Competing Manufacturers in a Retail Supply Chain: On Contractual Form and Coordination by Gerard Cachon and Gurhan Kok

8/3/2019 Competing Manufacturers in a Retail Supply Chain: On Contractual Form and Coordination by Gerard Cachon and Gurhan Kok

1/19

MANAGEMENT SCIENCEVol. 56, No. 3, March 2010, pp. 571589issn 0025-1909 eissn 1526-5501 10 5603 0571

informs

doi 10.1287/mnsc.1090.1122 2010 INFORMS

Competing Manufacturers in a Retail Supply Chain:

On Contractual Form and CoordinationGrard P. Cachon

The Wharton School, University of Pennsylvania, Philadelphia, Pennsylvania 19104,[email protected]

A. Grhan KkFuqua School of Business, Duke University, Durham, North Carolina 27708,

[email protected]

It is common for a retailer to sell products from competing manufacturers. How then should the firms managetheir contract negotiations? The supply chain coordination literature focuses either on a single manufacturerselling to a single retailer or one manufacturer selling to many (possibly competing) retailers. We find thatsome key conclusions from those market structures do not apply in our setting, where multiple manufacturers

sell through a single retailer. We allow the manufacturers to compete for the retailers business using one ofthree types of contracts: a wholesale-price contract, a quantity-discount contract, or a two-part tariff. It is wellknown that the latter two, more sophisticated contracts enable the manufacturer to coordinate the supply chain,thereby maximizing the profits available to the firms. More importantly, they allow the manufacturer to extractrents from the retailer, in theory allowing the manufacturer to leave the retailer with only her reservation profit.However, we show that in our market structure these two sophisticated contracts force the manufacturers tocompete more aggressively relative to when they only offer wholesale-price contracts, and this may leave themworse off and the retailer substantially better off. In other words, although in a serial supply chain a retailermay have just cause to fear quantity discounts and two-part tariffs, a retailer may actually prefer those contractswhen offered by competing manufacturers. We conclude that the properties a contractual form exhibits in aone-manufacturer supply chain may not carry over to the realistic setting in which multiple manufacturers mustcompete to sell their goods through the same retailer.

Key words : contracting; competition; retailing; wholesale-price contract; quantity discount; two-part tariffHistory : Received November 25, 2007; accepted November 6, 2009, by Ananth Iyer, operations and supply

chain management. Published online in Articles in Advance January 29, 2010.

1. IntroductionThe literature on supply chain coordination has stud-ied several contractual forms in settings with a sin-gle manufacturer and one or more retailers. One ofthe key results from this literature is that wholesale-price contracts lead to suboptimal decisions for thesupply chain (i.e., double marginalization), and moresophisticated contracts (like quantity discounts ortwo-part tariffs) can be employed by a manufacturerto achieve both supply chain coordination (i.e., max-imize the supply chains profit) and rent extraction(i.e., the ability to allocate a high share of the profits tothe manufacturer). Thus, they are viewed to work tothe advantage of the manufacturer and to the possi-

ble disadvantage of the retailer (coordination is goodfor the retailer but rent extraction is bad). Our objec-tive is to test these conclusions in a setting in whichmultiple manufacturers compete to sell their productsthrough a single retailer.

In our model, two manufacturers simultaneouslyoffer to the retailer one of three types of contracts: a

wholesale-price contract, a quantity-discount contract(i.e., a decreasing per-unit price in the quantity pur-chased), or a two-part tariff (i.e., a per-unit price anda fixed fee). We refer to the latter two as sophis-ticated contracts. The retailer determines the prod-ucts demand rates (by choosing a price for each ofthem) to maximize her total profit given the offeredcontracts and her inventory costs, which may exhibiteconomies of scale. This model represents a com-mon supply chain structure, as many manufactur-ers sell their products through retailers that also sellsimilar products from other manufacturers. A typi-cal example is Procter & Gambles Crest toothpasteversus Colgate-Palmolives Colgate toothpaste at asupermarket.

To understand how competition among multiplemanufacturers influences contracting with a retailer,consider an illustrative setting in which manufactur-ers A and B sell their products to a single retailer,as depicted in Figure 1. Focus on the supply chainformed by manufacturer A and the retailer. Without

571

notbepostedonany

otherwebsite,includingtheauthorssite.Pleasesendanyquestionsregardingthispolicytopermissions

@informs.org.

Additionalinformation,includingrights

andpermissionpolicies,isavailableathttp://journals.informs.org/.


2/19

Cachon and Kk: Competing Manufacturers in a Retail Supply Chain572 Management Science 56(3), pp. 571589, 2010 INFORMS

Figure 1 Four Profit Scenarios Holding Manufacturer Bs Contract

Fixed

9 2 17An unacceptableallocation of profit forthe retailer

Manufacturer

A

Manufacturer

B

Retailer

BA

12 8

13 5 3

Product A is not inthe assortment

Scenario

Retailers maximumprofit with A in theassortment

00

0

7 5 36An acceptableallocation of profit forthe retailer

Notes. Each boxrepresents a unit of profit.Product Asincremental value is 6

because that is the retailers maximum increase in her profit by adding prod-

uctA to the assortment. Scenario 3 is unacceptableto the retailer because the

retailer prefers Scenario 1 in which product A is not included in the assort-

ment. In Scenario 4, manufacturer A earns his incremental profit, and the

retailer is indifferent between including A in the assortment or not.

product A in the assortment, say the retailer earns12 units of profit given manufacturer Bs contract(Scenario 1). Hence, the retailers reservation profit is12 when considering As contract offer; that is, theretailer accepts As offer and add As product to theassortment only if the retailer can earn a profit of atleast 12. The supply chain is said to be coordinatedwhen their combined profit is maximized. This occurswhen A sells his product at cost to the retailerin thatcase, the retailer earns all of the supply chains profitand so chooses prices to maximize it. We refer to

this profit as the maximum profit, and in this example,say, it is 18 (Scenario 2). Finally, define product Asincremental profit as the difference between the supplychains maximum profit and the retailers reservationprofit, which in this example is 6 (18 12).

A products incremental profit is a useful con-struct because, among all possible contracts, it is theupper bound on how much a manufacturer can earn.To explain, if A were to offer a contract such thatA earned more than his incremental profit, then itmust be that the retailer is earning less than herreservation profit, and so she will reject the contract(Scenario 3). Thus, As objective is to try to earn asmuch of his incremental profit as possible. To earn allof the incremental profit requires supply chain coor-dination and substantial rent extraction (i.e., A earns6 units of profit, leaving the retailer with her reser-vation profit of 12), as in Scenario 4. This is truewhether A is a monopolist or is competing withanother manufacturer. For example, a wholesale-pricecontract does not allow a monopolist manufacturer toearn his incremental profit (because it fails on coordi-nation and has limited rent extraction), but the man-ufacturer can achieve this with a properly designed

sophisticated contract, hence the advantage of sophis-ticated contracts for a monopolist manufacturer. Infact, the same is true even when the manufacturersmust competeholding the other firms contract offerfixed, the sophisticated contracts allow a manufac-turer to capture a large fraction (possibly all) of its

incremental profit. In other words, holding Bs con-tract offer fixed, A prefers a well-designed sophisti-cated contract over the wholesale-price contract.

So what changes in the contracting processwhen manufacturers compete? The answer is thata manufacturers incremental profit is exogenouslydetermined when there is no competition andendogenously determined when there is competi-tion. Returning to our example, As incremental profitis 6 for a given contract offered by B. If B wereto offer a different contract, then As incrementalprofit could be different (possibly more, possibly less).Furthermore, we will show that As incremental profit

declines as B uses a more sophisticated contract (andvice versa), and it declines substantially when theproducts are close substitutes or if there are substan-tial economies of scale in inventory costs, or both.(This holds because adding product A to the retailersassortment increases the retailers profit more whenmanufacturer B offers a contract that does not max-imize the total profit from product B.) If they bothuse sophisticated contracts, they will both be ableto capture a large portion of their incremental prof-its (which is why they prefer these contracts hold-ing the other firms contract offer fixed), but if theproducts are close substitutes, their incremental prof-

its will be small. In contrast, if they both offerwholesale-price contracts, then they will both be ableto capture only a modest portion of their incrementalprofits, but their incremental profits will be large.In short, it can be better to have a piece of a large pie(with wholesale-price contracts) than to have all of asmall pie (with the sophisticated contracts). It followsthat when the manufacturers sell close substitutesthrough a common retailer, sophisticated contractscan work to their disadvantage, which is in sharp con-trast to the results with a single manufacturer. Putanother way, although the existing supply chain coor-dination literature suggest that a retailer should be

wary of a manufacturer offering sophisticated con-tracts (because this could lead the retailer to earnonly her reservation profit), our results with compet-ing manufacturers suggest that the retailer may pre-fer that they offer sophisticated contracts (because thesupply chain is coordinated but the manufacturershave small incremental values).

The rest of this paper is organized as follows.Section 2 reviews the relevant literature. Section 3describes the model. Section 4 presents our analysis ofthe retailers problem, 5 the analysis and comparison

notbepostedonany


@informs.org.




3/19

Cachon and Kk: Competing Manufacturers in a Retail Supply ChainManagement Science 56(3), pp. 571589, 2010 INFORMS 573

of the wholesale-price, quantity-discount, and two-part tariff games, and 6 the numerical study. Sec-tion 7 concludes this paper. All proofs are presentedin the appendix.

2. Relation to the LiteratureThe present paper is foremost a commentary onthe supply chain coordination literature. See Cachon(2003) for a review of this literature. As already dis-cussed, this literature focuses on either relationshipswith bilateral monopoly or models with one manufac-turer and multiple retailers. Wholesale-price contractsare nearly always found to be inefficient, and moresophisticated contracts can be used to eliminate thatinefficiency and reallocate rents arbitrarily betweenthe parties in the supply chain.1 There is an extensiveliterature on supply chain coordination with quantity-discount contracts and price-dependent demand (e.g.,

Jeuland and Shugan 1983, Moorthy 1987, Ingene andPerry 1995) and on lot-size coordination with fixeddemand (e.g., Monahan 1984, Corbett and de Groote2000), but they consider only one manufacturer.

Choi (1991), Trivedi (1998), Lee and Staelin (1997),and Martinez de Albeniz and Roels (2007) study sys-tems with multiple manufacturers and a commonretailer, but they only consider wholesale-price con-tracts. The literature on common agency (Bernheimand Whinston 1985, 1986) and vertical separation ineconomics is also relevant, because they study mod-els with multiple manufacturers and retailers thatcould sell more than one product. Mathewson and

Winter (1987) explore whether or not exclusive deal-ing arrangements lead to foreclosure of rivals andthe implications for antitrust laws considering onlywholesale-price contracts. Bernheim and Whinston(1998) and OBrien and Shaffer (1993, 1997) exploresimilar issues with two-part tariffs. Other variationsof the basic model include manufacturers that investin the retailers to reduce the marginal selling cost(Besanko and Perry 1994) and manufacturers thatassign exclusive territories to retailers to reduce com-petition (Rey and Stiglitz 1995). The general result isthat the manufacturers may prefer exclusive dealingdue to reduced competition at the retailer level even

though societal welfare and industry profits may behigher with common agency. This is essentially a com-parison of different supply chain structures, whereaswe are concerned with the effect of different contract

1 A wholesale-price contract can maximize profits in a system withone manufacturer and multiple quantity competing retailers. How-ever, it provides only one allocation of the systems rents, and it isnot even the manufacturers optimal wholesale price (see Cachonand Lariviere 2005). If retailers compete on price and quantity,then the wholesale price no longer guarantees coordination (seeBernstein and Federgruen 2003, 2005).

types in a given supply chain structure. Also relatedto two-part tariffs, there is a literature on slottingfees (which are essentially two-part tariffs with nega-tive payments to the manufacturer; see, for example,Marx and Shaffer 2010). Kuksov and Pazgal (2007)show that slotting fees do not occur in a setting with

simultaneous manufacturer competition and a singleretailer, and the same result applies in our model. Theliterature on strategic decentralization in marketing ishighly relevant. McGuire and Staelin (1983) study twocompeting supply chains under two structural forms:in each supply chain either the manufacturer sells toa dedicated retailer via a wholesale-price contract orthe manufacturer vertically integrates into retailing.In either structure, the products of the two manu-facturers are sold to consumers from different firms,whereas in our model the manufacturers productsare sold through a single independent retailer. Nev-ertheless, there are some similarities in our results.

McGuire and Staelin (1983) find that the manufactur-ers may prefer to sell via wholesale-price contracts,despite the fact that they do not coordinate the chan-nel nor allow the manufacturer to extract all rents,

because they dampen retail competition between thetwo products relative to the vertically integratedstructure. In our model, competition to consumers isheld constant, because we have a single retailer, sowhat changes is that the retailers reservation profitnow depends on the contract offers by the manufac-turers, enabling the retailer to gain more of the supplychain profits due to the manufacturers competition.Coughlan (1985) confirms the McGuire and Staelin

(1983) findings in an empirical study of the interna-tional semiconductor industry. In the context of theMcGuire and Staelin (1983) model, Moorthy (1988)provides conditions on the characteristics of the gameand the demand function for decentralization to be anequilibrium strategy, and Bonanno and Vickers (1988)show that decentralization is always the equilibriumwhen both manufacturers employ two-part tariffs. Ina different context with strategic consumers timingtheir purchase of a product, Desai et al. (2004) andArya and Mittendorf (2006) consider a monopolistmanufacturer and show that decentralization through

wholesale-price contracts can yield higher profits forthe channel.

3. The ModelThere are two products in the market supplied by twodifferent manufacturers. The products are partial sub-stitutes and are sold through a common retailer. In thefirst stage of the game, the manufacturers simultane-ously announce the payment schemes for their prod-ucts, i.e., their contract offers. In the second stage, theretailer chooses prices, which determine the products

notbepostedonany


@informs.org.




4/19


demand rates, to maximize her profit. In addition tothe payments to the manufacturer, the retailer incursoperating costs that depend on the average volumesold of each product. The manufacturers incur con-stant marginal production costs.

The retailer faces price sensitive customers. The rev-

enue from product i is

Rid = piddi

where di is the demand rate of product i, d is the pairof demand rates, and the inverse demand function is

pid = i idi idj and i > j > 0 for all ij

We elect to work with the inverse demand functionfor expositional simplicity. The formulation with thedemand function is equivalent to the above.

Let Gidi be the retailers inventory related opera-

tional costs of product i,

Gidi = Kidi Ki 0 0 < < 1

where Ki and are exogenous constants. This func-tional form, which exhibits economies of scale, is ageneral representation of the inventory costs that arisein common inventory replenishment models such asa base-stock model2 or an economic order quantity(EOQ) model.3

Let i denote the retailers profit from product iand = i + j, the retailers total profit. It followsthat

i = Rid Gidi Tidiand Tidi is the payment made to the manufacturer

based on the retailers demand rate and their agreed-upon contract.4 Manufacturer is profit is

i = Tidi cidi (1)

where ci is the manufacturers cost per unit.

2 In a periodic review model where demand follows a Normal dis-tribution with mean di and standard deviation id

i , the total inven-

tory related costs with the optimal base-stock level is given by

b + hzidi , where b is the backlog penalty per unit and h is

the inventory holding cost per period. Defining Ki = b + hzileads to the Gi function.3 In the EOQ model, the retailer incurs a fixed cost ki per orderand a holding cost hi per unit of inventory held for one period.The well-known EOQ formula suggests ordering every

2ki/hidi

periods. The resulting optimal total inventory and ordering cost isgiven by

2kidihi. Defining Ki =

2kihi leads to the Gi function

with = 1/2.4 The retailers payment Tidi to a manufacturer can be interpretedas a yearly (average) payment based on yearly (average) volume.Many manufacturerretailer purchasing contracts are based on theyearly volume rather than on the volume of individual shipments.

We focus on three different types of contracts:wholesale-price contracts, quantity-discount con-tracts, and two-part tariffs. In the first stage, the man-ufacturers strategy set includes one or more of thesecontractual forms. For expositional clarity, we beginwith a limited strategy space (e.g., just wholesale-

price contracts, or just quantity-discount contracts,etc.) and then later consider a broader strategy space.With a wholesale-price contract, the payment

function isTidi = widi

where wi is the wholesale price chosen by manufac-turer i. Wholesale-price contracts are common in prac-tice and serve as a theoretical benchmark.

Quantity discounts are common in many industries(see Munson and Rosenblatt 1998 for a fieldstudy). We consider the following family of quantitydiscounts:

Tidi =

widi vid2i /2 if di wi ci/viTiwi ci/vi + cidi wi ci/vi

otherwise,

(2)

where wi and vi are parameters, and vi 0 v andv = 2i i + j. Note that wholesale-price con-tracts are a subset of these quantity discountsawholesale-price contract can be obtained by settingvi = 0. Although we only consider a subset of possi-

ble quantity discounts, this is not overly restrictive.Our quantity discounts are continuous, differentiable,concave, and the manufacturer does not sell even the

marginal unit for less than its production cost (whichis the reason for the breakpoint at wi ci/vi. Theupper bound on vi implies that the quantity discountis not too aggressive in the sense that the marginalprice paid does not fall too rapidly as the purchasevolume increases, i.e., Ti di vi. In fact, it can beshown that our quantity discounts are optimal for themanufacturer (holding the other manufacturers con-tract offer fixed) among all concave and increasingpayment functions given the Ti di vi constraint(see Proposition 2 in the appendix). Furthermore,Ti di vi ensures that the retailers profit func-tion excluding the operational costs is strictly con-

cave in d1 d2. This naturally raises the question ofwhether the manufacturer could do better by offer-ing an even more aggressive quantity discount. In asingle-product environment, the answer is no: as viapproaches v, the supply chains profit is maximizedand the manufacturer earns all of that profit. In a two-product environment, the answer is not clear becauseretail profits are no longer strictly concave with vi v.However, the next contract we describe can be inter-preted as the most aggressive quantity discount, andwe do have results for those contracts.

notbepostedonany


@informs.org.




5/19


Our third contract type is the two-part tariff, whichis characterized by a fixed fee Fi and a marginalcost wi:

Tidi = Fi1di>0 + widi (3)where indicator function 1di>0 = 1 if di > 00 otherwise. The two-part tariff is an aggressive

quantity-discount contract because the marginal costof the first unit is Fi, whereas the marginal cost of allsubsequent units is only wi (which is generally muchsmaller than Fi.

In a symmetric game across manufacturers, the datafor the two products are identical, i.e., ci, i, i, i, Ki,and vi are the same for any i. The subscript i will bedropped in those cases. In a symmetric solution, thedecisions (di at the retail level, wi or Ti at the manu-facturer level) are identical across products.

In the following sections, we solve the prob-lem using backward induction. We analyze theretailers decision first and then the game between the

manufacturers.

4. The Retailers DecisionThis section studies the retailers quantity decisionsgiven the two contract offers from the manufacturers.We present the analysis assuming the manufacturersoffer the retailer quantity discounts, which is suffi-cient to understand the retailers decision under anyof the three contract forms we consider: wholesale-price contracts are quantity discounts with vi = 0,and, from the perspective of choosing demand ratesfor the products, two-part tariffs are identical to

wholesale-price contracts (because the fixed fees donot matter for the retailers pricing decisions given anassortment).

Define

didj = arg maxdi

idi dj (4)

di dj = arg maxdidj

di dj didj > 0 (5)

= di dj (6)where didj is the retailers optimal demand for prod-

uct i given a demand choice for product j,

di

dj

is the pair of optimal demands conditional that eachproduct is included in the assortment (with some pos-itive sales), and is the retailers resulting profit.

The retailers optimization problem can now bewritten as

max 1d10 0 20 d20 (7)Hence, the optimal solution to the retailers prob-lem is

d1 d2 d1 d2d10 0 0 d20

For expositional simplicity, we assume that theretailer breaks ties in favor of carrying a full productline over a single product.

Consider the problem of maximizing the systemsprofit (i.e., the total profit across the three firms). It isequivalent to the retailers problem if the manufactur-

ers charge only their production cost, Tidi = cidi fori = 1 2. Define 1 = 1d10 0, 2 = 20 d20, and12 = d1 d2 under those contracts. These profit lev-els are, respectively, the maximum profit the systemwould earn if it were to carry only product 1, onlyproduct 2, and both products. We assume

12 > i > 0 for i = 1 2 (8)which implies that it is always optimal for the systemto carry both products.

Holding manufacturer 2s contract offer fixed, theretailers reservation profit for accepting manufac-turer 1s offer is 20 d20, i.e., the retailer acceptsmanufacturer 1s offer only if adding product 1 to theassortment increases her profit over what she couldearn without it in the assortment. To evaluate manu-facturer 1s incremental profit, assume manufacturer 1charges only his production costs, T1d1 = c1d1 whileholding manufacturer 2s offer fixed. In that case,manufacturer 1s incremental profit is

max 1d10 0 20 d20where the first term is the most the product 1 supplychain can earn (conditional on product 2s contract),and the second term is the retailers reservation profit.

Now let us consider the retailers demand decisionwith independent manufacturers. Let Hi denote thefirst derivative of with respect to di:

Hi = /di = i 2idi i + jdj Gidi Ti dij= i (9)

Consider the case with no economies of scale (i.e.,K1 = K2 = 0). The retailers profit function is jointlyconcave in d1 d2, so the unique solution to Hi = 0i = 1 2 is the unique optimal solution. If an inte-rior solution does not exist, then the optimal solu-tion is either d10 0 or 0 d20, where di0 is theunique solution to Hi = 0 with dj = 0. Because iswell behaved, the optimal solution d1 d

2 can be eas-

ily characterized, and it is a continuous, differentiablefunction of the problem inputs such as the parametersof the manufacturers contracts.

The case with economies of scale, however, is rathercomplicated. Observe that 2/d2i = 2i Gi di Ti di is positive at di = 0 and then decreasing in diThus, is convex-concave in di for fixed dj. There areup to two solutions to Hi = 0 and the larger of the twosolutions is a local maximum. The corner solution,

notbepostedonany


@informs.org.




6/19


di0, is either the larger solution or zero. Evidently, is not jointly concave in di dj. As a result, theremay be multiple solutions to Hi = 0 i = 1 2, andwe do not know which one could be the interior opti-mal solution. Furthermore, the global optimal solu-tion may be at one of the boundaries di = 0. A final

technical note is that the profit function is not neces-sarily unimodal (in one or two dimensions).The next theorem shows that there can be at most

one interior local maximum and that the optimal solu-tion is either that interior solution or at one of the

boundary lines; that is, there are at most three candi-date optimal solutions and each is characterized by aset of first-order conditions. Furthermore, in a sym-metric problem, the unique interior maximum is asymmetric solution.

Theorem 1. The retailers optimal solution is d1 d2

d1 d2d10 0 0 d20, where d1 d2 is theunique interior optimal solution to Hi = 0 i = 1 2, anddi0 is given by the larger of the two solutions to Hi = 0st dj = 0. In a symmetric problem, d1 d2 = d d,where d is the larger of the two solutions to

2 + d Gd Td = 0In summary, there are at most three local maxima

for a retailers problem: one interior solution in whichthe retailer carries both products, and two bound-ary solutions in which the retailer carries only oneproduct.

Although the retailers profit function is generallycomplex in the presence of economies of scale, thefollowing conditions ensure that it is jointly concave

in d1 d2. We state and prove this result in Lemma 1in the appendix.

1 = 2 = 1 = 2 = (10)Ti di vi where 0 vi < v = (11)

2Ri/Gi ii for all i

where ii = j

ij ijpidi

(12)

Condition (10) requires the own- and cross-pricecoefficients to be symmetric. This is not a very restric-tive assumption, because we allow nonidentical i,

which implies different demand rates and price elas-ticities for the products. Condition (11) is stricter thanour earlier assumption: it requires the quantity dis-count to be less concave to guarantee the concavity ofthe retailers profit function. Condition (12) stipulatesthat the own-price elasticity (ii is less than two timesthe revenuestoaverage inventory costs ratio of theproduct. (It is similar to the conditions Bernstein andFedergruen (2003) developed for decentralized retail-ers, and as they point out, the conditions hold formost retailers based on the industry data in Dun andBradstreet (2006) and Tellis (1988).)

5. Competition Betweenthe Manufacturers

In this section, we study the game between the manu-facturers in contract offers. We begin with some obser-vations that apply no matter what type of contract isoffered. In 5.1, we characterize the games equilibria

when both manufacturers choose quantity-discountcontracts, and 5.2 provides the analogous analysiswhen the manufacturers choose two-part tariffs. Sec-tion 5.3 expands the manufacturers strategy set toinclude all three contract types that we consider. Sec-tion 5.4 compares the equilibria under different con-tract types for the more analytically tractable caseof symmetric products and no economies of scale ininventory costs.

In the game between the manufacturers, each onechooses his own best response TiTj given the othermanufacturers contract offer Tj:

TiTj = argmaxTidi

id for all i

where d = argmax (13)

and it is understood that Tidi is chosen from adefined strategy space (e.g., quantity-discount con-tracts with a fixed v An equilibrium of the game is apair of contracts Ti T

j such that neither manufac-

turer has an incentive to offer a different contract.The following remark demonstrates how the con-

tracting problem with multiple manufacturers is dif-ferent from that with a single manufacturer.

Remark 1. For any fixed contract offered by manufac-turer 2 such that 20 d20 > 0:

1. Consider the set of contracts such that the retailerspayment to manufacturer 1 is a nondecreasing function ofd1. There does not exist a contract in this set such that themanufacturer can extract all of the profit from his product(i.e., it is not possible to have 1 > 0 and 1 = 0).

2. The retailer accepts manufacturer 1s contract offerand stocks both products only if

1di dj + 2di dj 20 d20 (14)

Unlike in a supply chain with a single manufac-

turer, the first statement implies that a manufacturermust leave the retailer with some profit to induce theretailer to carry the manufacturers product (see theappendix for a detailed proof). However, this doesnot mean that a single reservation profit exists. Asdescribed in 1, the right-hand side of (14) is theretailers reservation profit with respect to manufac-turer 1s contract offer, and it depends on the partic-ular contract offered by manufacturer 2. Hence, theredoes not exist a single reservation profitthe reserva-tion profit is endogenously determined by the actions

notbepostedonany


@informs.org.




7/19


of the other manufacturer. Furthermore, it is not pos-sible to replicate the dynamics of this model witha carefully designed serial supply chain and a fixedreservation profit. To explain, via a simple manipula-tion of (14), the retailers reservation profit for prod-uct 1 alone is the right-hand side of

1di dj 20 d20 2di djwhich depends on manufacturer 1s contract offer(even if the other contract is held fixed), and thereforecannot be represented by a single value.

Before considering the equilibrium contract offers,we add an observation that rules out equilibria inwhich the retailer carries only one product in sym-metric games.

Remark 2. In a symmetric game between the manufac-turers, there does not exist an equilibrium where di = 0 forsome i.

The result is due to (8), which guarantees that the

inclusion of a manufacturer strictly increases systemprofit. Suppose there were an equilibrium in whichmanufacturer i is excluded. Regardless of the fractionof j the retailer earns, manufacturer i can offer to sellto the retailer at ci + for an arbitrarily small , andthen the retailers profit increases if it carries prod-uct i. If j is excluded as a result, it will react similarlyand get its product included.

5.1. Wholesale-Price and Quantity-DiscountContracts

In this section, we consider the game between themanufacturers in which they make quantity-discount

contract offers with a given vi = vj. Hence, thetwo manufacturers simultaneously announce their wiparameter to the retailer, then the retailer decideshow much to sell of each product. Recall thatwholesale-price contracts have v1 = v2 = 0, and theyare a special case of this analysis. In the pres-ence of economies of scale (i.e., if Ki > 0 for anyi, we assume that conditions (10) and (11) hold,and we restrict our attention to a region defined

by (12). Define wiwj = maxwi dj wi wj = 0,the maximum wi that makes the retailer exclude prod-uct j, and wiwj = minwi di wi wj = 0, the min-imum wholesale price of i that makes the retailer

exclude product i. Note that wiwj may not exist forevery wj. In that case, set wiwj = ci. Define wiwjas the best response of manufacturer i, which can befound via a line search between wi wi.

The following theorem characterizes the uniquesymmetric equilibrium of the contract offer game.

Theorem 2. Consider a symmetric game in which themanufacturers offer quantity discounts. There exists aunique symmetric solution to

di w1 w2 + wi vi cidiwi

= 0 for all i

denoted w1 w2 , which is the unique candidate to be a

symmetric equilibrium.

As can be seen in the proof Theorem 2, showing theunimodality of a manufacturers profit in wi requiresthe use of the second-order properties of the retailers

optimal solution d1 d2 . In the presence of economiesof scale, we have shown that the retailers decisionproblem is complex (e.g., not concave, not unimodal),and this creates significant challenges to the anal-ysis of the game between the manufactures, whichis built on top of the retailers decision problem.These challenges are not present in other competitionpapers in the literature.

We cannot guarantee that the candidate pointdescribed in Theorem 2 is an equilibrium. We showthat at w1 w

2 , w

i is a local optimum for manufac-

turer i, and i is concave for wi > wi . However, the

optimal solution wiwj may be different than w

i in

the range wi wi . If wiwj = wi , then w1 w2 isindeed an equilibrium point. If not, then there existsno symmetric equilibrium.

Now consider the situation in which there are noeconomies of scale. This substantially simplifies theanalysis of both the retailers demand decisions andthe manufacturers contract offer problem. For anyv1 v2 and asymmetric products, we can now guar-antee joint concavity of the retailers profit and theexistence and the uniqueness of the equilibrium with-out the symmetry assumptions. The next theoremprovides the closed form solutions for the demand

rates and the contract parameters by solving the first-order conditions given in Theorem 2.

Theorem 3. With no economies of scale (i.e., Ki = 0),there exists a unique equilibrium of the game in which themanufacturers offer quantity discounts. It is characterizedby the following reaction functions and the optimal demandrates:

di =2jvji wii +jjwj

2i vi2jvji +j2

wi wj=

ji i +jjwj+vij+cijj2+vij

where j2jvj and 2i vi2jvji +j2.

5.2. Two-Part TariffsIn this section, we consider the game between manu-facturers who offer two-part tariff contracts.

Theorem 4. The equilibrium of the two-part tariff gameis Fi = 12 j and wi = ci for i = 1 2, j= 3 i. Hence,i = 12 j and = 1 + 2 12.

notbepostedonany


@informs.org.




8/19


In the two-part tariff equilibrium, each manu-facturer charges wi = ci so that the retailer maxi-mizes their combined profits. The manufacturer thenextracts his full incremental profit via its fixed fee,and the retailer earns her reservation profit on eachproduct. However, this does not mean that the man-

ufacturers earn a large profit or even more than theretailer. If the two products are close substitutes, theneach manufacturers incremental profit, 12 j, can

be quite small. To explain, imagine the products wereperfect substitutes. In that case, the retailer can earnas much from selling just one as she can earn fromselling both, i.e., 12 = 1 = 2, and each manufac-turers incremental profit is then zero, leaving themanufacturers with zero profit and the retailer with1 > 0. This logic extends to the case of economiesof scale. As economies of scale become stronger, 12decreases relative to 1 or 2: selling both prod-ucts is relatively less attractive than selling just one

product when economies of scale increase becauseit becomes costly to fragment demand across mul-tiple products. Hence, an increase in economies ofscale should decrease each manufacturers incremen-tal profit, thereby resulting in lower profits for themand higher profits (in a relative sense) for the retailer.

5.3. Contract ChoiceIn the previous sections, we assumed that themanufacturers offered contracts from a limited set:wholesale-price contracts, quantity discounts witha fixed v parameter, or two-part tariffs. This sec-tion considers the manufacturers equilibrium contract

choice when the set of contracts available to them isexpanded.

Suppose the manufacturers could choose any con-tract type to offer the retailer, i.e., the unrestrictedcontract offer game. According to the next corollary(whose proof is actually part of the proof of Theo-rem 4), restricting the firms to the set of two-part tar-iffs does not actually change the outcome of the game.

Corollary 1. Consider the contract choice game inwhich the manufacturers are free to choose any contracttype and parameters. It is optimal for each firm to offera two-part tariff with wi = ci, no matter what contractoffer the other firm makes. Furthermore, the equilibrium ofthe two-part tariff game described in Theorem 4 is also anequilibrium of the unrestricted contract choice game.

Now suppose that two-part tariffs are not in thecontract space and the manufacturers simultaneouslyoffer a quantity discount wi vi where they are freeto choose any vi 0 v for some v < v. In otherwords, they can choose to offer a wholesale-price con-tract (vi = 0 or a quantity discount (vi > 0 Thenext proposition indicates, as in supply chains witha single manufacturer, that a manufacturer prefers to

offer a quantity-discount contract and, in particular,prefers more aggressive quantity discounts. Quantitydiscounts allow the manufacturer to improve supplychain coordination (i.e., reduce double marginaliza-tion) and to extract rents, so they are the preferred con-tract when the other manufacturers contract is held

fixed even when the retailer can adjust her demandallocations between the two products in response.

Proposition 1. In the quantity-discount game wherethe manufacturers can choose from among the set of quan-tity discounts, manufacturer is profit strictly increaseswith vi at the optimal wholesale price w

i ; that is, if a

manufacturer is given the option to choose between threecontractual forms with vi 0ab such that 0 < a < bthen iw

i vi = 0 < iwi vi = a < iwi vi = b.

The immediate implication of this proposition isthat if the manufacturers are able to choose thequadratic parameter of their quantity discount as well

as the starting wholesale price, each manufacturersbest response is to choose the most aggressive con-tract in our consideration set. The next corollary statesthat any equilibrium of this game would then havethe most aggressive contracts by both manufacturers.

Corollary 2. In the contract choice game in which themanufacturers simultaneously offer wi vi and they are

free to choose any vi 0 v for some v < v the equilib-rium contracts are such that vi = v for both manufacturers.

To summarize, in equilibrium, the manufacturerschoose the most aggressive contract in their strategyset, be it a two-part tariff or a quantity discount.

5.4. Comparison of EquilibriaThe previous section established that if the manufac-turers are free to choose any contract they want tooffer the retailer, in equilibrium they choose to offertwo-part tariffs. We also characterized the equilib-rium contract offers when the manufacturers strategyset is restricted to only wholesale-price contracts oronly quantity-discount contracts. In this section, wecompare these various outcomes. To provide analyti-cal tractability, we consider a system with symmetricproducts (i.e., identical i, i, i, ci, and vi and noeconomies of scale at the retailer i.e., Ki

=0 Fur-

thermore, among the set of quantity discounts, weprovide results in the limit as v goes to 2 , theupper bound. (Recall that Corollary 2 indicates thatthe manufacturers would choose those quantity dis-counts if they are allowed to do so.)

Table 1 presents the equilibrium profit terms underthe three contract types. The derivation of the profitexpressions are presented under Theorem 7 in theappendix.

As indicated earlier, two-part tariffs maximize thesystems combined profits, and Table 1 reveals that

notbepostedonany


@informs.org.




9/19


Table 1 Equilibrium Profits Under Wholesale-Price,

Quantity-Discount, and Two-Part Tariff Contracts in the Case

with Symmetric Products and No Economies of Scale

Manufacturer Retailer Supply chain

Contract i + 1 + 2

Wholesale price

c2

2 + 2 2

c22

2 + 2 2

c23

2

2 + 2 2

Quantity discount c2

4 + 2 c2 + 2

c22 +

as v 2

Two-part tariff c2

4 + c22 +

c22 +

our chosen quantity discounts do as well (this resultdoes not generalize to other quantity-discount con-tracts). Nevertheless, the manufacturers are not ableto extract their full incremental profit with quantitydiscounts (they earn less with those contracts than

the two-part tariff), which implies the retailers is bet-ter off with quantity discounts than with two-parttariffs. Furthermore, it is straightforward to show thatthe wholesale-price contract fails to maximize the sys-tems combined profits.

Suppose the products are perfect substitutes, i.e.,= . In this case, the incremental profit of each man-ufacturer is zero under all contract types. Therefore,at equilibrium, the manufacturers offer marginal costpricing, resulting in system optimal profits, and theretailer captures all the profits; that is, all contracts areequivalent for all players.

When products are independent, i.e.,

=0, the sys-

tem is equivalent to two independent supply chains,and each manufacturer is able to achieve supplychain optimal profit and extract all of it either withquantity discounts or two-part tariffs, but not withwholesale-price contracts. This case replicates thecommon assumption in the supply chain contract-ing literature with a monopolist manufacturer. If isreduced further, < 0 then the products are com-plements. Now, 1 + 2 < 12 (i.e., selling both prod-ucts together earns more than the sum of selling themindividually). It follows that the retailer has no lever-age over the manufacturers, and she earns zero profit,

just as in the = 0 case. Theorem 9 in the appendixprovides further details for the equilibrium with com-plementary products.

Now we turn attention to the interesting situa-tion with imperfect substitutes, 0 . The nexttheorem compares the manufacturers profits withwholesale-price contracts to the other two types.

Theorem 5 (Manufacturers Perspective). (i) Themanufacturers profits are higher under quantity discountsthan wholesale-price contracts if and only if < 3 7 0354.

(ii) The manufacturers profits are higher under two- part tariffs than wholesale-price contracts if and only if < 2 2 0586.

Theorem 5 indicates that the manufacturers are bet-ter off with the sophisticated contracts only if theproducts are not close substitutes, and this effect

is stronger for the quantity discounts (in the sensethat quantity discounts are preferred over a nar-rower parameter range). Note, holding the othermanufacturers contract offer fixed, a manufactureralways prefers a quantity-discount contract over thewholesale-price contract and a two-part tariff overthe quantity discount. Hence, although they wouldchoose two-part tariffs if they are not restricted intheir contract choice, they would be better off hadthey been restricted to offer only wholesale-price con-tracts. In effect, the contract choice game is like theclassic prisoners dilemma.

The manufacturers may be worse off in equilibriumwith the sophisticated contracts because the degree ofproduct substitutability, , influences both a contractsincremental profit as well as the share of that profitthat the manufacturer can extract from the retailer. Toillustrate this explicitly, the following are the incre-mental profit of a manufacturer under the three con-tract types given the equilibrium contract offer fromthe competitor (derivations are presented in Theo-rem 8 in the appendix):

IPW = c22

2 + 2 2

IPQ = IPT = c2

4 +

where the superscripts W, Q, and T denote thewholesale-price, quantity-discount, and two-part tar-iff contracts, respectively. It is straightforward to showthat the wholesale-price contract has the highest incre-mental profit: IPW > IPQ for all 0 . Thisholds because adding product A to the retailersassortment increases the retailers profit more whenmanufacturer B offers a contract that does not max-imize the total profit from product B (a wholesale-price contract) relative to when manufacturer B offers

a contract that does maximize product Bs profit (thetwo sophisticated contracts). It is also worth notingthat the wholesale-price contracts relative advantagein incremental profit increases as the products becomemore substitutable ( increases):

IPW

IPQ= 4

2

2 2 > 1

The drawback of the wholesale-price contract is thatit captures only a fraction of its incremental profit,one-half of it to be precise, for all . Even the

notbepostedonany


@informs.org.




10/19


quantity-discount contract extracts only a fraction ofits incremental profit, / + , but it is greater thanone-half in the range of interest (though, it is alsodecreasing in ). Putting these results together, ifproducts are close substitutes, half of a large incre-mental profit (under wholesale-price contracts) may

be larger than a larger share of a smaller incrementalprofit (under sophisticated contracts).Now, let us consider the problem from the retailers

perspective. The next theorem compares the retailersequilibrium profits under the three contract types.

Theorem 6 (Retailers Perspective). (i) Theretailers profit is higher under quantity discounts thanwholesale-price contracts if and only if > 1

23

7

0177.(ii) The retailers profit is higher under two-part

tariffs than wholesale-price contracts if and only if > 1

23 5 0382.

With a single manufacturer, the retailer is worse offwith quantity discounts and two-part tariffs becausethey enable the manufacturer to extract rents from theretailer, leaving her with her reservation profit (possi-

bly zero). However, when the manufacturers compete,the outcome is quite different. Now the retailer is bet-ter off with the more sophisticated contracts when-ever the products are sufficiently close substitutes.These more sophisticated contracts increase the sys-tems total profit and as long as the products are closesubstitutes; the lions share of that profit goes to theretailer (because the manufacturers incremental prof-its are small).

Figure 2 illustrates under which contract equilib-rium the manufacturers and the retailer are better offfor different levels of substitutability. As already dis-cussed, for high values of , the retailer prefers thesophisticated contracts because they lead to low incre-mental profits for the manufacturers, and the man-ufacturers have the opposite preferencealthough

Figure 2 Equilibrium Preferences of the Firms for Different Levels of Substitutability

Wholesale-price vs. quantity-discount contracts

Wholesale-price contracts vs. two-part tariffs

< (0.586) > (0.586)

< (0.382) > (0.382)

Manufacturers

Retailer

Two-part tariff Wholesale price

Wholesale price Two-part tariff

< (0.354) > (0.354)

< (0.177) > (0.177)

Manufacturers

Retailer

Quantity discount Wholesale price

Wholesale

priceQuantity discount

total rents in the system are lower with wholesale-price contracts and those contracts cannot extractrents as efficiently (they earn only a fraction oftheir incremental profit), when is high they pre-fer the wholesale-price contract. When is low, themanufacturers prefer the rent extracting ability of

the sophisticated contracts because their incremen-tal profits remain reasonably high. The retailer isworse off with the sophisticated contracts becausethe manufacturers extract a large chunk of the sys-tems profit. Hence, for low and high levels of prod-uct substitutability, the manufacturers preference isthe opposite of the retailers preference. However, forintermediate values of , they both prefer the sophis-ticated contracts. As with all , the sophisticatedcontracts increase the systems total profit (i.e., coor-dinates the system), but now both the manufacturersand the retailer are better off because the manufactur-ers incremental profits are neither too high (as with

a low ) nor too low (as with a high ).As mentioned, the manufacturers face the clas-

sic prisoners dilemma when the products are closesubstitutes. In a single-period game context, theyalways offer the more sophisticated contracts at equi-librium, but they prefer the system in which they bothoffer wholesale-price contracts. In a repeated gamesetting, the Folk Theorem suggests that they couldtheoretically coordinate (on offering wholesale-pricecontracts) using trigger-type punishment strategies.(This requires the ability to observe the competitorscontract type, which may be theoretically inferredfrom the retail prices.) In this case, the retailer mayoffer side payments to bring the manufacturers tothe system-efficient outcome (sophisticated contracts).When products are not so close substitutes, the equi-librium under sophisticated contracts is preferred bythe manufacturers, but not by the retailer. In practice,the bargaining powers of the firms in the negotiation

notbepostedonany


@informs.org.




11/19


process determine which of these outcomes will beobserved.

Although our findings bear a resemblance to thosein McGuire and Staelin (1983), there are some impor-tant distinctions. In McGuire and Staelin (1983),two manufacturers sell through dedicated retailers

(i.e., the retailers only carry one of the manufactur-ers products). As in our model, the manufacturersmay prefer the equilibrium with wholesale-price con-tracts relative to the equilibrium under vertical inte-gration, which can be achieved with two-part tariffsor quantity discounts. But the mechanism for thisresult is differentin their model, wholesale-pricecontracts dampen competition between the retailers,which indirectly dampens competition between themanufacturers, whereas in our model wholesale-pricecontracts directly influence the competition betweenthe manufacturers while leaving constant the indi-rect competition they face with consumers. In addi-

tion, in their model, the retailers always prefer thewholesale-price equilibrium because two-part tariffsalways leave them with no profit, whereas in ourmodel the retailer may prefer the two-part tariffequilibrium.

6. Numerical StudyThis section presents a numerical study that comparesthe equilibrium solutions when the manufacturersoffer wholesale-price, quantity-discount, and two-parttariff contracts. Our primary goals are to study sce-narios with economies of scale (K > 0) and to measure

the magnitude of the effects discussed earlier.

Table 2 Average Percentage Change in the System, Retailer, and Manufacturer Profits in Equilibrium Under Quantity-Discount and

Two-Part Tariff Contracts Relative to the Equilibrium Under Wholesale-Price Contracts

K= 0 K= 1 K= 3Quantity discount (%) Quantity discount (%) Quantity discount (%)

Two-part Two-part Two-part

Change in profit = 05 = 095 = 18 tariff (%) = 05 = 095 tariff (%) = 05 = 095 tariff (%)

= 025 System 57 112 213 225 58 113 226 60 118 229

Retailer 33 60 147 234 32 59 201 33 61 112Manufacturers 73 146 258 531 75 148 502 77 152 436

Scenarios 0 0 0 0 0 0 0 0 0 0

manu. lose (%)

= 05 System 35 68 120 125 36 69 124 37 71 121

Retailer 62 138 385 125 66 147 163 71 158 256

Manufacturers 08 02 145 125 07 06 86 05 12 03Scenarios 0 100 100 0 0 100 0 0 100 417

manu. lose (%)

= 075 System 14 26 41 42 14 26 39 14 26 34

Retailer 59 124 281 173 61 129 195 63 131 230

Manufacturers 75 169 442 219 82 186 279 86 193 398Scenarios 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

manu. lose (%)

Note. manu., Manufacturers.

There are 108 scenarios formed by all combinationsof the following parameters: = 20 40, = 1 2 4, = 025 05 075 , c = 1 3, K = 0 1 3, and = 05. With each scenario, we searched for an equi-librium under wholesale-price, quantity-discount, andtwo-part tariff contracts. With the K = 0 scenarios,we included v = 0 05 095 18 , and withthe K > 0 scenarios, we included v = 0 05 095 . In total there are 360 scenario/contract com-

binations. In 8 of the 108 scenarios, we were unableto find an equilibrium for at least one of the con-tracts (including five scenarios with wholesale-pricecontracts). The best response functions in those sce-narios reveal that the effect of economies of scaleis very strong at the retailer, and the manufacturerscycle between undercutting prices to get the retailerto exclude the other manufacturer and being under-cut. In those scenarios, there also does not exist anasymmetric equilibrium. We report results for the

remaining 100 scenarios. As a validity check, we com-pared the average cost per unit that the retailer paysto the manufacturers, Td/d with wholesale-priceand quantity-discount contracts: the average cost is 5%lower in the quantity-discount equilibrium than thewholesale-price equilibrium when v = 05 , and11% lower when v = 095 . These results indicatethat in these scenarios the quantity discounts in equilib-rium are modest.

Table 2 compares the firms profits underwholesale-price contracts to the other contracts. Theresults in the table expand upon our intuition devel-oped analytically in the previous section with K

=0

and the most aggressive quantity discount (v at its

notbepostedonany


@informs.org.




12/19


upper bound). For any quantity discount (v is fixed),the manufacturers are better off if product substi-tutability, , is low, otherwise they are worse off.On the other hand, the percentage increase in theretailers profit is not monotonic in : the retailergains the most with an intermediate level of product

substitutability. When is too low, the retailer doesnot gain as much because the manufacturers are ableto extract a large portion of the systems increase inprofit. When is too high, the retailer does not gainas much because there is less to gain for everyone (i.e.,the systems profit does not increase by a substantialamount). Interestingly, even with a low , the retail-ers may earn a higher profit with quantity discountsthan with wholesale-price contractsbecause the sys-tem becomes much more efficient, there is more profitavailable to share even if they are unable to gain alarge fraction of it. Furthermore, these patterns areaccentuated as v increases or as K increases (the shifts

in profits become more dramatic as the manufac-turers use more aggressive quantity discounts or ifeconomies of scale become more pronounced).

According to Table 2, as with the quantity dis-counts, with two-part tariffs the manufacturers aresignificantly better off when product substitutabilityis low and substantially worse off when is high.However, the retailer is generally better off as increases (K = 3 is the only exception). In sharp con-trast to the case with quantity discounts, when islow, the retailers are worse off with two-part tariffsthe systems profit is increased substantially but therent extraction capability of two-part tariffs leaves the

retailer with even less than she had with the whole-sale price contracts.

In summary, our results are consistent with our con-jectures that more sophisticated contracts (i) increasethe total profit of the system, thereby making it pos-sible for all firms to gain, and (ii) lower the manu-facturers incremental profit, especially as the prod-ucts become more substitutable ( increases) or aseconomies of scale increase, or both, and (iii) thatproduct substitutability and economies of scale influ-ence the fraction of a manufacturers incrementalprofit that he can extract from the retailer. These threefactors combine to create complex dynamics. Never-

theless, we observe that the manufacturers are gener-ally worse off when sophisticated contracts are usedand product substitutability is high. The retailer has

just cause to fear two-part tariffs when product substi-tutability is low and otherwise can gain substantiallyfrom them.

7. DiscussionThis paper studies competing manufacturers whosell their products through a single retailer. In someways this supply chain structure is similar to the

commonly studied supply chain structure with asingle manufacturer. For example, the manufacturercan use sophisticated contracts (quantity discount ortwo-part tariffs) to (i) increase the systems prof-its relative to wholesale-price contracts (i.e., supplychain coordination) and (ii) extract a large fraction of

the manufacturers incremental profit. Consequently,in equilibrium, the manufacturers choose the mostaggressive contract available to them (aggressivein the sense that the retailers marginal cost fallsrapidly), just as a single manufacturer would. The keydifference is that, unlike the exogenous incrementalprofit with a single manufacturer, their incrementalprofits are endogenous with competition. Further-more, the manufacturers incremental profits decreaseas they use more sophisticated contracts, especiallywhen the products are close substitutes or there areeconomies of scale in retailing. Hence, while quantitydiscounts and two-part tariffs can work to the advan-

tage of a single manufacturer, they can work to thedisadvantage of competing manufacturers.

Our results provide guidance to firms as they con-sider the contractual form they wish to negotiate.Unlike when dealing with only a single manufacturer,a retailer may wish to encourage competing man-ufacturers to offer sophisticated contracts. A manu-facturer must be careful when choosing to offer asophisticated contract. If the industry standard is cur-rently wholesale-price contracts, then a manufacturercan gain from offering a sophisticated contract, but ifa competing manufacturer responds, then the ensu-ing sequence of offers and counteroffers can lead

them to an equilibrium in which they are worseoff if products are close substitutes. In that case,the retailer may induce the manufacturers to offerthe sophisticated contracts by using side payments.And conversely, the manufacturers may induce theretailer to accept sophisticated contracts when sub-stitutability is low. Furthermore, our results reinforcethe importance to manufacturers of investing to dis-tinguish their productsmanufacturers with strong

brands (i.e., products that are not easily substitutablewith a competitors product) can benefit from the useof sophisticated contracts. Manufacturers also benefitwhen operational improvements are made in the sup-ply chain to reduce the economies of scale associatedwith retailing.

We consider a diverse set of contracts, but othercontracts have been observed in practiced and stud-ied in the literature, such as buy backs, quantityflexibility, and revenue-sharing contracts. These con-tracts, like quantity discounts and two-part tariffs, aredesigned to improve supply chain coordination andextract rents. Consequently, we conjecture that evenwith those contracts, the concept of an incrementalprofit holds along with our qualitative results.

notbepostedonany


@informs.org.




13/19


Our findings also yield empirically testable hy-potheses. For example, the prevalence of quantity-discount contracts could be correlated with the substi-tutability of products within a product category andthe power of retailers and manufacturers to dictatecontractual forms. There is anecdotal evidence from

our conversations with executives from grocery retailchains (H-E-B in Texas and Stop & Shop in the north-eastern United States) that, in most categories, mostfirms offer the same type of contracts and the retailersmake most of their money from incentives in cate-gories with low brand loyalty (i.e., high substitutabil-ity). Buzzell et al. (1990) suggest that as major retailersgrow and increase their relative power, they use it toextract concessions from suppliers such as trade deals,merchandising support, and slotting allowances in theconsumer packaged goods industry. These observa-tions have the same tone with our main result thatretailers stand to gain more from the use of more

sophisticated contracts.

AcknowledgmentsThe authors thank Awi Federgruen, Leslie Marx, RichardStaelin, and the anonymous review team for their helpfulcomments.

Appendix. ProofsProof of Theorem 1. The proof holds for general quan-

tity discount contracts that satisfy

Ti0 = 0 Ti d ci vi Ti d 0Td 0 Td 0

(15)

The quantity-discount contract given by (2) satisfies these

conditions. The proof consists of two steps. The firststep proves that the retailers problem would not admitmore than one local maximum in di dj di > 0 fori = 1 2. The second step characterizes each of the solutionsd1 d2 d10 0 0 d20. The proof of the first step is

by contradiction. Suppose that there are two interior localmaxima: x y and x y. The line that connects x yand x y can be characterized by x + ty + t, where = x x , = y y, , t 0 1 represents the linesegment between the two points, and t represents thewhole line. Define t as the value of on that line,

t = x + ty + tWe have

t = di

dit

+ dj

dj

t=

di+

dj

Because t achieves local maxima at the points x y andy x, we have t = 0 and t< 0 at these points. Wenow derive higher-order derivatives:

t = 3Gi x + t 3Gj y + t 3Ti x + t 3Tj y + t

t = 4Gi x + t 4Gj y + t 4Ti x + t 4Tj y + t

It follows that t > 0 because Gi di = 1 2 3 Kid3i < 0 and Ti 0 by Equation (2). There-fore, t is increasing.

Recall that t = 0 and t< 0 at t 0 1. Given thatt is continuous in t this can only occur if there is at leastone segment in t 0 1 such that t is convex-concave,which requires that t is decreasing along some seg-

ment. However, we have established that t is increas-ing, hence, a contradiction.

For the second step, it suffices to say that the interioroptimal solution d1 d2 satisfies the first-order conditionsHi = 0 i = 1 2. The solution on the boundary d10 0satisfies the first-order condition H1 = 0 s.t. d2 = 0. Thesame holds for 0 d20.

It is easy to see that the profit function is convex-concavealong the di = 0 lines.

di 0/di = i 2idi Gidi Ti di2d i 0/d

2i = 2i Gi di Ti di

The second derivative is (i) decreasing in di, (ii) positive at

di = 0 (because Gi di = 1 Kid2i as di 0+,and (iii) negative for sufficiently large di. Thus, there can beat most one local maximum for each problem.

For a symmetric problem, it follows from the above argu-ment that the unique interior optimal solution is on thedi = dj line. (If di dj with di = dj is an interior optimalsolution, then there are at least two local maxima, becausedj di is also an optimal solution by the symmetry of theprofit function. This contradicts with the result above.) Sim-ilar to the boundary lines, the profit function is convex-concave on the di = dj line, and the solution is characterized

by the first-order condition

dd/d=

2+

d

Gd

Td=

0

Lemma 1. The retailer profit function d1 d2 is jointly con-cave in the region where d1 d2 satisfies (12) and Tidi satis-fies (11).

Proof. First consider the retailers problem underwholesale-price contracts from both manufacturers. Weshow that the Hessian is a negative semidefinite matrix,which guarantees joint concavity of the profit function. Notethat (12) implies that 2Ri/Gi = 2pidi/Gi > /2 2 pi/di First,

2/d2i = 2 + 1 Gid2i 0

if and only if 1/ipi/di 21

1 1

pidi/Gi which isimplied by (12), /2 2 > 1/ and 11 1 4.Second, we show that the Hessian is a diagonally dominantmatrix.

2/d2i = 2 1 Gid2i 2/djdi = 2

if and only if pi/di 2 11 1pidi/Gi whichholds under (12) if /2 2 > 1/22, or equivalently,2 > + .

The proof of the case with quantity-discount contracts issimilar. To show diagonal dominance, we split the right-hand side of the second derivatives to show > Gi

notbepostedonany


@informs.org.




14/19


and > Ti . The former inequality holds by condi-tion (12) and the latter by (11).

Proof of Remark 1. Suppose manufacturer 1 offersT1d1 and there exists a demand pair d

1 d

2 such that

1 = 0. By definition, 2d1 d2 = R2d1 d2T2d2G2d2.We have 2d

1 d

2 < 20 d

2 20 d20. The first in-

equality is due to R2d1 d2/d1 < 0 for all d1, d2, and

T1d1 is nondecreasing: the retailer can always increaseher revenue from product 2 by dropping product 1, andher payment to manufacturer 1 cannot increase. The sec-ond inequality follows by optimality of d20 Therefore,0 d20>d

1 d

2.

Proof of Proposition 1. The proof follows from theproof of Proposition 2.

Proposition 2. Among all quantity discounts that satisfyTi di vi the dominant strategy for manufacturer i has thefollowing functional form.

Tidi wi vi =

wid vid2i /2if di wi ci/vi

Tiwi ci/vi + cidi wi ci/viotherwise.

(16)

Thus, for given Tjdj wj vj and fixed vi the best responseof manufacturer i is given by Tidi w

i vi for some w

i

ci ivi/i + 1.Proof. Without loss of generality, let j = 2 and i = 1.

Take any T1d1 and T2d2 Suppose that the retailers opti-mal solution is an internal point d = d1 d2. (The proof issimpler if one of di is zero.) At the optimal solution, thefirst-order conditions are satisfied: H1 = 0 H2 = 0. Define such that

= H1d + T1 d1 = R1d + R2d G1d1d1

We have T1 d1 = The optimal discount scheme for man-ufacturer 1 among those that generate d is the solution to

max T1d1

subject to Hid = 0 for i = 1 2

The constraints guarantee that both first-order conditionsare satisfied at d1 d2. The retailer function is jointly con-cave everywhere in the absence of economies of scale atthe retailer as long as Ti satisfies T

i di vi. Thus, d1 d2

remains the optimal solution (and the unique local maxi-

mum) for the retailer. In the presence of economies of scale,d1 d2 remains the unique interior optimal solution for theretailer by Theorem 1 as long as Ti satisfies (11). Because, theretailer profit is jointly concave when (10)(12) are satisfied,d1 d2 remains the optimal solution in the region defined

by (12).The manufacturers problem can be written as

max T1d1 T

1 d1 = because the condition impliesH1d = 0, and we already have H2d = 0. Therefore, theobjective is to increase T1d1 while keeping the marginalcost at d1 the same. This can be achieved by reducingT1 d1 as little as possible for all d1 < d1 which can be

achieved by setting T1 d1 = v1 Together with T1 d1 = this implies that

T1 d1 = v1d1 d1 + for all d1 0 d1That is, the marginal cost to the retailer is decreasing asslowly as possible and equals at d We have T1 d1 = v1.The derivative payment T

1

d1

can be specified in any wayfor d1 > d1 as long as it satisfies (15). We use the same func-tional form to specify T1 d1 (which implies that T

1 d1 =

v for all d1 > d1 to make sure that the argument appliesto all d1. Another condition in (15) requires T

1 d1 c1. The

derivative payment T1 is decreasing linearly in d1, and itwill reach c1 at some finite value. Rewriting T

1 , we replace

v1d1 + with w1 to obtain

T1 d1 =

w1 v1d1 for d1 w1 c1/v1c1 for d1 > w1 c1/v1

Integrating T and recalling the boundary value T 0 = 0 by(15), we obtain the quantity-discount schedule

T1d1=

w1d1 v1d2

1 /2 if d1 w1 c1/v1T1w1 c1/v1+c1d1 w1 c1/v1 otherwise

Note that this is not the optimal discount scheme over allpossible discount schemes. It is the best scheme among theones that produce d In other words, the functional formdominates other functional forms of T1d1 Thus, the opti-mal quantity-discount scheme can be found by consideringthe functions of this type only. We will see in Theorem 2that the constant part of T1 d1 is not relevant. A technicalnote is needed here that (15) requires continuous and dif-ferentiable payment functions, but the suggested solutionhas a nondifferentiable point. This can be circumvented byreplacing the two-piece marginal cost function with a dif-

ferentiable convex decreasing function that approximates itvery closely.The manufacturer profit is zero if w1 c1 It is also zero if

w1 1v1/1 + 1, because the marginal cost to the retailerw1 v1d1 > w1 v11/1 > 1v1/1 + 1 v11/1 = 1which implies zero demand. The first inequality is due tod1 1/1 to keep p1 nonnegative. If the retailer charges1 for product 1 the demand cannot be positive, whichimplies that i = 0.

Proof of Theorem 2. Recall that vi = 0 when the man-ufacturers employ wholesale-price contracts. Furthermore,while evaluating the equilibrium, we need to only con-sider the quadratic part of the payment functions. Sup-pose by contradiction that the retailer chose di such that

di > wi ci/vi The manufacturer can increase wi such thatdi = wi ci/vi and increase his profit without affecting theretailers or the other manufacturers decisions.

We show in the proof that i is concave in wi at thesymmetric solution to the first-order condition stated in thetheorem. Hence, the solution is (at least) a local maximum.

Using the quadratic part of Ti we obtain the first-ordercondition for a manufacturer as follows:

iwi

at wi = di + wi ci vidi diwi

= 0

Applying the implicit function theorem on the retailersfirst-order conditions Hi = 0 for all i, we can derive the

notbepostedonany


@informs.org.




15/19


impact of the wholesale prices on the optimal demand rates.Define Ai = 2i + Gi + Ti , Ai = Ai/di = Gi = 1 2 d3i = 2 d1i Gi > 0, B = i + j, and = AiAj B2. Note that > 0 because of the diagonal dom-inance of the Hessian as shown in Lemma 1.

di /wi

dj /wi=

1

AiAj B2 Aj

i + j As expected, di decreases with wi and increases with wjThe second derivative of i yields

2diwi

v

diwi

2+ wi ci vidi

2diw2i

which is negative when d1 = d2 because we have A1 = A2and

2diw2i

= 2AjB1 + AjAiAjB1 + AiAjAj1

= 3AjB + AiAjAjB AiA3j= 3AjBAiAj + AjB3 + AiAjAjB AiA3j= 3AiA3j + AjB3 < 0

Hence, the first-order condition above has a unique solutionwhen d1 = d2 . The solution w1 w2 satisfies the first- andsecond-order conditions for both manufacturers; that is, wiis a local maximum of i for fixed wj, and vice versa. Ifwi > w

i then d

1 decreases and d

2 increases, A

i increases, Aj

increases, and Aj decreases, implying that 2di /w

2i remains

negative. Hence, i is concave in wi for wi > wi for fixed w

j .

Thus, there can be no other local maximum greater thanwi but a linear search in wi w

i is necessary to find the

optimal wholesale price.

Proof of Theorem 3. The optimal demand rates can be solved as the unique solution to first-order conditionsHi = 0 i = 1 2 for given w1 w2. We have di/wi =2j vj/, 2di/w2i = 0, and 2di/wiwj = 0 di/wj =i + j/. Substituting these in the first-order conditionsin Theorem 2, we verify that the profit function of manu-facturer i is concave in wi. This guarantees the existence ofequilibria in the quantity-discount game between the man-ufacturers. The best response wiwj is the explicit solutionto each first-order condition. Differentiating, we obtain

wiwj

= 1 vii + j/2 + vi2

=1

vii

+j

2j vj2 v > 0 and < 1

The second inequality is due to vi < 2i i + j. Hence,we have increasing reaction functions with a slope lessthan 1. This implies that there is a unique equilibrium ofthe game between the manufacturers.

Proof of Theorem 4. The first part of the proof showsthat for any set of contracts by the manufacturers, manu-facturer i can do better by switching to a two-part tariffFi ci. Suppose manufacturer 2 is offering some contract tothe retailer (it can be any contract that depends only on d2We will show that manufacturer 1 can do no better than

to offer a two-part tariff. The retailer will accept any con-tract from manufacturer 1 as long as her profit is at least20 d20 0 Suppose manufacturer 1 offers to sell hisproduct at marginal cost and F1 = 0 so 1 = 0. In that case,define to be the retailers maximum profit given the twocontract offers:

= 1 + max 1d10 0 20 d20= max 1d10 0 20 d20

Hence, the retailer maximizes the combined profits ofthe two firms. manufacturer 1s incremental profit is 20 d20 We now show, by contradiction, that manu-facturer 1 cannot earn more than his incremental profit.Suppose manufacturer 1 offers a contract, T1, such that heearns more than his incremental profit,

1 > 20 d20and the retailer earns at least her reservation profit (so sheis willing to accept manufacturer 1s contract offer),

20 d20Add the above two equations to get

1 + > which contradicts the fact that is the upper bound onthe firms combined profits. Hence, manufacturer 1 can-not earn more than his incremental value. However, he canearn precisely his incremental value by choosing F1 = 20 d20 and w1 = c1 Therefore, he can do no better withany other contract.

The second part characterizes the equilibrium feesF1 F

2 . For any F1 c1F2 c2 we have i = Fi1di >0

and =

max1

F1

2

F2

12

F1

F2 0. Let us focus

on manufacturer 1. If F2 12 1, then 1 F1 12 F1 F2 and = max 2 F2 12 F1 F2, and manu-facturer 1s best response to F2 is to set F1 as high aspossible while ensuring that 12 F1 F2 2 F2; hence,F1 F2 = 12 2 If 2 F2 > 12 1 then 1 F1 >12 F1 F2 and = max 1 F1 2 F2, and manufac-turer 1s best response to F2 is to set F1 as high as possi-

ble while ensuring that 1 F1 2 F2 Hence, F1 F2 =1 2 + F2 for arbitrarily small If F2 > 2 then1 F1 > 12 F1 F2 and = max 1 F1 0, and man-ufacturer 1s best response is to set F1 = 1. The bestresponse function of manufacturer 2 is similarly obtained:F2 F1 = 12 1 if F1 12 2, 2 1 + F1 other-wise}. The unique equilibrium point is F1 F

2

=12

1

12 1 which are precisely the incremental profits ofthe manufacturers. At equilibrium, the retailers profit is= 12 12 1 12 2 = 1 + 2 12 The retailersprofit is nonnegative, because 12 1 + 2 follows fromthe substitutability of the products.

Proof of Theorem 5. Comparing the manufacturersprofit expressions given in Table 1, we see that profits underquantity discounts are always lower than those under two-part tariffs because < + . Profit under wholesale-price contracts is less than that under quantity discounts ifand only if 2 + < 2 2 which is equivalent to0 < 22 6+ 2 The right-hand side is equal to 22 at

notbepostedonany


@informs.org.




16/19


= 0 and 32 at = and it is decreasing in between,crossing zero only once at 3

7. Similarly, profit under

wholesale-price contracts is less than that under two-parttariffs if and only if 22 < 2 2 which is equivalentto + . Profit under wholesale-pricecontracts is less than that under quantity discounts if andonly if 2 + 22 2 < 0 The left-hand side ispositive for all 0 it is equal to 0 at = , and it is con-vex in between, implying at most a single crossing point in0 , which would be the smallest root of this third-degreepolynomial, i.e., 1

23

7. Profit under wholesale-price

contracts is less than that under two-part tariffs if and onlyif 3 2 2 < 0 The left-hand side is positive for all 0 is equal to 0 at = and it is convex in between,implying at most a single crossing point in 0 whichwould be the smallest root, i.e., 12 3

5.

Theorem 7. Equilibrium profits of the firms under each con-tract type are given in Table 1.

Proof. Start with the centralized system. The optimaldemand rates and the profit are as follows:

di

dj

= 1

42 2

2 2

2 2

i + cij + cj

di =i ci j cj

22 2 pi ci =

i ci2

ij =i ci2 +jcj2 2 jcji ci

42 2

if symmetric ij = 2 c2

4 +

If only product i were carried, then

pi ci = i ci/2 di =i ci

2 and i =

i ci24

Now consider the equilibrium under two-part tariffs. ByTheorem 4, manufacturer is optimal strategy is of a Fi citype two-part tariff.

i = Fi = ij j

=i ci2 + j cj2 2j cji ci

42 2 j cj2

4

= 2i ci2 + 2j cj2 2j cji ci

42 2

if symmetric i = c2

4 +

The retailers profit in the symmetric case is

= i + j ij

= 2 c2

4 2 c

2

4 + = c22 +

Consider the equilibrium under quantity-discount andwholesale-price contracts. Based on the demand rates andreaction functions derived in Theorem 1, we can solve theequilibrium for the symmetric problem. Recall that 2 v2 22 The unique symmetric equilibrium givenvi = vj = v is characterized by

w = 2 v 2 + v2 v + c2 v22 v 2 + v2 v2 v 2

= + v2 v + c2 v2 v + 22 v + 222 v 2 + v2 v

w c = c2 v 2 + v2 v22 v 2 + v2 v2 v 2

= c + v2 v2 v + 222 v 2 + v2 v

Substituting v = 0 yields the equilibrium wholesale prices,and taking the limit as v 2 yields the equilibriumunder the most aggressive quantity discounts.

At v = 0

w = + c2 w c =

c 2

w = c2

As v 2

w = + c2 + w c =

c +

w = c2 +

Similarly, the demand rates are given by

d = 2 v 2 w2 v2 22 =

w2 v + 2

= 22 v 2 + v2 v2 v 2 2 v 2 + v2 v c2 v

22 v 2 + v2 v2 v 21 1

2 v + 2

= c2 v22

v

2

+v2

v2

v

2

1

2

v

+2

= c2 v 22 v22 v 2 + v2 v2 v 2

= c2 v2 v + 222 v 2 + v2 v

At v = 0 d = c2 + 22

As v 2 d = c2 +

p w = + w2 v + 2 w =

w v + 2 v + 2

notbepostedonany


@informs.org.




17/19


The retailers profit at equilibrium is

= 2pd wd +vd2/2= 2wv+2v+2 d+vd

2

= 2 c22v+22v2

2v+222v2 +v2v2v+2v+2

+v c2

2v2

2v+222v2 +v2v2

= c22v22v+2

2v+222v2 +v2v2 (17)

At v=0 = c22

2+222

As v2

= 2wv+2v+2 +vd

2

= 2c2/+v+2v+2

c2+ +v

c24+2

= 22v+2v+2 c

2

2+2 +v c2

4+2

= c2

+2

The manufacturers profit at equilibrium is

= wcd vd2/2= w cvd/2d= c2v2+v2v

v/2c2v22v c2v22v

22v2 +v2v2v22

=

c22

v

222

v+

v/22

v

22v2 +v2v2v22

= c22v+v/22v

2v+222v2 +v2v2 (18)

At v=0 = c22

2+22222

= c2

2+22

As v2 = c222

42 +222

= c2

4+2

Theorem 8. The incremental profit of the manufacturer ineach contract type given the equilibrium contract offer of the othermanufacturer is as follows:

IPW = c22

2 + 2 2 IPQ = IPT = c

2 4 +

Proof. We know that the two-part tariff extracts the fullincremental profit. Hence, IPT is given by the manufac-turers profit at the two-part tariff equilibrium. For thewholesale-price equilibrium, we have v = 0 and

w = c2

First, evaluate the retailers outside option at this contract.

0d20=maxp2 wd2 =maxwd2d2

d2 =w

2 p2 =

w2

0d20=c2

422

Now, manufa

Documents

Competing Manufacturers in a Retail Supply Chain: On Contractual Form and Coordination by Gerard Cachon and Gurhan Kok