PRODUCTION AND OPERATIONS MANAGEMENT

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Vol. 17, No. 5, September–October 2008, pp. 532–542

Coordinating a Supply Chain System with Retailers

Under Both Price and Inventory Competition

Xuan Zhao

School of Business andEconomics, WilfridLaurier University, Waterloo, Ontario N2J 3C5, Canada, xzhao@wlu.ca

W

e investigate a supply chain system with a common supplier selling to downstream retailers who are engagedin both price andinventory competition. We establish the existence anduniqueness of the pure-strategy Nash equilibrium for the retailer game and study how a supplier can coordinate the system to achieve the best performance. Our main conclusions are as follows: First, a buyback contract can be usedto coordinate retailers competing on both price andinventory in a sense that optimal retail prices andinventory levels arise as the Pareto-dominant equilibrium. With symmetric retailers, the system optimum arises as the unique sym-metric equilibrium. Second, the particular type of competition experienced by retailers (price versus inventory competition) affects the characteristics of the contract. Specifically, strong price competition leads to a coordi-nation mechanism with a positive buyback rate, where the supplier subsidizes retailers for leftover inventories; however, strong inventory competition leads to a negative buyback rate, where retailers are punished for over-stocking. Using a linear expecteddemandfunction, we further explore the impact of system parameters on the coordination contract andthe competitive equilibrium. We also findthat the performance of the supplier’s optimal contract is asymptotic to the system optimal coordination contract as competition becomes fierce.

Key words: pricing; inventory; competition; coordination

History: Received: June 2005; Revised: July 2006, March 2007, May 2007, and September 2007; Accepted:

September 2007 by Suresh Sethi.

1. Introduction

Advances in information technology and the increas-ing prevalence of global trade have made multiple andalternative distribution channels ubiquitous. A car may now be purchasedthrough a dealership or online. A watch may be purchasedin a depart-ment store, in a single-brandboutique, or over the Internet. Increasing the number of distribution chan-nels inevitably increases interchannel competition, andconsequently, this interrelation between multiple channels gives rise to complexities in managing and coordinating supply chains. This paper will inves-tigate the following questions: How can a supplier coordinate a multichannel supply chain system with competitive retailers in order to achieve the best per-formance of the entire supply chain system? How does the coordination contract relate to the nature of competition at the market level? How does the sup-plier’s incentive to use the coordination contract relate to the existing degree of competition?

Competition can be modeled in many ways. In this paper, we assume that retailers compete not only on price but also on inventory, in the sense that unsatisﬁedcustomers look aroundandmight switch

(spill over) to other stores that stock similar products. This consumer behavior is very common andcan be illustratedby the following example (Voice of Small Business 2002):

Talbot’s [clothing retailers], for instance, has a tele-phone in each of its outlets so that customers can speak

with representatives familiar with in-store andcata-log merchandise. Customers can order directly through these representatives when a store doesn’t have a spe-ciﬁc item in stock. Also, Sears andother large retailers long have had dedicated in-house catalog representa-tives who work with customers to locate items that are not available in the store

Similarly, according to a survey provided by P&G, approximately 50% of its customers switch to another retailer after a stockout (Tierney 2004). This consumer switching behavior results in the so-called“inventory competition” between stores. How does the combined price andinventory competition affect the pricing and stocking behavior of retailers, andhow does it inﬂu-ence a supplier’s coordination strategy?

This paper will focus on one well known and widely studied contract: the buyback contract, which is commonly usedin some industries. Car manufac-turers, for example, take back unsold old-model cars from dealerships and recast them into new ones; sim-ilar observations apply to the publishing industry. In a noncompetitive context, studies in channel coordi-nation suggest that double marginalization between a retailer and a supplier leads to overcharging or under-stocking at the retailer level. For this reason, special incentive arrangements are needed in order to align the manufacturer andthe retailer. A buyback contract

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can coordinate the retailer whose decision relates to procurement only andcan achievesystem optimal per-formance, but it cannot achieve the ﬁrst-best result if the retailer also sets the retail price (e.g., Emmons and Gilbert 1998). Much of the marketing andeconomics literature has considered coordination contracts in the case of retailers making only pricing decisions under deterministic demand (Shapiro 1989). In this paper, however, we will investigate how a supplier should set buyback contracts in a supply chain system where retailers facing stochastic demand compete on both price andit inventory.

The main ﬁnding of this paper is that buyback contracts can perfectly coordinate the supply chain system: the retail prices andinventories that achieve optimal supply chain performance arise as the Pareto-dominant equilibrium for the system and as the unique Pareto-dominant equilibrium if the system optimum is unique andproﬁts are transferrable among supply chain members. In the case of sym-metric retailers, the system optimal retail prices and safety stocks arise as the unique symmetric equilib-rium for the retailer game under the coordination contract. Interestingly, the nature of the coordina-tion buyback contract is closely relatedto the nature of the competition. For example, if inventory com-petition outweighs price comcom-petition in the market, the buyback contract will stipulate that the retailer must pay the supplier for overstocking, in contrast to the common buyback arrangement. Conversely, if price competition outweighs inventory competition in the market, the buyback contract will stipulate that the supplier must buy back extra inventories from the retailer.

Using a linear demand function for the determinis-tic portion of demand, we further explore the proper-ties of the equilibrium andthe coordination contract. We ﬁndthat the wholesale price slightly increases but the buyback rate decreases with the spill rate and that both terms increase with the degree of price compe-tition. In our numerical study, we examine the sup-plier’s optimal buyback contract andcompare it with the system optimal contract. We ﬁndthat the two con-tracts become extremely close as the degree of the price competitiveness andthe degree of spill increase. In other words, the value of coordination actually

decreases with the degree of both types of competi-tion. Along with these main ﬁndings, we will also demonstrate that retail-price-maintenance contracts andquantity-forcing (service-level-maintenance) con-tracts can also achieve optimal system performance.

The paper is organizedas follows. Section 2 reviews the related literature. Section 3 introduces details of the model. Section 4 provides the main results of the paper using the general model. Section 5 uses

the special case of a linear demand function to fur-ther examine the properties of the equilibrium and the coordination contract, and §6 provides numerical results. In §7, other contracts (such as service-level-maintenance andretail-price-service-level-maintenance contracts) are discussed. We conclude in §8.

2. Related Literature

The ﬁeldof channel coordination has attractedcon-siderable attention in the past decade. Cachon (2003) gives an excellent review of the literature anddis-cusses a variety of supply chain contracts. Kouvelis et al. (2006, p. 456) provide a more recent discussion on supply chain research, commenting that “[coor-dination and contracts are] particularly important in a competitive environment in which entire supply chains are competing for customers.” Here, we limit this review to works that consider coordination and contracting in a competitive environment.

The first stream of research considers issue of con-tracting for supply chains where a manufacturer sells products through both direct (manufacturer-owned) and indirect (competitor-owned) channels, so the manufacturer is both a supplier anda competitor of the retailer. Tsay andAgrawal (2004) analyze the inef-ficiency of such a system and provide recommenda-tions about how to adjust the manufacturer–retailer relationship. Boyaci (2005) explores the impact of channel inefficiency on stocking decisions and devel-ops contract schemes to coordinate the supply chain. Cattani et al. (2006) analyze pricing strategies to ease the channel tensions as a result of the manufacturer entering into the market andcompeting with its sup-ply chain partner. Different from this literature, apart from considering multiple competitive retailers, we also allow retailers to compete in both price and inventory.

There has been very little research that considers coordination and contracting for supply chains with a manufacturer selling to multiple competitive retailers. van Ryzin andMahajan (1999), for instance, examine the performances of vendor and retailer management inventory under horizontal inventory competition among retailers. They findthat the two schemes tend to achieve first-best profits as the inventory compe-tition increases. Cachon (2003) considers coordina-tion for supply chains with multiple newsvendors. To illustrate the trade-offs, he has designed two mod-els: The first model assumes that the uncertain total demandis allocatedto each retailer according to the ratio of its inventory to total system inventory, with the retail price being fixed. The externality between retailers in this model is that each has an incentive to stock higher in order to be assigned a higher demand. To mitigate this effect, the supplier can either charge

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a wholesale price greater than the marginal cost or use a buyback contract to adjust the retailers’ incen-tives. The secondmodel assumes that, in the case of market-clearing prices, higher inventory levels leadto lower prices. The externality between retailers in this model drives them to establish lower inventory lev-els to ensure higher market prices. Both the buyback contract andthe retail-price-maintenance contract can achieve coordination in this setting. Anupindi and Bassok (1999) consider a system with only inven-tory competition andﬁndthat a wholesale price plus a holding-cost subsidy contract can coordinate the system.

In this paper, we will provide a more general model in the sense that it captures the externalities between retailers introduced by both inventory competition andprice competition. Using a novel approach, Zhao andAtkins (2008) establish the existence of pure-strategy Nash equilibrium for the retailer game with combinedcompetition under an exogenous wholesale price contract. They also examine the properties of the equilibrium andinvestigate the impact of com-petition on retailers’ behavior. Different from them, we focus on exploring how a manufacturer/supplier shoulddesign the contract to align the competing retailers in a way that creates the best performance for the whole system, andwe will investigate the prop-erties that coordination contracts should have under competition as well. Bernstein andFedergruen (2007) consider coordination mechanisms for supply chains under price and service competition. They treat ser-vice competition in the same way as price competi-tion, with both affecting the deterministic portion of the demand. In contrast, we assume that a proportion of a retailer’s unsatisﬁeddemandswitches to its com-petitors, affecting the stochastic portion of a retailer’s demand.

3. Model Details

Consider a supplier selling to N oligopoly retailers in a market. Before the selling season, the supplier announces the wholesale pricew_iandbuyback rate_i

to each retailer ifor i=1 toN. These may be differ-ent for each retailer, but the differences will be based mainly on product characteristics or costs of pro-duction. Retailers then make decisions about order-ing quantities andprices simultaneously before they know the demand. Products are delivered immedi-ately, demands are then realized, and ﬁnally, proﬁts are collected.

3.1. The Retailer Game

3.1.1. The General Model. Following the litera-ture that considers the price-sensitive newsvendor model (e.g., Petruzzi and Dada 1999, Agrawal and

Seshadri 2000), we useL_ip +_i to model a retailer’s direct demand, where p=p1 p2 pN . The ﬁrst part, Lip , represents the deterministic portion of the demand, having Li

i p def=dLip /dp i≤0, Lj i p def=

dL_ip /dp _j≥0. The secondpart, _i, represents price-independent uncertain demand, having a general probability density function (PDF) fi· andcumula-tive distribution function (CDF) Fi· . Retailers need to decide on retail pricespi andsafety stocksyibefore knowing the demand. So the total order of a retailer is Y_i=L_ip +y_i, where y_i is usually viewedas a “hedge” of uncertain demand, determining the avail-ability or service level. We buildon the above model by assuming that a proportion ji of lost sales from retailerjwill switch toi. By doing this, we have incor-poratedinventory competition between retailers, as some of the demand lost by j will be “stolen” by i

if i has enough inventory (a similar assumption is usedby Netessine andRudi 2003 andthe literature therein). As a result, each retailer’s total demand con-sists of three parts:Lip +i+j=ijij−yj +, with the thirdpart, j=ijij−yj +, representing indirect stochastic demand of i, andcapturing the fact that a proportion of lost sales from a competitor will switch to retaileri.

We use Ds

i =i+j=ijij −yj + to simplify the notation; the associatedPDF isf_Ds

i·, andthe CDF is FDs

i· . So a retailer’s problem is to make a decision

aboutp_iandy_ito maximize_i=p_i−w_i L_ip −w_iy_i+

piEminDis yi+iEyi−Dsi+. By collecting terms, we have

i=id−wi−i yi+pi−i EminDsi yi (1) where d

i =pi−wi Lip . We assume that retailers’ strategy sets are compact: pi yi wi≤pi≤pmaxi 0≤

yi ≤ yimax, where pmaxi and yimax are large enough numbers andnever restrict the players (Cachon and Netessine 2004). It can be shown that the game with decisions pi yi is equivalentto the game with deci-sionsp_i Y_i .

There are two extreme cases for our general model. We call a game an inventory competition game if

Lj _i p =0 and Lip =Lipi , anda game aprice

com-petition gameif_ji=0 for alli jand Ds

i=i.

3.1.2. A Special Case. To explore the compara-tive statics concerning the equilibrium andthe coor-dination contract, we return to a special case where the deterministic portion of demand has a linear form; i.e., Lip =a−b+ pi+j=ipj−pi . This linear demandform has been commonly usedto study issues related to operations/marketing inter-faces (Tsay andAgrawal 2000, Boyaci andRay 2003). When it is not possible to derive managerial insight with a general pure-strategy Nash equilibrium, we sometimes use a symmetric equilibrium concept,

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which is commonly usedby economists for com-plex models (Fudenberg and Tirole 1991, Tirole 1988). A game is symmetric in the sense that the players have identical parameters and identical strategies and payoffs (Cachon andNetessine 2004). In other words, payoff functions are permutable. We use=jifor all

i j when a symmetric game is considered. In a sym-metric equilibrium, all of the players choose the same actions.

3.2. The Supplier’s Problem

Our primary interest is to investigate whether the het-erogeneity among retailers is a barrier for coordination when retailers compete on both price andinventory. Therefore, our goal is to ﬁnda supplier contract that maximizes the total supply chain proﬁt (c₌

iic,

where c

i =id−ciyi+piEminDis yi and ci is the production cost of producti). A supplier’s problem is to set (wi i) so that the system optimality solution arises as an equilibrium outcome. A supplier might be interestedin using a system optimal contract for sev-eral reasons: she might want to builda long-term rela-tionship with the retailer, or she might care most about the product being delivered to the market efficiently. It couldalso be the case that the supplier wouldfollow the recommendation of a parent firm whose interest is in the performance of thetotal supply chain. So the goal of the supplier is to make the total supply chain per-form in the best possible way. However, we will still investigate the contract that maximizes the supplier’s own profit andthen compare the two.

4. Results with the General Model

4.1. Equilibrium Analyses with Fixed Contract Parameters

Given the contract parameters w_i and _i, we ﬁrst study the competitive retailer game under both price andinventory competition. We start by investigating the existence of the pure-strategy Nash equilibrium of the game, andthen we introduce some results con-cerning the uniqueness of the equilibrium.

4.1.1. Existence of Nash Equilibrium in Pure Strategies. Two conditions are needed for establish-ing the existence of a Nash equilibrium (Theorem 1):

(A) !2d

i/!p2i<0 and!3id/!p3i ≤0. (B) The distribution ofDs

i has an increasing failure rate (IFR r_Ds

i def

=f_Ds

i· /1−FDsi· ). (This is guaranteed

but not restrictedby the assumption that s have IFR distributions and are independent.)

Theorem 1. Assume(A)and(B) hold.(i)There exists a pure-strategy Nash equilibrium.

(ii) The best response of retaileriis the unique solution

of

!i/!pi=!id/!pi+EminDis yi=0 (2)

!i/!yi= −wi−i +pi−i PrDsi> yi =0 (3)

The proof is similar to Zhao andAtkins (2008), where a novel methodis providedandassumptions are justiﬁed. We include the proof in the Online Sup-plement, available at http://www.poms.org/journal/ supplements, for the convenience of the reader. An immediate result from this theorem is that a symmet-ric pure-strategy Nash equilibrium exists in the game (Theorem 2, Cachon andNetessine 2004) with sym-metric retailers.

If we consider the extreme case of the inventory

competition game whereL_ip =L_ip_i , the above con-ditions are not needed.

Proposition 1. The duopoly inventory competition game is supermodular, and there exists a pure-strategy Nash equilibrium.

4.1.2. Sufﬁcient Conditions for Uniqueness of the Nash Equilibrium.

Theorem 2. Sufﬁcient conditions for uniqueness are

−!2d

i/!p2i >j=i!2id/!pi!pj +1 and j=iji +1/

w_i−_i r_Ds

i0 <1, where rDsi0 is the failure rate at y_i=0.

A similar result is provedin Zhao andAtkins (2008); we include the proof in the Online Supplement for the convenience of the reader. The conditions are restrictive. However, if we consider identical retailers, the conditions can be relaxed. Condition (C) below is commonly usedto establish the uniqueness of a Nash equilibrium for a deterministic pricing game by economists (Vives 1999):

(C)!2d

i/!p2i+

j=i!2id/!pi!pj<0.

Proposition 2. With N identical retailers and

con-ditions (B) and (C), there exists a unique symmetric

equilibrium.

4.2. System Optimal Coordinating Contracts

After studying pricing and inventory equilibrium under exogenous contract parameters, we investigate supply chain coordination. In this section, the key question to be addressed is: in a decentralized sup-ply chain system with both price andinventory com-petition between independent retailers, how should a supplier set a contract to achieve system optimal prices andsafety stocks as an equilibrium andmaxi-mizesupply chain-wideproﬁts?

In an integratedsystem, the common supplier owns all retailers: c ₌

iic, where ic = id − ciyi +

piEminDis yi. Netessine and Rudi (2003) study a similar system but with no consideration of pric-ing decisions; analytically they ﬁnd that the objective function might not be jointly quasiconcave in stock quantities, but numerically with reasonable demand variability they do not ﬁnd multiple solutions. With 2N decision variables on both prices and stocks, c

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might not be jointly quasiconcave either, but numeri-cally no multiple optimal solution is found(see §6).

However, it can be shown (as in part (ii) of Theo-rem 1) that a global optimal solution ofc _{is interior} andsolves !c_/!p i =Lip +pi−ci Li i p +EminDis yi + j=i pj−cj Li j p =0 (4) !c_/!y i = −ci+piPrDis> yi − j=i p_j_ijPrDs j< yj i> yi =0 (5) By comparing this with (2) and(3), we identify the difference between a global optimum and a competi-tive equilibrium andthe source of inefficiency of the competitive equilibrium. Subtracting (2) and(3) from (4) and(5), we have !i/!pi=!c/!pi− j=i pj−cj Li j p −w_i−c_i Li i p (6) !_i/!y_i=!c_/!y i+ j=i _ijp_jPrDs j< yj i> yi −wi−ci−i PrDsi< yi (7) By comparing the first-order conditions of the com-petitive retailers’ game andthe joint optimization problem, we can deconstruct the impact of vertical conflict (double marginalization) and horizontal con-flict (competition) on pricing andinventory decisions. The secondterm of Equation (6) reflects the horizon-tal externality causedby price competition, whereas the secondterm of (7) represents the horizontal exter-nality of inventory competition. Both of them dis-tort competing retailers’ decisions regarding price and safety stock. A competitive retailer tends to set lower

pi than the system optimum because an increase in

p_i benefits rival retailers, resulting from the nature of demand substitutability. A competing retailer also has a higher level of safety stock y_i because a decrease in yi results in more spill demand to the competi-tor. The thirdterms of both equations reflect the ver-tical externalities introduced by double marginaliza-tion. A decentralized retailer tends to set higher pi than the system optimum because a lower p_i hurts the retailer but boosts demandandbenefits the sup-plier. Similarly, a decentralized retailer has a lower level of safety stock yi than the system optimum if the buyback rate i is small. To fully coordinate

inde-pendent and competitive retailers, the supplier should design the contract so that the vertical and horizon-tal conﬂicts balance each other; i.e., the sums of the secondandthirdterms become zero for both equa-tions. Under some conditions on the parameters, a

set of wholesale prices (with i=0) can coordinate the system, but the restrictive parameter conditions make the coordinating wholesale-price-only contract less interesting.

Let (pc_yc_{) be a global optimum that maximizes} the total supply chain proﬁt. Theorem 3 gives a sys-tem optimal coordinatingbuyback contractw∗

i ∗i that

achieves the best supply chain performance in a sense that the system optimal retail prices andinventory levels, pc_yc _{, arise as a Pareto-dominant} equilib-rium for the system. Note that an equilibequilib-rium X is saidto Pareto-dominateYif every player prefersXto

Y (Varian 1992), i.e.,iX ≥iY for alli, including the supplier andall of the retailers.

Theorem 3. There exists a unique w∗

i ∗i with ∗i ≤

w∗

i < pci and wi∗≥ci that perfectly coordinates the supply

chain system, i.e., pc_yc _{arises as an equilibrium of the}

retailer game, where w∗ i =ci− j=i pc j−cj Li j pc /Li i pc and (8) ∗ i =pci−pci−wi∗ /PrDis< yci (9)

Furthermore, pc_yc _{is a Pareto-dominant equilibrium}

for the supplier–retailer game. If payoffs are transferrable under coordination, then it is the unique Pareto-dominant equilibrium if the global optimum for the supply chain sys-tem is unique.

Note that, in general, there might be multiple equi-libria underw∗

i ∗i . However, Schelling’s (1960) the-ory of “focal point” suggests that in some “real-life” situations players may be able to coordinate on a particular equilibrium by using information that is abstractedaway by the strategic form. For example, if players are able to communicate to one another before the game is played, then it is reasonable that players will coordinate on the Pareto-dominant equilibrium, a commonly usedfocal point of a strategic form game (Fudenberg and Tirole 1991). If pc_yc _{is the unique} global optimum for the total supply chain system, and if a transfer payment can be arrangedamong supply chain members under coordination, thenpc_yc _arise as the unique Pareto-dominant equilibrium for the supplier–retailer game. The transfer payment could be enforcedunder the request of the parent ﬁrm or the supplier as a requisite for the supplier to use the coordination contractw∗

i ∗i insteadof the sup-plier’s optimal contract, in the format of lump-sum payments between supply chain members. However, a detailed investigation on the arrangement of the transfer payments is outside the scope of this paper.

In the case of symmetric retailers, according to Proposition 2, the symmetric system optimal retail prices andsafety stocks, pc_yc _{, will arise as the}

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See Mahajan andvan Ryzin (2001) for an argument that a symmetric equilibrium is more focal than an asymmetric equilibrium.

Proposition 3. With price competition only, ∗

i >0

and w∗

i > ci; with inventory competition only,∗i <0 and

w∗

i =ci.

Bernstein and Federgruen (2007), considering both price andservice competition, foundthat perfect coor-dination can be achieved if a wholesale price is com-binedwith a backlogging penalty, andthe sign of the penalty can be either positive or negative. Cachon (2003) has positive buyback rates for both models. In contrast, in our contract, ∗

i may be a subsidy (posi-tive) or a penalty (nega(posi-tive) for the retailers; interest-ingly, whether it is a subsidy or a penalty depends on the particular type of competition taking place—price or inventory competition.

With price competition only, the system-coordi-nation contract shouldhave the supplier charging wholesale prices greater than the production cost and paying retailers for residual inventories. The idea is that price competition induces retailers to set lower prices andlower levels of safety stock. By charging higher wholesale prices, the supplier forces the retail-ers to set higher prices, thus offsetting the damag-ing effect of price competition on retail prices. At the same time, by offering to buy back or subsidize leftovers, the supplier encourages the retailers to set higher levels of safety stock.

Under the theoretical extreme case of only inven-tory competition, the coordination contract has the supplier set the wholesale price at a level equal to the marginal cost of production in order to avoid double marginalization, charging fees for leftover inventories to reduce the effect of overstocking under inventory competition. Under this contract, the supplier does not earn revenue through direct procurement; instead, the supplier collects revenue if the retailers have leftovers, ending up with a positive proﬁt. Because the retailers have to pay the supplier for overstock-ing, it seems that this wholesale-price-plus-leftover-penalty contract wouldrequire the supplier to under-take more inventory monitoring; this has been made possible with the current advances in information technology. We will discuss this more later.

5. Results with the Special Case of

a Linear Demand Function

To further explore the properties of a competitive equilibrium andthe coordination contract, in this section, we consider a special case where the deter-ministic portion of demand has a linear form, i.e.,

Lip =a−b+ pi+j=ipj−pi , andwe focus on a symmetric equilibrium. We ﬁrst explore the proper-ties of an equilibrium for any buyback contract, and

then we examine the properties of the coordination buyback contract (w∗

i ∗i).

5.1. Comparative Statics for an Equilibrium Underwi i

How does the equilibrium change with the contract parameters andthe competition parameters? Using (p∗_y∗_{) to denote a symmetric equilibrium, the} follow-ing propositions address this.

Proposition 4. For a symmetric equilibrium,

(i) dp∗_{/d >}₀_and_dy∗_{/d >}₀_.

(ii) The only situation that cannot occur isdp∗_{/dw <}₀

anddy∗_{/dw >}₀_.

This result actually holds for not only the linear demand case, as shown in the Online Supplement. A higher buyback rate set by the supplier induces higher retail prices andsafety stocks from the retail-ers. So consumers are less likely to face a stockout, but at a cost of paying more. As shown in the previous literature, a higher wholesale price (double marginal-ization) leads to two opposite effects: a higher retail price (Spengler 1950) but a lower safety stock (Cachon 2003). When retailers compete on both price and inventory, the two effects work against each other, resulting in three possible outcomes. It is straightfor-wardto see that the system optimal (pc_yc_{) is} insen-sitive to the contract terms.

Using a methodsimilar to that in Proposition 4, we have that increaseddemandspill leads to a higher equilibrium retail price andsafety stock, but increased price competition drives the equilibrium price and safety stock down, echoing Zhao and Atkins (2008).

5.2. Properties of the Coordination Contractw∗

i ∗i

How does the coordination contract w∗

i ∗i change with the degree of price competition andthe spill rate ()? An analytical result concerning is not tractable; we numerically study this in §6. Proposi-tion 5 addresses the impact of.

Proposition 5. w∗

i increases with, and so does ∗i. As price competition becomes ﬁercer, the coordina-tion buyback contract has a higher wholesale price andbuyback rate. The intuition is that as price compe-tition becomes stronger, retailers tendto reduce prices (which, in turn, woulddamage the proﬁt of the entire supply chain) but that a higherw∗

i prevents the price reduction. A higher wholesale pricew∗

i usually results in a higher buyback rate∗

i as compensation.

Proposition 6. For the symmetric price competition

game, under coordinating contractw∗ ∗ _,

i/ic=1−

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As the number of retailers increases, a retailer’s proﬁt share under coordination decreases, and the proﬁt share of the supplier increases. The same result is observedas the price competition becomes stronger (a larger). In general, however,_i/c

i does not have such a nice form; an arbitrary division of channel proﬁt has to rely on a ﬁxedside payment between retailers andthe supplier.

Proposition 3 examines the contract terms under two extreme cases, but what happens when price and inventory competitions coexist? It turns out that the sign of ∗

i is determined by the relative strength of price competition anddemandspill (to be conﬁrmed in §6). This point becomes clearer if we use a linear demandfunction andconsider a symmetric equilib-rium, where pc

i yic is indifferent to . From (8) and (9), if pc

iPrDsi> yic < ci−j=ipjc−cj /b+ , then

∗

i >0. Note that the right-handside of this inequality increases with . If price competition is very strong (a relatively large), then it is more likely that∗

i >0

will holdtrue, andthe contract will require the sup-plier to pay the retailers for leftover inventories. On the other hand, if price competition is very weak (a relatively low ), then the reverse of the above inequality is more likely to hold, which means that the contract will require the retailers to pay the sup-plier for the leftover inventories (∗

i <0). Buyback contracts with positive buyback rates are commonly implemented in different industries. For example, in the publishing andmusic industries, retailers usually return unsoldproducts; in computer andapparel, the supplier usually subsidizes unsold products. For the situations where a negative buyback rate is neces-sary (where retailers pay back), the implementation of supplier’s inspection might be needed. “Inspection rights” of the supplier are usually written clearly in a supplier contract. For example, quotedfrom §4.6 of “HardDisk Drive Supply Agreement” by and between Quantum Corporation andTiVo Inc. (Sample Business Contracts 1998):

Quantum may cause, at Quantum’s expense except as below provided, an independent, certiﬁed pub-lic accountant selectedby Quantum andacceptable to TiVo in its reasonable discretion to examine such

true and complete books of account at all

reason-able times TiVo shall promptly pay to Quantum all amounts due to Quantum hereunder, plus interest thereon

6. Numerical Results

The purpose of the numerical experiments is to gain some insights into questions that cannot be answered analytically. We use normal, uniform, andexponen-tial distributions and different sets of parameters; the results obtainedare consistent. Here we represent the results concerning a symmetric equilibrium when a

normal distribution, a symmetric linear demand func-tion, andthe following parameter setup are used:

a=200,b=12,c=6,&=100,=0 b/2 b2b3b4b

(incorporating a broadrange of the degree of price competition), '=2050 (resulting in coefﬁcient of variation 0.2 and0.5, respectively),=02095, and

N=3.

First, how do the terms of the coordination contract change with system parameters such as the degree of price competition () andthe spill rate ()? Consis-tent with Proposition 5, bothw∗

i and∗i increase with price competition factor . Furthermore, w∗

i slightly increases with , but ∗

i decreases with (Figures 1 and2). The buyback rate is usedto encourage retailers to stock more. So if a market has a higher spill rate, then a lower buyback rate shouldbe usedto achieve coordination. Both w∗

i and ∗i are quite insensitive to the coefﬁcient of variation for the stochastic portion of demand.

Second, how do the system optimal retail prices and safety stocks (i.e., the equilibrium under coordi-nation) change with system parameters such as,, andthe coefﬁcient of variation? From Figures 3 and4, we see that both decisions increase with the spill rate

but are insensitive to. The former decreases but the later increases with the coefﬁcient of variation.

How do the supplier’s and retailers’ profits under coordination change with the parameters? From Fig-ures 5 and6 we see that the supplier’s profits increase with both and . The retailers’ profits, on the other hand, decrease withbut increase very slightly with. So most of the profit gain obtainedfrom coor-dination goes to the retailer when price competition

Figure 1 System Optimal Buyback Contract

6 7 8 9 10 11 12 13 14 15 0 6 12 18 24 30 36 42 48 w θ γ = 0.95, cv = 0.2 γ = 0.95, cv = 0.5 γ = 0.2, cv = 0.5

Figure 2 System Optimal Buyback Contract

– 9 – 6 – 3 0 3 6 9 12 15 0 6 12 18 24 30 36 42 48 β θ γ = 0.95, cv = 0.2 γ = 0.95, cv = 0.5 γ = 0.2, cv = 0.5

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Figure 3 System Optimal Retail Price 14.9 15.0 15.1 15.2 15.3 15.4 15.5 0 6 12 18 24 30 36 42 48 p θ γ = 0.95, cv = 0.2 γ = 0.95, cv = 0.5 γ = 0.2, cv = 0.5

Figure 4 System Optimal Safety Stock

103 104 105 106 107 108 109 110 111 0 6 12 18 24 30 36 42 48 y θ γ = 0.95, cv = 0.2 γ = 0.95, cv = 0.5 γ = 0.2, cv = 0.5

is weak; however, most of the proﬁt gain belongs to the supplier when price competition is strong. As expected, low demand uncertainty leads to high prof-its for both the supplier andthe retailers.

The next question is, how much does this coor-dination contract differ from the supplier’s optimal contract? The system optimal buyback contracts per-fectly coordinate the supplier and the competitive retailers. However, if the supplier needs to negotiate with retailers or with its parent ﬁrm for compensation (in the case that the supplier uses a system optimal

Figure 5 Supplier’s Proﬁt Under System Optimal Buyback Contract

0 700 1,400 2,100 2,800 0 12 18 24 30 36 42 48 Pi ( S ) 6 θ γ = 0.95, cv = 0.2 γ = 0.95, cv = 0.5 γ = 0.2, cv = 0.5

Figure 6 Retailer’s Proﬁt Under System Optimal Buyback Contract

100 300 500 700 900 0 6 12 18 24 30 36 42 48 Pi ( R ) θ γ = 0.95, cv = 0.2 γ = 0.95, cv = 0.5 γ = 0.2, cv = 0.5

Figure 7 Supplier’s Optimal Buyback Contract

13.5 14.0 14.5 15.0 15.5 0 6 12 18 24 30 36 42 48 w θ γ = 0.95, cv = 0.2 γ = 0.95, cv = 0.5 γ = 0.2, cv = 0.5

Figure 8 Supplier’s Optimal Buyback Contract

0 5 10 15 0 6 12 18 24 30 36 42 48 β θ γ = 0.95, cv = 0.2 γ = 0.95, cv = 0.5 γ = 0.2, cv = 0.5

contract under the interference of the parent ﬁrm), how much wouldthe supplier expect? To answer this question, we needto examine the buyback con-tracts that maximize the supplier’s proﬁt—that is, the

w_i and _i that maximize _iw_i −c_i L_ip +y_i −

_iEy_i−Ds

i+.

The wholesale price of the supplier’s optimal buy-back contract increases with , but at a slower rate than the wholesale price under a system optimal con-tract does (Figure 7). It also increases with . The buyback rate under the supplier’s optimal contract increases with but decreases with spill rate (Fig-ure 8). An interesting observation is that the buy-back rate is always positive, so the supplier will never let the retailers pay for overstocking. Both contract terms decrease with the coefﬁcient of variation of the stochastic portion of demand.

In Figure 9, the supplier’s efficiency loss under the system optimal contract diminishes dramatically as the degree of price competition grows. (Effi-ciency loss of the supplier’s profit = (supplier’s profit under her optimal contract−supplier’s profit under coordination contract)/supplier’s profit under her optimal contract.) When = 3b, the supplier loses only approximately 6% by using a coor-dination contract. On the other hand, the effi-ciency gain of the total supply chain profit (Fig-ure 10) under a system optimal buyback con-tract drops dramatically (approaching zero) as the degree of price competition increases. (Efficiency gain of the total supply chain profit=c ₋s_/

c_{, where} s _{is the total supply chain proﬁt when} the supplier chooseswi and i to maximize her own

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Figure 9 Efﬁciency Loss of the Supplier Proﬁt Under System Optimal Contract 0 0.2 0.4 0.6 0.8 1.0 0 6 12 18 24 30 36 42 48 Supplier’s efficiency loss θ γ = 0.95, cv = 0.2 γ = 0.95, cv = 0.5 γ = 0.2, cv = 0.5

Figure 10 Efﬁciency Gain of Total Supply Chain Proﬁt Under System Optimal Contract 0 0.05 0.10 0.15 0.20 0.25 0 6 12 18 24 30 36 42 48 Efficiency gain of coordination θ γ = 0.95, cv = 0.2 γ = 0.95, cv = 0.5 γ = 0.2, cv = 0.5

proﬁt.) This result conveys the managerial insight that system-wide coordination provides less value to sup-ply chains if the market is highly competitive. Both terms also slightly decrease with the degree of spill but are quite insensitive to demand variability.

7. Extensions: Discussion on

Other Contracts

Next we discuss two other types of contract. One is the retail-price-maintenance contract (price ceiling or price ﬂoor); the other is the quantity-forcing (service-level-maintenance) contract. When retail-price main-tenance was legal in North America, it was widely used in preventing discount stores from free riding. In an environment where retail-price maintenance is per-mitted, the supplier charges retailers wholesale prices

wp_i =c_i+_j₌_i_ijpc

jPrDjs< ycj i> yci andsets retail prices equal to the integratedlevel of retail prices. Then the distortion from horizontal and vertical exter-nalities can be adjusted. To see this, when w=w_ip, the sum of the secondandthirdterms in (7) is zero, andthe retailers will choose safety stocks equal to the centralizedsolutions.

Quantity-forcing contracts (or service-level-main-tenance contracts) are another widely used strat-egy. The supplier charges wholesale prices ws

i =ci−

j=ipcj−cj Li j pc /Li i pc andsets a quota equal to the centralizedsafety stock. The quota can be seen as a requiredservice level forcedon the retailers as a prerequisite of the partnership. This scheme can perfectly coordinate the system: as with ws

i, Equa-tion (6) becomes !i/!pi =!c/!pi, (i.e., price dis-tortion under combined competition can be fully

adjusted). So the retailers will set retail prices that are essentially the same as the centralizedretail prices.

Retail-price-maintenance contracts have been used in a wide range of markets and have often been dis-cussedin the economics literature in the context of vertical control. Winter (1993) provides detailed exam-ples andjustiﬁcations. Service-level-maintenance con-tracts (Mathewson andWinter 1984) are widely observedin the automobile industry, where car sup-pliers usually set certain levels of quota in their con-tracts with car dealers, who are then advised to stock at least a minimum number of those suppliers’ cars for a given sales season.

8. Conclusions and Future Research

We have studied a two-echelon supply chain sys-tem with a common supplier selling to retailers en-gagedin both price andinventory competition. Each retailer’s total stochastic demand depends on their own retailpriceandinventorylevel, as well as those of their competitors. In such a system, incentive conﬂict exists not only in the vertical dimension of the supply chain (between the supplier anda retailer), but also in the horizontal dimension, i.e., between competing retailers. How can a supplier coordinate such a mul-tichannel supply chain system in order to achieve the

bestperformance of the entire system? How does the vertical control arrangement (coordination contract) relate to the nature anddegree of horizontal compe-tition between retailers?

We first address these research questions with the general model. We provide sufficient conditions on the existence anduniqueness of a pure-strategy Nash equilibrium, andwe findthat there exists a unique optimal contract that perfectly coordinates the system, in the sense that the system optimal retail prices and safety stocks arise as the Pareto-dominant equilib-rium, a focal point that is usually usedby economists to predict the players’ behavior when multiple equi-libria exist. If a transfer payment is allowedunder coordination, then the Pareto-dominant equilibrium is unique given that there is a unique global optimum for the whole supply chain system. Furthermore, if we consider reasonably homogenous retailers, then the system optimal retail prices andsafety stocks arise as the unique symmetric equilibrium of the retailer game.

The interaction between vertical externalities (dou-ble marginalization) andhorizontal externalities (price-inventory competition) enriches the context of coordination. When both types of competition coexist, a coordination contract has a wholesale price greater than the production cost. However, the buyback rate,

∗

i, can be either positive or negative, depending on the relative strength of the two types of competition.

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The buyback rate is positive when price competition is stronger, which encourages retailers to stock higher than the safety stock levels under competition. Alter-natively, it is negative when inventory competition is stronger, which discourages retailers from stocking higher-than-optimal levels. In addition to the tradi-tional function of the buyback rate, which is usedto motivate the retailer’s incentive for procurement, the buyback rate in this context is also usedto adjust the competitive behavior of the retailers.

To further explore the problem, we then consider a special case for the deterministic portion of demand. In this special case, we findthat retailers tendto set higher safety stocks but also charge more if the upstream supplier promises a higher buyback rate. However, if the supplier charges a higher wholesale price, retailers will not respondby decreasing retailer prices but increasing safety stocks simultaneously. For the optimal contract, we findthat both the whole-sale price andbuyback rate increase with the degree of price competition factor. Numerically, the former increases but the latter decreases with the proportion of demand spill. Both terms are quite insensitive to the coefficient of variation of the stochastic portion of demand. By comparing the system optimal contract with the supplier’s optimal contract, we findthat the contract terms become reasonably close as competi-tion intensifies. As a result, the performances of the two contracts are asymptotic to each other as compe-tition grows.

The coordination of supply chain systems has been an important focus of research in the area of opera-tions management. Whereas most research has exam-ineda single-monopoly supply chain, in this paper we have considered supply chains with downstream retailers competing in the same market. Two main insights that supply chain managers can derive from this are the following: First, the particular type of competition in the market (price competition versus inventory competition) inﬂuences the design of con-tracts between the supplier andretailers. Second, as price competition becomes ﬁercer, it is not necessary to implement the system optimal contract, because the supplier’s optimal contract will not damage the system-wide performance of the supply chain.

There are many avenues for future research. One possibility is to design a coordination contract that has nicer properties such as being free of system parameters and providing arbitrary proﬁt division. A buyback contract can coordinate a system with arbi-trary proﬁt division if the system has only stochas-tic demand, but not for a hybrid demand system with both stochastic and deterministic demand com-ponents. It will be interesting but also challenging to explore a simple contract form having those nice properties to coordinate such a complicated system.

Another possibility wouldbe to consider a sequen-tial retailer game rather than a simultaneous price andinventory retailer game. If an open-loop equilib-rium concept (Fudenberg and Tirole 1991) is used, then the results of this paper holdtrue. However, with a closed-loop equilibrium concept, the analysis will be much more complicated, if tractable. Finally, rather than assuming a ﬁxedspill rate, a further step wouldbe to see how it relates to retail price by explicitly modeling consumer switching behavior. However, this consideration could make the model intractable. In fact, our study of the marketing liter-ature (e.g., Anderson et al. 2006, Bell and Fitzsimons 2000, Campo et al. 2000) on consumer behavior after a stockout leads us to conclude that when consumers face a stockout at the retailer they chose initially, price appears to drop sharply down the hierarchy of decision criteria they use when switching to another retailer.

Acknowledgments

This research was supportedby the Natural Sciences and Engineering Research Council of Canada, Grant 312572-05. The author thanks the anonymous reviewers andthe senior editor for their constructive comments and suggestions.

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