierto announce its cost, which imposes communicationand evaluation burdens on the supply chain firms. Sowhat is the value of using a complex menu in supplycontracting? To answer this question, we study a singletwo-part tariff {wo ,To} in this section, i.e., the buyerpays a wholesale price wo and a lump-sum payment To

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to the supplier. This contract is simpler than the opti-mal contracts in Section 4 because it only involves asingle, linear payment function. Next, we derive theconditions for such a simple contract to be optimal;then we examine the performance of this contract inmore general settings (i.e., even when it is suboptimal).

5.1. When Simple Contracts Are OptimalRecall from Proposition 2 that the buyer’s second-bestsolution can be implemented by a menu of two-parttariffs {wo(x),To(x)}, where wo(x) � pH(x)/(1+ H(x)).If H(x) is a constant (H′(x) � 0), then both wo(x) andTo(x) will be independent of x. This indicates that theoptimal contracts may reduce to a single two-part tar-iff under certain conditions. The following propositionconfirms this conjecture.

Proposition 5. If H′(x)� 0 for all x ∈ [0, c], then the opti-mal menu of contracts reduces to a single two-part tariff{wo ,To}. This happens if and only if the supplier’s cost hasa density f (x)� δxη−1, δ > 0, η > 0. Additionally, if co ≤ c,then wo � co and To � 0, i.e., the optimal menu of contractsreduces to a single wholesale price.

Proposition 5 states that a two-part tariff consistingof only two numbers {wo ,To} could be optimal for thebuyer when H′(x)� 0 for all x ∈ [0, c]. Further, if co ≤ c,then there must be To � 0 and the {wo ,To} contract fur-ther reduces to a single wholesale price wo � co thatcan be easily determined. Note co ≤ c means the buyerwishes to exclude some high cost types, which tendsto happen if the supplier has too wide a cost support(or c is relatively large compared to p). In this case,an exceedingly simple arrangement is optimal for thebuyer: The two parties agree upon a unit price, and thesupplier decides on how much capacity to invest. Thisis essentially a pull contractwidely observed in practice,under which the buyer pulls inventory from the sup-plier and meanwhile pays a wholesale price. Cachon(2004) discusses an example of a pull contract from thehigh-end bicycle industry. In the semiconductor equip-ment industry, buyers often share a demand forecastwith their suppliers. The demand forecast serves asan informal or soft order intended to guide the sup-plier’s production decisions. After observing the actualdemand, a buyer is able to cancel or renegotiate thesoft order (see Cohen et al. 2003 and Johnson 2003).Similarly, Kurt Salmon Associates Reports (1993) indi-cate that 80% of the wholesale transactions in the gro-cery industry are “forward buy” in nature, where buy-ers place tentative orders in anticipation of demandgrowth. They are prone to cancelling their orders if theactual demand falls short of what they expected. In allof these examples, wholesale price arrangements areused, and the buyer simply pulls inventory from thesupplier when demand arrives.

To help the buyer choose the optimal contract for-mat, we examine the distributions satisfying the con-dition in Proposition 5, i.e., f (x) � δxη−1 with δ > 0and η > 0. Such a condition may hold under eitherincreasing (η > 1), decreasing (η < 1), or constant (η � 1)density functions. In particular, f is a uniform dis-tribution when η � 1. A uniform distribution meansthat the buyer has no reason to believe that certaincost realizations are more likely to happen than theothers. Hence, Proposition 5 indicates that a two-parttariff or even a pull contract could achieve the opti-mal profit if the buyer has extremely limited informa-tion about the supplier’s capacity cost. Interestingly,the adoption of a uniform cost distribution is in linewith the Laplace principle on decision making underuncertainty. The principle states that in the absence of areason to do otherwise, a decision maker may assumeeach of the possible outcomes of the random event hasan equal likelihood of occurrence. This suggests thatif the buyer does not know the supplier well (whichcould be quite common, especially when the supplieris new or located far away), then the buyer may followsuch a principle to assume a uniform cost uncertaintydistribution, and our result indicates that the simplepull contract is optimal for the buyer.

The significance of Proposition 5 would be moreclear if we posit it within the existing literature on theperformance of simple contracts. It has been shownthat under conditions such as limited liability (Garrett2014) or minimax criterion (Chung and Ely 2007,Carroll 2015), a single contract could be optimal in cer-tain circumstances. We show that the optimality of asingle contract may hold in the classical supply chainunder study, without any assumption on the limitedliability or minimax criterion. Such a result has notbeen previously reported in the literature, which mayprovide a new explanation of the wide use of sim-ple contracts in supply chains (e.g., wholesale priceor pull contracts). A few papers have tried to obtainlower bounds on the performance of simple contracts.Rogerson (2003) studies performance of the fixed pricecost reimbursement (FPCR) contract in Laffont andTirole’s framework. Assuming that just one unit of aproduct is required, he shows that if the agent’s costfollows a uniform distribution, an optimally designedFPCR contract can secure at least 75% of the optimalcomplex contract. Chu and Sappington (2007) extendRogerson’s analysis to the case of power distributionwhere high cost agents are more likely. They show thatthe good performance of simple FPCR contract mayno longer hold. Instead they identify another type ofsimple contracts that can capture at least 73% of theoptimal profit. Later, Chu and Sappington (2015) con-sider amultiunit framework and show that in thismorecomplex setting, a simple FPCR contract still capturesat least 75% of the optimal attainable profit. Our paper

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offers a stronger result in our supply chain setting:for both uniform and power distributions, 100% of theoptimal performance can be secured through a singletwo-part tariff contract.

5.2. Performance of Simple ContractsWe proceed to investigate the performance of the{wo ,To} contract inmore general settings, evenwhen itis suboptimal. We first derive analytical bounds on theperformance for specific settings and then use a com-prehensive numerical study to confirm the robustnessof the findings. Specifically, consider a family of densityfunctions for capacity cost uncertainty: f (x)� α+ δxη−1,where α > 0, δ >−α/cη−1, η > 1, and x ∈ [0, c]. This func-tional format represents a reasonable approximation ofmany monotone density functions. To maintain analyt-ical transparency, we assume that the random demandis uniformly distributed over a support [β − θ, β + θ],where β is themean and θ/

√3 is the standard deviation

of demand. Let πOMb and πS

b denote the buyer’s profitsunder optimal mechanism and simple contract, respec-tively. Then the ratio (πOM

b − πSb )/πS

b (i.e., the percent-age profit gain by using the optimal mechanism) can beused tomeasure the value of using amenu of contracts,which, equivalently, also measures the performance ofthe simple contract. The following proposition charac-terizes the effect of different parameters on the value ofusing the optimal menu of contracts.

Proposition 6. Consider the density functions and uniformdemand distribution described above. The performance of thesimple contract for the buyer (i) improves in β, and (ii) dete-riorates in θ.

Proposition 6 indicates that a higher mean demand(measured by β) would improve the performance ofthe simple contract, while a larger demand variability(measured by θ) would deteriorate its performance.These results are also consistent with the observa-tions from later numerical studies (i.e., Tables 1–4).They are not surprising because the buyer’s expectedprofit increases in mean demand while it decreases indemand uncertainty.We have also tried to analyze howthe performance of the simple contract depends on thecapacity cost distribution. It turns out that the effect ofcapacity cost distribution is ambiguous: In the numer-ical studies, the performance may be either increas-ing or decreasing in the coefficient of variation of thecapacity cost distribution.While for the above density function we can solve for

the performance of the simple contract in closed-form,we still need to put more structure on this distributionto derive analytical bounds on the performance. Thefollowing proposition assumes the linearity of the costdensity.

Proposition 7. Suppose f (x) � α + δx on a support [0, 1]and Φ is uniform on [0, 1]. Assume co ≥ 1 and wo ≥ 1, i.e.,

no supplier type is excluded under both contracts. Then theratio (πOM

b − πSb )/πS

b has an upper bound 1/48 (≈2.08%)for 0 ≤ δ ≤ 2, and an upper bound 5/208 (≈2.40%) for−1≤ δ < 0. In addition, the ratio decreases in p in both cases.

Proposition 7 indicates that the simple contractyields close-to-optimal performance in this specialcase, since the value of using a menu of contracts isbounded by a small number (2.40%). In addition, theprofit difference tends to decrease in market price (orprofit margin). This proposition indicates that evenwhen the simple contract is suboptimal (the condi-tions in Proposition 5 do not hold), its performancemight be close to optimal. Note the bound (2.40%) ismuch tighter than the ones reported in the literature(e.g., Rogerson 2003; Chu and Sappington 2007, 2015),despite the challenging setting that involves both costand demand uncertainties.

An analytical comparison between the simple andoptimal contracts is challenging for general distribu-tions (see the analysis in the online appendix). There-fore, it is instructive to evaluate the performance ofsimple contracts numerically for more general set-tings. In the first numerical study, we generalize thelinear density in Proposition 7 to the densities men-tioned earlier: f (x)� α+ δxη−1. We construct 1,200 sce-narios in total using the following parameter values.The market demand D follows a gamma distributionwith mean µ � 50 and coefficient of variation CV ∈{0.2, 0.4, 0.6, 0.8, 1.0}. We vary the CV while fixing µto investigate the impact of demand variability. Therest of the parameters are allowed to range from verylow values to very high values so that the numeri-cal study covers a wide range of practical situations:the supplier’s cost c has a support [0, c], where c �

{1, 5, 10}, η ∈ {1.5, 2.0, 2.5, 3.0}, and δ � η(1−αc)/cη. Toensure δ > 0 (or 1 − αc > 0), we set α � δα/c, whereδα � {0.01, 0.1, 0.5, 0.9}. Since the lower bound of thecost support is fixed at zero, the supplier’s mean costincreases in c. Thus we set market price p � δp c withδp ∈ {0.5, 1, 1.5, 2, 2.5}, where δp measures the magni-tude of market price relative to the cost (or the profitmargin of the product).

For each of the above scenarios, we calculate the ratio(πOM

b − πSb )/πS

b . Table 1 reports the performance of thesimple contract, measured by (πOM

b − πSb )/πS

b , acrossall scenarios. We group the data by different levels of

Table 1. Performance of Simple Contracts ((πOMb − πS

b )/πSb )

for Cost Density Function f (x)� α+ δxη−1 (δ > 0)

Demand CV Average (%) 90th percentile (%) Maximum (%)

0.2 0.09 0.20 0.390.4 0.10 0.23 0.680.6 0.15 0.28 1.100.8 0.21 0.41 1.651.0 0.28 0.58 2.34

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demand uncertainties (i.e., CV values). For instance,under CV � 0.2, the buyer enjoys a 0.20% or less profitincrease for 90% of the scenarios with an average of0.09% among all scenarios. We highlight several obser-vations from this numerical study. First, the value ofusing a menu of contracts is quite small: the aver-age profit increase is 0.17% with a maximum of 2.34%among the 1200 tested scenarios. Second, we find thatthe simple contract tends to perform better for largep values and low CV values. Third, consistent withProposition 5, we observe that To � 0 as long as co ≤ c inthis numerical study, i.e., the two-part tariff reduces toa pull contract when the optimal mechanism excludessome high cost types. This happens among 55% of thetested scenarios.Tables 2 to 4 present similar findings from three ad-

ditional numerical studies. These numerical studies aresimilar to the first except that the supplier’s cost fol-lows different distributions. In Table 2, the cost densi-ties are given by f (x) � α + δxη−1 with δ < 0, i.e., thedensity is decreasing in x. Due to the negative sign ofδ, we have used δα � {1.01,0.5η/(η − 1),0.9η/(η − 1)}in this numerical study. (To ensure δ < 0, we need1 − αc < 0, or δα > 1. Further, we need f (x) � α +

δxη−1 ≥ 0 for all x, which yields α ≤ η/(c(η − 1)), orδα ≤ η/(η−1).) There are 900 scenarios in Table 2. Sincewe show that both uniform and power distributions


b )/πSb )

for Cost Density Function f (x)� α+ δxη−1 (δ < 0)


0.2 0.07 0.14 0.330.4 0.11 0.24 0.420.6 0.14 0.34 0.610.8 0.16 0.38 0.831.0 0.19 0.44 0.96


b )/πSb )

for Normal Cost Distribution


0.2 0.23 0.39 0.660.4 0.40 0.76 0.910.6 0.60 1.19 1.490.8 0.83 1.68 2.231.0 1.08 2.31 3.13


b )/πSb )

for Gamma Cost Distribution


0.2 0.29 0.48 0.730.4 0.55 0.96 0.980.6 0.84 1.53 1.600.8 1.18 2.20 2.391.0 1.54 2.97 3.37

can secure the second-best solution, the good perfor-mance of these distributions is expected. To furthercheck for robustness of our findings, we have testedmany more log-concave cost distributions, includingnormal, gamma, logistic, exponential, extreme value,chi-square, Laplace, and beta distributions. This widerange of distributions allows for nonmonotone distri-butions. Tables 3 and 4 present our findings for nor-mal and gamma distributions with 375 scenarios each.The results for the remaining cost distributions aresimilar and are thus given in the online appendix.In Table 3, we consider (two-sided) truncated normalcost distributions. The unimodal normal distribution isquite natural from a practical perspective. The normaldistribution has a mean of c/2 and takes CV valuesfrom {0.1, 0.3, 0.5, 0.7, 0.9}; the truncated probability isevenly distributed on the support [0, c]. The numeri-cal study in Table 4 considers (truncated) gamma costdistribution, which generalizes the normal distributionto allow skewed density curves. In sum, through com-prehensive numerical studies, we show that the sim-ple contract (i.e., two-part tariff) yields close-to-optimalperformance under a wide range of circumstances.

5.3. Why Simple Contracts Perform WellThe above analysis and numerical studies show thatthe two-part tariff {wo ,To} (or sometimes a pull con-tract wo) is nearly optimal for the buyer (i.e., it yieldsalmost the same profit as the optimal menu of con-tracts). Why does the simple contract perform so well?Cachon andZhang (2006) study a contract design prob-lem in which the supplier is modeled as a make-to-order manufacturer with private capacity cost. Theydemonstrate that the buyer’s profit is very flat in capac-ity cost under the optimal mechanism; therefore, somesimple contracts with fixed parameters yield nearlyoptimal performance. However, in our problem setting,the buyer’s profit in the optimal contract is found tobe highly variable in capacity cost. So the insight fromCachon and Zhang (2006) does not apply here.

The first observation that contributes to the goodperformance of a simple contract is that it inducessimilar expected capacity as the optimal contract. Thecapacity plays a key role in our model setting becauseit determines both the supply chain’s profit as wellas the supplier’s profit. Let Ks(x) � Φ−1((wo − x)/wo)denote the capacity level induced by the simple con-tract. Then, E(|(Ko(x)−Ks(x))/Ks(x)|) denotes expectedabsolute percentage capacity difference between theoptimal and the simple contracts. The following propo-sition shows that this value can be bounded above bya small value under specific circumstances.

Proposition 8. Suppose f (x) � α + δx on a support [0, 1]and Φ is uniform on [0, 1]. Assume co ≥ 1 and wo ≥ 1,i.e., no supplier type is excluded under both contracts. Thenthe ratio E(|(Ko(x) − Ks(x))/Ks(x)|) has an upper bound

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≈ 5.6% for 0 ≤ δ ≤ 2, and an upper bound ≈ 7.8% for −1 ≤δ < 0. In addition, the ratio decreases in p in both cases.In our numerical experiments, we find the mean,

90th percentile and maximum values of the ratio tobe 2.84%, 8.70%, and 17.53%, respectively. This showsthat in such a supply chain setting, the simple contractinduces quite close capacities as those stipulated by theoptimal menu. To further investigate this observation,notice that Equation (18) satisfied by the optimal wo

can be rewritten as follows:∫ min{c , co }

0

xφ(K(x))(wo)2 [CFo(x) −CFS(x)] dx � 0,

where CFo � 1− (x + F(x)/ f (x))/p and CFS � 1− x/wo

are the newsvendor critical fractiles that determineKo(x) and Ks(x), respectively, for the optimal and sim-ple contracts. Since (x/(φ(K(x))(wo)2)) > 0, there mustbe Ko(x) > Ks(x) for some x, whereas Ko(x) < Ks(x)for the others. In other words, compared to the opti-mal mechanism, the simple contract would result inboth over-investment and under-investment in capac-ity, depending on the supplier’s type. This implies thatthe ratio bounds in Proposition (8) would be evensmaller if one considers the expected percentage capac-ity difference E((Ko(x) − Ks(x))/Ks(x)) (i.e., by takingout the absolute sign).To visualize this observation, we present an example

in Figure 1. It can be seen that the newsvendor crit-ical fractile for the optimal contract is quite close tothat induced by the single contract (Figure 1(a)). It isnot surprising that this results in quite similar capaci-ties and therefore small expected absolute percentagecapacity differences between optimal capacity and theone induced by the simple contract (Figure 1(b)). Alsonotice that the curve of Ks(x) can be either above orbelow that of Ko(x) (i.e., over- and under-investments).Since the supply chain profit only depends on the

induced capacity, the above observation on Ko(x) andFigure 1. The Capacity Difference Between the Optimal Mechanism and the Simple Contract

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(a) Newsvendor critical fractile (b) Capacity

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3.0 3.5

Note. Market demand follows a gamma distribution with mean 50 and CV � 0.5; capacity cost follows a normal distribution N( 52 ,

54 ), p � 15/2

and c � 5.

Ks(x) leads to a conjecture that the supply chain per-formances under the optimal and simple contracts areclose to each other. Let πOM

sc and πSsc denote the sup-

ply chain profits under the optimal and simple con-tracts, respectively; then the ratio (πOM

sc − πSsc)/πS

sc maybe used to measure the improvement in supply chainperformance by adopting the optimal mechanism. Wefind that among all four numerical studies, the ratiois −0.41% on average and 0.10% at the maximum.Thus, we can see that the simple and optimal mech-anisms perform closely for the supply chain as well.Note that these ratios are smaller than the ratios forthe expected absolute percentage capacity difference,which can be explained as follows. Recall that thereare Ko(x) < K∗(x) (due to the distortion effect as shownin Proposition 1) and Ks(x) < K∗(x) (due to doublemarginalization). Thus, the simple contract yields ahigher supply chain profit than the optimalmechanismif Ks(x) > Ko(x) and vice versa. Since there are bothKs(x) > Ko(x) and Ks(x) < Ko(x), the effects of over-investments and under-investments on supply chainprofit will cancel out each other in expectation, whichleads to even smaller ratios that measure the differencein supply chain profit under the two contracts.

The buyer’s profit under the optimal contract can beexpressed as supply chain profit minus the supplier’sprofit (information rent). Thus, to complete the expla-nationwhy simple contracts performwell for the buyer,we are left to show that the simple contract does notleave the supplierwith excessive profit compared to theoptimal one. FromProposition 1,we know that the sup-plier’s profit can be written as ∫ c

x Ko(u) du. Similarly, wemay write the supplier’s profit under the simple con-tract as a function of Ks(x) according to the next lemma.

Lemma 2. The supplier’s expected profit under a singletwo-part tariff contract {wo ,To} can be written as πS

s �

∫ cx Ks(u) du.

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Thus, the difference in the supplier’s profits underthe optimal mechanism and simple contract dependson the two capacity functions Ko(x) and Ks(x). Again,based on the above observation on Ko(x) and Ks(x),one may expect the difference in the supplier’s prof-its under the two different mechanisms to be small.Indeed, our numerical experiments show that the aver-age percentage difference is 2.09% (more details re-garding the difference in the supplier’s profits betweenthe two different mechanisms are provided in theonline appendix).A closer scrutiny reveals another subtle relationship

that further contributes to the excellent performance ofthe simple contracts. Recall that ex post the simple con-tract will cause both over- and under-investments incapacity compared to the optimal mechanism (there isKs(x)> Ko(x) for some x and Ks(x)< Ko(x) for the rest).For a cost realization x with Ks(x) > Ko(x), the sup-ply chain’s profit is higher under the simple contractthan under the optimal mechanism (note both Ko(x)and Ks(x) are lower than the centralized optimal capac-ity K∗(x)); however, the fact Ks(x) > Ko(x) implies thatthe adjacent range around x would contribute more tothe supplier profit under the simple contract (note thesupplier profits are ∫ c

x Ks(u) du and ∫ cx Ko(u) du under

these two contracts). An analogous observation canbe made for x with Ks(x) < Ko(x). Thus, from the expost perspective, there is a positive correlation betweenthe supply chain’s profits and the supplier’s profits,whose difference determines the buyer’s profit. There-fore, even when there is a gap between the capac-ity functions Ko(x) and Ks(x), its effect on the buyer’sprofit will be dampened due to such a positive correla-tion. This helps explain the bounds derived in Propo-sition (7) and (8) (i.e., the bounds on profit ratios aretighter than the bounds on capacity ratios).We have identified a couple of reasons to explain the

excellent performance of simple contracts in our sup-ply chain setting. The first observation indicates thatexpected induced capacities are quite close under thesimple and optimal contracts. As a result, both the sup-ply chain’s profits and supplier’s profits are also closeunder these contracts. Second, the positive correlationbetween the supply chain profit and supplier profitsuggests that any effect of over-investment and under-investment in capacity on buyer’s profit under the sim-ple contract would be dampened. Since the buyer’sprofit is just the difference between the supply chain’sprofits and supplier’s profits, these two observationsindicate that the simple and optimal contracts woulddeliver close profits for the buyer.

5.4. Managerial ImplicationsThe above results have useful implications for sourc-ing contract design, especially in global supply chainsettings. It has been widely acknowledged that when

procuring globally, firms would be less certain aboutwhether contractual agreements can be supportedby effective legal enforcement. In such environments,proactive arrangements designed to avoid and resolveproblems in the first place provide a superior alterna-tive to reliance on formal dispute-resolution strategies.One suggestion is to tie payment schedules to actualdelivery and acceptance of the buyer (see, e.g., Pierce2012). In this study, we show that this is an effectivesolution for the buyer and even a single, linear con-tract can yield close-to-optimal profit. Thus, the single,linear contract represents an excellent option for ourcontracting problem in several respects: it has a simpleformat, does not require the capacity cost informationas an input, and is enforcement-free.

The above analysis also reveals that simple contractsperform well because the induced capacity is close tothat under the optimal mechanism. As a result, thesupply chain performances are very close under thesimple and optimal contracts as well. In fact, we findthat in the majority of numerical scenarios, the sup-ply chain profit is higher under the simple contract(this does not hold for only 367 out of 2,850 scenarios;these exceptions tend to happen for low demand CVs(coefficient of variation)). This suggests that in general,the simple contracts help improve supply chain coor-dination. An implication of this finding is that in thepresence of asymmetric cost information, supply chaincoordination may not be an appropriate benchmarkbecause there is a conflict between the supply chain’sobjective and the buyer’s objective (to minimize theinformation rent). Moreover, the supplier prefers thesimple contracts because they do not screen the sup-plier type and hence leave him more information rent.Overall, the simple contracts are attractive to all par-ties in the supply chain. This bodes well for the buyerbecause she can easily convince the supply chain part-ner to adopt such contracts.

We are now in a position to discuss the impact ofthe two incentive issues studied in this paper. Section 4shows that noncontractibility of capacity does not re-sult in efficiency loss for the buyer. Through numericalexperiments we may also quantify the impact of costinformation asymmetry on supply chain and buyerprofits. Let πFB

b denote the buyer’s expected profitunder the first best solution (i.e., the capacity cost infor-mation is known to the buyer). Notice that πFB

b alsodenotes the supply chain profit, since the buyer cancoordinate the supply chain and leave zero profit tothe supplier. Therefore, (πFB

b −πOMb )/πFB

b measures theprofit loss for the buyer due to private capacity costinformation, whereas (πFB

b − πOMsc )/πFB

b measures thelost supply chain efficiency; the difference of these tworatios measures the gain for the supplier due to safe-guarding his capacity cost information.

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In our numerical experiments, we find that the lostefficiency for the supply chain due to asymmetric infor-mation is 15.97% on average. This is caused by thedistorted capacity function Ko(x) tominimize the infor-mation rent the buyer needs to pay. More importantly,the average profit loss for the buyer is 42.17%, in con-trast to 0.31%, the average value of (πOM

b − πSb )/πS

b(more details about these performance measures fromour numerical experiments are provided in the onlineappendix). Thus, we can conclude that the gap betweenπFB

b and πOMb (i.e., the value of eliminating information

asymmetry) is much higher than that between πOMb

and πSb (i.e., the value of using the optimalmenu of con-

tracts). These results imply that information asymme-try provides a strong shield for the supplier’s profit—even a well-designed complex screening contract doesnot extract much surplus from the supplier.To summarize, this research provides useful mes-

sages for managers who want to ensure supply whilefacing demand risk, uncertain supplier capacity cost,and noncontractible capacity. Such sourcing contexthas become typical in today’s global supply chains.First, the buyer should try every means to obtain bet-ter information about the supplier’s cost structure.This corroborates industry evidence where firms exertrelentless effort to educate themselves on their suppli-ers’ cost structures (see Ask and Laseter 1998, Kaya andÖzer 2009). However, if reducing the uncertainty in thesupplier’s cost is not a viable strategy, then the buyermay adopt a simple two-part tariff contract: it yieldsnear-optimal profit for the buyer but does not need thesupplier’s cost information or the contractibility of thesupplier’s capacity.

6. ConclusionThis paper studies a contracting problem in a sup-ply chain setting consisting of a newsvendor buyerand a supplier. Two incentive issues that may arisein practice have been considered when studying thebuyer’s contracting strategy: first, there is asymmet-ric cost information in the supply chain (i.e., adverseselection issue); second, the supplier may shirk fromhis responsibility for capacity investment (i.e., capac-ity contractibility issue). These issues have becomemore prominent in today’s global sourcing environ-ment where trading partners are often new and distantfrom each other. First, we find that lack of contractibil-ity on capacity does not necessarily reduce the buyer’sprofit, but it may require a more complex contract for-mat. Although the buyer’s optimal (second-best) mech-anism is generally complex, it may reduce to a menuof linear contracts under certain conditions (i.e., eachcontract in themenu represents a two-part tariff, wherethe buyer pays the supplier a wholesale price for each

product delivered and charges the supplier a lump-sum payment). Second, we show that a single two-part tariff can be optimal for the buyer when the sup-plier’s cost distribution represents a constant modifiedreverse hazard rate. Interestingly, even when the sin-gle two-part tariff is suboptimal, it performs nearly aswell as the optimal mechanism for a wide range of sit-uations. That is, a single two-part tariff contract candeliver either optimal or near-optimal profit for thebuyer under both cost uncertainty and contractibilityissues. These findings indicate that in a global sourc-ing environment, using a complexmenu and achievingenforceability may provide little value to the buyer.

Our results bode well for managers facing the aboveincentive issues, especially in a global sourcing con-text: for both ease of implementation and excellentperformance, a buyer can simply use a two-part tariffcontract (or sometimes just a wholesale price contract)under the presence cost information asymmetry andcontractibility issues. In addition, our findings sug-gest that the buyer should try all means to reduce theuncertainty about the supplier’s cost structure, whichcould lead to significant losses both for the buyer andthe supply chain. This is because such uncertaintyprovides a robust protection of the information rentthe buyer has to pay to the supplier; even the com-plex, optimal menu of contracts does not help much inreducing the information rent.

More research is needed to study supply chain con-tracting that involves informational and incentive is-sues. There are several potential directions for futureresearch in this area. First, in this paper, we focus onadverse selection and contractibility issues in sourcingproblems; it would be interesting to study a signal-ing game (rather than screening game) under moralhazard. Second, one may consider the impact of non-contractibility under the presence of both asymmet-ric demand information and asymmetric cost informa-tion. That is, the current model setting can be extendedto include private demand information for the buyer.Finally, the supply chain studied in this research rep-resents the simplest setting with one buyer and onesupplier. It would be interesting to extend our modelto more general supply chain structures (e.g., there areeither multiple suppliers or multiple buyers).

AcknowledgmentsThe authors thank Morris Cohen, Brian Tomlin, the associateeditor, and two anonymous referees for their constructivecomments that helped improve the paper. All inquiries on thepaper can be directed to the corresponding author, TianjunFeng ([email protected]).

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https://www.cnet.com/roadshow/news/tesla-q1-model-x-production-delayed-by-severe-parts-shortages-hubris/



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