is channel coordination all it is cracked up to be?

Is Channel Coordination All It Is Cracked UpTo Be?

CHARLES A. INGENEUniversity of Washington

MARK E. PARRYUniversity of Virginia

A fundamental task for supply-chain managers is to determine wholesale-prices. Suchdetermination is a core theme in the marketing science literature on distribution chan-nels—which seems to have concluded that channel coordination—setting wholesale-pricesto maximize total channel profit—represents the best of all possible objectives. Thisjudgment is based on analyses ofbilateralmonopoly models, within which profit may beredistributed to the benefit of all channel members. However, many manufacturers dealwith multiple, competing retailers. The pure logic of bilateral monopoly models holds inthe presence of multiple retailers if and only if the manufacturer can price discriminatebetween retailers. Although mechanisms for price discrimination exist, in many situationsthey are infeasible, illegal, or both.

When retailers compete there are two feasible and legally permissible methods ofachieving channel coordination: an explicit quantity-discount schedule or a menu oftwo-part tariffs. (The latter is derived in detail in this paper.) A feasible and legallypermissiblealternative to channel coordination is for the manufacturer to utilize asophisticated Stackelberg two-part tariff—itself a form of a quantity-discount schedule.Although such a tariff cannot coordinate the channel, it is the best of all possible two-parttariffs from the viewpoint of maximizing manufacturer profit.

Because manufacturers are ultimately interested in their own profitability, it follows thatchannel coordination ismanufacturer-optimalonly if it generates at least as great a levelof profit for the manufacturer as does noncoordination. In this paper we determine theconditions under which a channel-coordinating wholesale-price strategy will manufactur-er-profit dominate a sophisticated Stackelberg two-part tariff. We show that the optimalpolicy is dependent on (1) the retailers’ fixed costs, (2) the relative size of the retailers, and(3) the degree of inter-retailer competition. We conclude that, from the perspective of a

Charles A. Ingene, University of Washington, School of Business, Box 353200, Seattle, WA 98195-3200(e-mail: [email protected]). Mark E. Parry, University of Virginia, Colgate Darden Graduate Schoolof Business Administration, Box 6550, Charlottesville, VA 22906-6550 (e-mail: [email protected]).

Journal of Retailing, Volume 76(4) pp. 511–547, ISSN: 0022-4359Copyright © 2000 by New York University. All rights of reproduction in any form reserved.

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manufacturer, channel coordination is often undesirable relative to utilizing a noncoor-dinating, sophisticated Stackelberg price-strategy. Therefore, channel coordination can nolonger be regarded as the ultimate goal which supply-chain managers should uncriticallypursue.

INTRODUCTION

One of the more basic duties of supply-chain management is determination of an intra-channelwholesale-price at which a product should be sold. The marketing science literature ondistribution channels provides a major source of guidance on how to perform this task. Anessential conclusion of these writings is that the wholesale-price charged by a manufacturershould be set to assure that the channel is “coordinated.” To marketing scientists the term“coordination” means setting all marketing mix variables to maximizetotalchannel profit. Thefundamental logic of coordinationseemsintuitively appealing: if a previously uncoordinatedchannel can be coordinated, then the additional profit can be divided among channel membersso that all of them are better off than they were prior to coordination.1

This conventional wisdom has been bolstered by analytical research that identified twocommonly used price-mechanisms—quantity-discount schedules and two-part tariffs—that can be used to achieve channel coordination (Jeuland and Shugan, 1983; Moorthy,1987). But the fact that coordination is Pareto-optimal in a bilateral monopoly model—amodel of one manufacturer selling through one retailer—does not assure that coordinationbenefits a manufacturer that sells throughmultiple retailers.

Does the logic of channel coordination extend to a manufacturer serving multipleretailers?If a manufacturer can charge different wholesale-prices to different retailers—ifit can price discriminate between them—then a multiple-retailer channel may be concep-tualized as a set of bilateral monopolies within which channel coordination is optimal.Butthere are reasons why the multiple-retailer scenario should not be cast as a set of bilateralmonopolies. First, survey evidence suggests that manufacturers regard retailer-specificpricing arrangements as infeasible due to administrative, bargaining, and contract devel-opment costs (Lafontaine, 1990). Hence they tend to employ a single wholesale-priceschedule, common across retailers, rather than treating each retailer separately. Second,there are legal restraints on channel pricing, at least in the United States. Section 2(a) ofthe Robinson-Patman Act “. . . prohibits sellers from charging different prices to differentbuyers for similar products where the effect might be to injure, destroy, or preventcompetition, in either the buyers’ or sellers’ markets” (Monroe, 1990, p. 394). In short,price discrimination is infeasible and is generally impermissible.2

If the manufacturer cannot employ a unique price-scheme for each retailer, anotherquestion arises: Are there wholesale-price schedules, common to all retailers, that arefeasible, legally permissible, and that induce coordination when there are multiple retail-ers? The answer is yes. Concerning feasibility, either a quantity-discount schedule3 or amenu of two-part tariffs4 can be structured to achieve coordination. (A two-part tariff,which consists of a per-unit fee and a fixed fee, is anonlinearquantity-discount schedule.With such a tariff the retailer’s average, per-unit acquisition cost declines with increases

512 Journal of Retailing Vol. 76, No. 4 2000

in quantity purchased. Two-part tariffs are used explicitly in franchising, whereas non-linear quantity-discount schedules are common in nonfranchise retailing. With a menu theretailer is allowed to select between a high fixed fee paired with a low per-unit charge ora lower fixed fee paired with a higher per-unit price.) Concerning legality, the Robinson–Patman Act requires every retailer to have the right freely to select from the wholesale-price schedule: “the seller must offer the discount structure to all buyers” (Monroe, 1990,p. 339). Because the retailer determines its own quantity, a quantity-discount schedulemeets the legality criterion. Similarly, because any element from a menu may be selected,a menu meets the legality criterion. Therefore channel coordination is both feasible andlegally permissible.

This brings us to our central question: Is it in amanufacturer’sinterest to establish awholesale-price policy that will induce coordination in a multiple retailer channel? Thereis limited evidence that it is not. Ingene and Parry (1995b) showed that a variety ofnoncoordinating two-part tariffs might manufacturer-profit dominate a channel-coordinat-ing quantity-discount schedule. (The terminology “manufacturer-profit dominate” meansthat a particular wholesale-price strategy generates manufacturer profit that is at least asgreat as could be generated by the strategy to which it is being compared.) Their resultsdepended on the model’s parameters, of which the most critical was the difference inretailers’ fixed costs. Subsequently the same authors (1998) derived the manufacturer-optimal two-part tariff by simultaneously determining the per-unit priceand the fixed fee.They demonstrated that this feasible and legally allowed “sophisticated Stackelberg”two-part tariff manufacturer-profit dominates all other two-part tariffs—including thosethat dominate the quantity-discount schedule. It follows that under some parametric valuesthe manufacturer prefers sophisticated Stackelberg noncoordination to quantity-discountcoordination. What we do not know is whether the manufacturer prefers coordination tonon-coordination in a multiple-retailer channel.

Should a manager seek to coordinate a multiple-retailer channel? The answer to thisquestion is not merely of academic interest. For several reasons it is of considerablepractical importance as well. First, virtually all supply-chains for consumer goods dis-tribute through multiple outlets. Second, supply-chains regularly establish wholesale-priceschedules that are common across many of their retailers. Thus, realistic supply-chainsgenerally arenot examples of bilateral monopoly models. Third, many supply-chaindecision-makers understandably base their wholesale-price policies on knowledge that iscurrently accepted as state-of-the-art. Fourth, such knowledge has been based on insightsgained from bilateral monopoly models. It logically follows that if “state-of-the-artknowledge” is applicable only under limited circumstances, then managerial decisionspredicated on this knowledge may be applied inappropriately. In extreme situations thesedecisions may even have a deleterious impact on profit. Therefore, it is of practical as wellas academic value to ascertain if a manufacturer’s best interest is served by coordinatinga multiple-retailer supply-chain.

In this paper we investigate the optimality of channel coordination (1) when themanufacturer distributes through competing retailers and (2) when price discriminationbetween the retailers is infeasible and/or illegal. We address two specific questions. First,“under what set of parametric values (if any) does the sophisticated Stackelberg two-parttariff manufacturer-profit dominate a channel-coordinating menu of two-part tariffs?”

Channel Coordination 513

Second, “under what set of parametric values does the sophisticated Stackelberg two-parttariff manufacturer-profit dominate a channel-coordinating quantity-discount schedule?”These comparisons permit us to make a definitive statement about the optimality ofchannel coordination. As a subsidiary insight, we determine the conditions under which achannel-coordinating menu of two-part tariffs manufacturer-profit dominates a channel-coordinating quantity-discount schedule.

Answers to the questions raised in the preceding paragraph have not appeared in theliterature previously; indeed, the questions themselves do not seem to have been voiced.We shall see that the answers pivot on three factors: (1) the magnitude of the differencein retailers’ fixed costs, (2) the relative size of the retailers and (3) the degree ofcompetition between them. By “fixed cost” we mean the retailer’s opportunity cost inaddition to its actual, quantity-independent dollar outlay. Opportunity cost is the profit theretailer could obtain by devoting its scarce resources (such as shelf space) to selling themerchandise of a different manufacturer. Covering fixed cost is essential for retailer’svoluntary participation in the channel. By “relative size” we mean the relativeunit salesof the retailers. In the case of two retailers this may range from nearly zero (the smallerretailer is tiny compared to the larger one) to one (they are equal sized). We use unit, ratherthan dollar, sales because the former is common to the competitors—they sell an identicalproduct—whereas the latter is convoluted by the fact that they may charge different prices.By the “degree of competition” we mean the willingness of consumers to switch betweenretailers on the basis of price. Each of these factors is formalized in the fourth Section. Thebottom line is that the manufacturer-optimal wholesale-price schedule depends on retail-ers’ costs, demands, and competitive interaction, all of which are factors over which themanufacturer has little or no control.

We organize the paper as follows. In the next Section we describe our model and ourassumptions and we review and extend key results from the relevant literature. Then we derivein detail the channel-coordinating menu of two-part tariffs.5 In the following Section we definethe theoretical conditions under which the sophisticated Stackelberg two-part tariff, thechannel-coordinating menu of two-part tariffs, or the channel-coordinating quantity-discountschedule maximizes manufacturer profit. We also connect our research to three classic articlesfrom the marketing science literature. In the subsequent Section we offer a range of practicalillustrations in order to give a flavor to the parametric conditions under which channelcoordination is, or is not, manufacturer-optimal. In the final Section we discuss our results andsuggest directions for future research. A lengthy Technical Appendix that contains themathematical details of each proof is available from the authors on request. To presage ourresults, we find that over almost all parametric values channel coordination rarely benefits themanufacturer. And, when it does benefit the manufacturer, coordination out-performs non-coordination only by a narrow margin. If a supply-chain manager (a manufacturer) seeks a ruleof thumb, it should be that maximization of manufacturer profit is generally superior tomaximization of channel profit.


THE MODEL AND RELEVANT RESULTS FROM THE LITERATURE

In this Section we establish our model of a distribution channel and briefly overview somerelevant results from the marketing science literature. In the first sub-Section we presentthe model while in the second sub-Section we evaluate a vertically integrated manufac-turer in order to define channel-coordinating values for prices, quantities and profits. Thefinal two sub-Sections detail the channel-coordinating quantity-discount schedule and thesophisticated Stackelberg analysis with competing retailers.

The Model

We model the distribution channel as a single manufacturer selling its product throughtwo competing retailers. In conformity with previous research we make six key assump-tions. (1) The channel has only two levels: A manufacturer and its retailers. (2) Channelbreadth is fixed at two competing retailers. (3) All decision-makers are profit-maximizers.(4) All decision-makers have full information. (5) Retailers cannot resell merchandise toeach other. (6) Each retailer has a linear, downward-sloping demand curve of the form:

Qi 5 ~ Ai 2 bpi 1 upj! s.t. 0# u , b, i, j [ ~1, 2! (1)

In equation (1), pi denotes the price charged by the ith retailer while the intercept Aimeasures the ith retailer’s demand at prices of zero (its “base demand”). Parameters b andu reflect the retailer’s own-price and cross-price sensitivity of demand, respectively. Forthere to be competition between the retailers requiresu . 0; for aggregate demand to bedownward sloping requiresu , b.

Our retailers may be envisioned as intra-type or as inter-type competitors. The formercase (metaphorically, Macy’s versus May Company) is compatible with a high degree ofcompetition (u close to b) and roughly equal sized retail outlets. The latter case (meta-phorically, department stores versus apparel boutiques) is compatible with a moderatedegree of competition and retail outlets of substantially different size. Of course, inter-type competitors could be of roughly equal size. The point is that our logic is applicableto a wide variety of competitive supply-chain management situations.

Our second assumption—fixed channel breadth—merits additional commentary. Weknow that determination of optimal channel breadth6 is inordinately complex; the follow-ing pages will demonstrate that the topic addressed herein is also quite complicated.Therefore, in this paper we seek the manufacturer-optimal wholesale-price policy atconstant channel breadth, leaving the convolution of breadth and wholesale-price opti-mality as a topic for future research.


The Vertically-Integrated Manufacturer

A vertically integrated manufacturer obtains the totality of channel profit: therefore, itwill seek tomaximize channel profit. Such maximization leads to the Nash-equilibrium,channel-coordinating margins and quantities:

m*i ; ~ p*i 2 ci 2 C! 5 FbQ*i 1 uQ*jb2 2 u2 G (2)

Q*i 5 FAi 2 b~ci 1 C! 1 u ~cj 1 C!

2 G (3)

In the preceding equations C, ci, and cj are per-unit variable costs of the manufacturer, andthe ith and jth retailers respectively. These prices and quantities coordinate the channel byinternalizing the inter-retailer externality implicit in demand curve (1). The relationshipbetween retailers’ outputs is determined by:

~ Ai 2 Aj! x ~b 1 u !~ci 2 cj! f Q*i x Q*j (4)

To ease our exposition we will henceforth assume Q*i . Q*j. This can occur due todifferences in base demand (Ai 2 Aj), differences in weighted per-unit costs ((b1 u)(ci 2cj)), or some combination thereof. Although our ensuing analysis is completely general,for expositional simplicity we will treat the retailers as having equal per-unit costs. Hencethe ith retailer will be discussed as having the greater base demand. Finally, it can beproven that if Q*i . Q*j, then the ith retailer is larger under all of the wholesale-pricepolicies investigated in this paper.

A decentralized methodof achieving coordinated prices and quantities is with atransfer-pricing scheme in which the ith retailer pays a marginal per-unit wholesale-priceof:

W*i 5u

b FuQ*i 1 bQ*j~b2 2 u2! G 1 C (5)

Whether coordination is determined by centrally dictated prices or through use ofretailer-specific transfer-prices7 (5), the behavioral results are identical. The vertically-integrated manufacturer, charging prices defined by Equation (2), generates channel profitof:

P*C 5 S @b~Q*i !2 1 2uQ*iQ*j 1 b~Q*j !

2#

~b2 2 u2! D 2 f i 2 f j 2 F ; R*C 2 f i 2 f j 2 F (6)

The terms F, fi, and fj represent fixed costs for the manufacturing arm and the ith and jth

retail outlets respectively and R*C is the channel’s net revenue—total dollar sales minus all


variable costs. We make three observations. First, no wholesale-price scheme withindependent retailers can induce channel coordination unless it reproduces the prices andquantities given by equations (2) and (3). Second, these values can only be obtained bywholesale-price schedules that generatemarginal wholesale-pricesof W*i and W*j —givenby Equation (5). Third, from (5) we see that the optimal per-unit marginal cost must belarger for the retail outlet with the smaller optimal output. Therefore, no single two-parttariff can generate channel coordination. Rather, a per-unitdiscountis required.

The Channel-Coordinating Quantity-Discount Schedule

For a quantity-discount schedule to be channel coordinating requires that it generate amarginal wholesale cost to the ith retailer precisely equal to the W*i of Equation (5). Themost mathematically tractable of such schedules8 is linear in quantity; this leads to a totalcost to the ith retailer of: (WQD 2 wQDQi)Qi 1 f. With this schedule each retailer paysan identical fixed fee (f) but a quantity-dependent per-unit wholesale-price. Hence theper-unit marginal cost to the ith retailer is (WQD 2 2wQDQi). Channel-coordinating valuesfor WQD and wQD are:

WQD* 5 Fu~Q*i 1 Q*j !

b2 2 u2 G 1 C (7)

wQD* 5 F u

2b~b 1 u !G (8)

Note that wQD is the per-unit quantity-discount; if it were zero we would have a two-part tariff.This quantity-discount schedule yields the same channel-coordinating retail prices and

quantities as obtained under vertical integration, while generating profit for the ith retailer of:

p iQD* 5 @~1 2 bwQD*!~Q*i !

2/b# 2 f i 2 fQD ; RiQD* 2 f i 2 fQD (9)

In Equation (9) RiQD* is the ith retailer’s net revenue and fi is its fixed cost (including the

minimal economic profit—its opportunity cost—required for it to participate in thechannel).

To retain both retailers in the channel, the (common) fixed fee cannot leave eitherretailer with insufficient profit for it to be willing to distribute the manufacturer’s product.That is, it cannot exceed the lesser of (Ri

QD* 2 fi) and (RjQD* 2 fj) if both retailers are to

be retained as channel members. And, to maximize manufacturer profit, the fixed feecannot be less than the maximum value that is compatible with retaining both retailers.Thus we have:

fQD* 5 min$~RiQD* 2 f i!, ~Rj

QD* 2 f j!% (10)


There are three possibilities. (1) The ith retailer is more profitable, sofQD* 5 (RjQD* 2

f j) andpiQD* . 0 5 pj

QD*. (2) The jth retailer is more profitable, sofQD* 5 (RiQD* 2

f i) andpiQD* 5 0 , pj

QD*. (3) The retailers are equally profitable, sofQD* 5 (RiQD* 2

f i) 5 (RjQD* 2 f j) and pi

QD* 5 0 5 pjQD*.

Because the difference in retailer profitability is (piQD* 2 pj

QD*) 5 (RiQD* 2 Rj

QD*) 2(fi 2 fj), the more profitable retailer is given by the relationship:

~p iQD* x p j

QD*!3 dQD* ; ~RiQD* 2 Rj

QD*! (11)

5 S ~2b 1 u !~Q*i 1 Q*j !~Q*i 2 Q*j!

2b~b 1 u ! D x ~ f i 2 f j!

The termdQD* is the difference in retailer net revenues. By virtue of our assumptionconcerning relative retailer size (i.e., Q*i . Q*j ) we havedQD* . 0. Hence, we may thinkin terms of three “Zones” in (fi 2 fj) space that are defined as:

Zone ZjQD*: dQD* . ~ f i 2 f j!

Zone ZijQD*: dQD* 5 ~ f i 2 f j!

Zone ZiQD*: dQD* , ~ f i 2 f j! (12)

The subscript on each ZQD* defines which retailer nets zero economic profit after payingthe fixed fee. In game-theoretic terms, the magnitude of (fi 2 fj) determines the retailer forwhom the channel participation constraint is binding.

The impact of a change in fi on manufacturerprofit varies by Zone:

]PMQD*~ f i, f j!

]f i5 H $0 ; ~ f i 2 f j! , dQD*

2$2 ; ~ f i 2 f j! . dQD* (13)

At (f i 2 fj) , dQD* increases in fi lower the ith retailer’s profit but have no effect onfQD*

or on manufacturer profit—the ith participation constraint is not binding. At (fi 2 fj) .dQD*, retention of the ith retailer requires that a $1 increase in fi be matched by a $1decrease infQD* for bothretailers—a $2 cost to the manufacturer. Figure 1 illustrates thisgraphically.

Summarizing, the quantity-discount schedule [Equations (7) and (8)] coordinates thechannel atall (fi 2 fj) values and generates total channel profitP*C [Equation (6)]. Themanufacturer obtains all ofP*C only along the line in (fi 2 fj) space that is defined by thekink (fi 2 fj) 5 dQD*. On this lineno noncoordinating wholesale-price schedule canmanufacturer-profit dominate the channel-coordinating quantity-discount schedule.


The “Sophisticated Stackelberg” Two-Part Tariff

A manufacturer using a “naı¨ve Stackelberg” tariff optimizes its per-unit fee but accepts thefixed fee as a residual that extracts all profit from the less-profitable retailer. A naı¨ve two-parttariff can never coordinate a channel, although it leads to manufacturer profit that is defined interms of three “Zones” (the middle one defined at a kink (call itdnS)—similar to the Zonesderived above). In fact,everypossible two-part tariff will describe the same general shape ofmanufacturer profit (horizontal for (fi 2 fj) , dnS, declining at a rate of $2 for (fi 2 fj) . dnS).Only the value ofdnS and the manufacturer profit at the kink differ with the per-unit fee.

In contrast to a naı¨ve tariff, a “sophisticated Stackelberg” two-part tariffsimultaneouslyoptimizes the per-unit fee and the fixed fee in light of the difference in retailers’ fixedcosts. The resulting per-unit fee and the fixed fee in the sophisticated tariff are a functionof (fi 2 fj), varying as this difference varies.9 The former is constant for all values of (fi 2fj) less than a lower boundary LSS*, it declines at a linear rate until an upper boundarycondition (fi 2 fj) 5 USS* is met, and then it is constant again. The lower boundary isdetermined by the profit conditionspi

SS* $ 0 5 pjSS* whereas the upper boundary is

Legend: P*C [ Total profit of a coordinated channel and PMQD* [

Manufacturer profit with a channel-coordinating quantity-dis-count schedule. dQD* [ Point at which manufacturer extractsall profit from both retailers.

FIGURE 1

Manufacturer Profit with a Channel-CoordinatingQuantity-Discount Schedule


determined bypiSS* 5 0 # pj

SS*. In short, the boundaries are the points at which bothparticipation constraints are binding. These boundaries are defined as:

LSS* ; 2~Q*i 2 Q*j !@3uQ*i 1 ~4b 2 u !Q*j #/~2b 1 u !2 (14)

USS* ; 2~Q*i 2 Q*j !@~4b 2 u !Q*i 1 3uQ*j #/~2b 1 u !2 (15)

By virtue of our assumption that Q*i . Q*j, we have USS* . LSS* . 0. The fixed fee isequal to the net revenue of the jth retailer for all (fi 2 fj) , LSS* and is equal to the netrevenue of the ith retailer for all (fi 2 fj) . USS*. Between the two boundaries the fixedfee is equal to the net revenue of both retailers.

Legend: P*C [ Total profit of a coordinated channel; PMSS* [ man-

ufacturer profit with a sophisticated Stackelberg tariff; andPC

SS* [ channel profit with a sophisticated Stackelbergtariff. LSS* [ Lower bound on fixed cost fi below which theith retailer obtains a positive profit. USS* [ Upper bound onfixed cost fi above which the jth retailer obtains a positiveprofit.

FIGURE 2

Manufacturer Profit with a Sophisticated StackelbergTwo-Part Tariff


Manufacturer profit under the sophisticated Stackelberg two-part tariff can be describedgraphically by connecting together the kinked points ofall possibletwo-part tariffs (seeFigure 2). In game-theoretic terms, this defines the envelope of manufacturer profits fromall possible two-part tariffs that retain both retailers as channel members. As can be seen,the manufacturer extracts all profit from both retailers provided (fi 2 fj) lies between LSS*

and USS*. Below LSS* and above USS* one retailer is able to retain an excess economicprofit. In short, there are three “Zones” in (fi 2 fj) within which wholesale price, totalchannel profit, and the distribution of profit between manufacturer and retailers differ. TheZones are:10

Zone ZjSS*: LSS* . ~f i 2 f j! (16)

Zone ZijSS*: LSS* # ~f i 2 f j! # USS*

Zone ZiSS*: ~f i 2 f j! . USS*

In Zone ZjSS* the jth (the less profitable) retailer obtains zero economic profit, while in

Zone ZiSS* the ith retailer earns zero economic profit. In Zone Zij

SS* both retailers net zeroeconomic profits. Within Zone Zj

SS* manufacturer profit is unaffected by changes in the ith

retailer’s fixed cost. In Zone ZijSS* manufacturer profit declines at an increasing rate with

changes in retail fixed costs ($0, ]PMSS*/]fi , 2$2) and in Zone Zi

SS* manufacturer profitis a constant, negative function of the ith retailer’s fixed cost (]PM

SS*/]fi 5 2$2).The sophisticated Stackelberg manufacturer is more profitable than is the naı¨ve Stack-

elberg manufacturer—or any other manufacturer utilizing a single two-part tariff. Ifanynoncoordinating two-part tariff can ever manufacturer-profit dominate a channel-coordi-nating wholesale-price policy, it will be the sophisticated Stackelberg policy. Nonetheless,with a sophisticated Stackelberg two-part tariff,channel profit (defined asPC

SS*) isgenerally less than what is earned by a channel-coordinating, vertically-integrated mo-nopolist:PC

SS* , P*C.11

THE CHANNEL-COORDINATING MENU OF TARIFFS12

We now turn to the determination of a channel-coordinating menu of two-part tariffs.Because our primary purpose in devising a menu is to achieve coordination of a competingretailer channel, the menu must induce both retailers (1) to participate in the channel and(2) to sell their channel-optimal outputs. We begin with five observations. First, the fixedfee must leave each retailer with a non-negative profit; that is, the participation constraintmust be met for both of them. Second, the per-unit fee must equal the optimal per-unitwholesale-price [defined by Equation (2.5)]; if it does not the channel will not becoordinated. Third, because the optimal per-unit wholesale-price differs by retailer, theremust be a unique two-part tariffintendedfor each retailer. Fourth, because retailers mayselect whichever tariff they want, the fixed fee must prevent “defection” to the tariff


intended for the other retailer: each retailer must prefer its “own” tariff of the menu. Fifth,within these constraints, fixed fees may be set to maximize the manufacturer’s profit.

The Model

We model the decision process as a three-stage game. In Stage 1 the manufacturercreates a menu of two-part tariffs and offers that menu to the retailers. In Stage 2 eachretailer selects the tariff that gives it the higher profit. In Stage 3 retailers set prices tomaximize their own profit. To conserve space we sketch our argument for the first twostages of the game (the third stage involves a standard set of maximizations). Details ofthe mathematical logic for all three stages, and the resulting equilibrium values of allvariables are in a Technical Appendix that is available from the authors.

To simplify our presentation we denote the mth two-part tariff astm [ {W *m, fm}, m 5i or j. We express the menu itself ast [ { ti, tj}. To assure coordination W*m must bedefined by Equation (5); only the fixed fee is a choice variable for the manufacturer.

The Manufacturer’s Ideal Fixed Fees

For the manufacturer theideal result would be for the menu to allow the manufacturer toextract all profit from a coordinated channel. There are two steps that must occur for thisoutcome to obtain. First, the manufacturer must set the mth fixed fee precisely equal to the mth

retailer’s maximum possible net revenue minus its fixed cost (i.e.,f*m [ (R*m 2 fm), m 5 ior j). If a lower fixed fee is set the manufacturer will not receive all channel profit. Second, theith retailer must select the ith tariff and the jth retailer must select the jth tariff when the fixedfee components of the menu aref*i andf*j. If there is “defection”—if a retailer selects thewrong element of the menu—the channel will not be coordinated. The only menu of two-parttariffs that satisfies the first step—provided the second step is met—is:

t* ; $t*i, t*j % ; $~W*i, f*i !, ~W*j, f*j !% (17)

Under what conditions will both retailers behave in the requisite fashion if (17) is themenu? There are two necessary and sufficient conditions. First, theparticipation con-straint, which sets an upper limit on the fixed fee, must be met for both retailers:

fk # $@~Q*kmn!2/b# 2 f %k ; ~R*kmn 2 f k!, k [ ~i, j ! (18)

In Equation (18) the term R*kmn denotes the net revenue of the kth retailer given that itselects the mth element of the menu when its rival selects the nth element. Second, bothretailers must prefer the tariff that is intended for them. Each will do so if and only if theintended tariff generates the higher profit; this is theself-selection constraint. In ourcontext channel coordination in the presence of self-selection requires that (1) the ith


retailer to pickt*i (rather thant*j) when the jth retailer selectst*j and that (2) the jth retailerpick t*j (rather thant*i) when the ith retailer selectst*i. These statements reduce to thecomparative profit requirements:

~R*iij 2 f i! . ~R*ijj 2 f j! and~R*jji 2 f j! . ~R*jii 2 f i! (19)

These channel-coordinating tariff selections are an equilibrium set of choices providedneither retailer can improve its revenue by choosing an alternative tariff. However, there areactually three13 possible responses to menu (17)—or to any other menu of two-part tariffs:

Case I: both retailers choose tarifft*j,Case II: the ith retailer chooses tarifft*i and the jth retailer chooses tarifft*j, and

Case III: both retailers choose tarifft*i.

Only Case II satisfies condition (19); thus only Case II leads to a coordinated channel withthe manufacturer extracting all channel profit. Neither the Case I nor the Case IIIequilibrium leads to channel coordination or full profit extraction. Thus, two questionsarise. (1) What is required for the Case II equilibrium to obtain? (2) If a Case I or a CaseIII equilibrium does occur, can the manufacturer modify the fixed fee to precludedefection, thereby restoring channel coordination?

In response to the first question, the Case II equilibrium occurs if and only if:

LMenu* # ~f i 2 f j! # UMenu* (20)

where:

LMenu* ;h@bhQ*i 1 ~h 1 2u ~2b2 2 u2!!Q*j #~Q*i 2 Q*j !

~b 1 u !2~4b2 2 u2!2 . 0 (21)

UMenu* ;h@~h 1 2u ~2b2 2 u2!!Q*i 1 bhQ*j #~Q*i 2 Q*j !

~b 1 u !2~4b2 2 u2!2 . LMenu* . 0 (22)

and where h[ (4b2 1 2bu 2 u2) . 0. These boundary conditions were derived bysubstituting the fixed fee definitionsf*m 5 (R*m 2 fm) into the comparative profitrequirement (19).

Case I equilibrium occurs if LMenu* . (fi 2 fj) and Case III equilibrium occurs if (fi 2fj) . UMenu*. In short, just as with the previous wholesale-price policies, there are three“Zones” in (fi 2 fj) space. These Zones are:

Zone ZjMenu*: LMenu* . ~f i 2 f j! f Case I (23)

Zone ZijMenu*: LMenu* # ~f i 2 f j! # UMenu* f Case II

Zone ZiMenu*: ~f i 2 f j! . UMenu* f Case III


We now turn to the second question: modifying the fixed fee to preclude defection.

Adjusting the Fixed Fees to Assure Channel-Coordination

How can the fixed fees be modified to assure channel coordination when either a CaseI or a Case III defection occurs? Because the problem is symmetric, it is sufficient for usto focus on Case I as illustrative of both Cases. In Case I both retailers select the jth tariffwhen offered the menu {t*i, t*j}. If the i th retailer were not to defect it would obtain zeroeconomic profit since its profit before paying the fixed fee (R*i 2 fi) would be equal to thefixed feef*i 5 [R*i 2 fi]. By defecting the ith retailer is able to pay a lower fixed feef*j 5[R*j 2 fj] , f*i. But defection also entails a higher per-unit fee; thus the ith retailer netrevenue is reduced from R*i to {R*i 2 Bj}. The net revenue reduction is:

Bj ;u ~2b2 2 u2!~Q*i 2 Q*j !

b~b 1 u !2~4b2 2 u2!2 @~8b3 1 6b2u 2 2bu2 2 u3!Q*i (24)

1 u ~2b2 2 u2!Q*j ] . 0

The important point is that in a Case I equilibrium the ith retailer defects to the jth tariffbecause its sacrificed profit from a reduction in net revenue is less than its saving from alower the fixed fee. As a result, the ith retailer’s profit rises. Formally we have:

p ijj ; ~$R*i 2 Bj% 2 f i! 2 @R*j 2 f j# . ~R*i 2 f i! 2 @R*i 2 f i# ; p iij 5 0 (25)

Expression (25) simply states that profit due to defection is greater than the zero profit thatis earned without defecting. On both sides of the inequality the [bracketed] term is therelevant fixed fee and the (parenthetical) term is retailer profit prior to paying the fixed fee.The term in {braces} on the LHS is the ith retailer’s net revenue when it defects.

To assure coordination the manufacturer must eliminate the ith retailer’s incentive todefect. How can it achieve this? It cannot adjust the per-unit fees because doing so wouldmake coordination impossible. Thus, it must alter a fixed fee. If the manufacturer were toraisef*j it would encourage the jth retailer to defect; thus the manufacturer has only oneoption: f*i must be lowered. But how far? The answer is implicit in expression (25). Weknow that when it defects the ith retailer pays a fixed fee off*j andsacrifices net revenueof Bj. Thus, the maximum fixed fee that the manufacturer can extract from the ith retailerwithout causing defection is (f*j 1 Bj). As a result, instead of usingt*i [ {W *i, f*i 5 (R*i 2fi)} as the ith element of the menu, the manufacturer must instead offerti

L [ {W *i, fiL 5

((R*j 2 fj) 1 Bj)} as the ith element. With this modification the tariff intended for ith retailer(ti

L) and the tariff intended for the jth retailer (t*j ) generate equal profit for the ith retailer.We break such “ties” in favor of the channel-coordinating solution. In summary, in a CaseI equilibrium the manufacturer-optimal menu of two-part tariffs is {ti

L, t*j}. This menuleads to the ith retailer earning a positive economic profit while the jth retailer “breakseven.”


Similarly, in a Case III equilibrium, which occurs when (fi 2 fj) . UMenu*, bothretailers select tarifft*i. Defection by the jth retailer can be prevented if the jth element ofthe menu is set totj

U [ {W *j, fjU 5 ((R*i 2 fi) 1 Bi)}, where Bi is defined as:

Bi ;u ~2b2 2 u2!~Q*i 2 Q*j !

b~b 1 u !2~4b2 2 u2!2 @u ~2b2 2 u2!Q*i (26)

1 ~8b3 1 6b2u 2 2bu2 2 u3!Q*j ] . 0

Now both the tariff intended for the ith retailer (t*i) and the (modified) tariff intended forthe jth retailer (tj

U) yield the same positive profit to the jth retailer (R*j 2 fj) 2 fjU),

namely:

p jii ; ~$R*j 2 Bi% 2 f j! 2 @R*i 2 f i# 5 ~R*j 2 f j! 2 @~R*i 2 f i! 1 Bi# ; p jji . 0(27)

In expression (27) the LHS is the jth retailer’s profit from defecting when the menu is {t*i,t*j} and the RHS is the profit fromnot defecting when the menu is {t*i, tj

U}. On both sidesthe [bracketed] term is the relevant fixed fee, the (parenthetical) term is profit beforepaying the fixed fee, and the term in {braces} on the LHS is the jth retailer’s net revenuewhen it defects. The term Bi is the net revenue reduction to due defection.14

To summarize, in Case I or Case III equilibrium the manufacturer-optimal, channel-coordinating menu of two-part tariffs varies across Zones. The respective per-unit fees(W*i and W*j ) are the same in every Zone—as is necessary to assure coordination—but themanufacturer must vary the fixed fees to ensure that both retailers select the properelement of the menu. The manufacturer-optimal fixed fees are:

In Zone ZjMenu* the jth (the less profitable) retailer obtains zero net economic profit,

whereas in Zone ZiMenu* the ith retailer is the less profitable—and it obtains zero net

economic profit. In Zone ZijMenu* both retailers net zero economic profits after paying their

fixed fees. We conclude that an appropriately specified menu of two-part tariffs cancoordinate the channelregardlessof the actual difference in retailers’ fixed costs. The“cost” of coordination is lower manufacturer profit under some distributions of retailers’fixed costs; this is detailed graphically in Figure 3.

Note that in Zone ZjMenu* manufacturer profit is invariant with the ith retailer’s fixed cost

because an increase in fi lowers the profit of the ith (the more profitable) retailer. In Zone

Zone Zonal Definitions

Fixed Fees

ith Retailer jth Retailer

ZjMenu* LMenu* . (fi 2 fj) [(R*j 2 fj) 1 Bj] R*j 2 fj

ZijMenu* LMenu* # (fi 2 fj) # UMenu* R*i 2 fi R*j 2 fj

ZiMenu* (fi 2 fj) . UMenu* R*i 2 fi [(R*i 2 fi) 1 Bi]


ZijMenu* manufacturer profit declines at a constant rate (]PM

Menu*/]f1 5 2$1) because a $1increase in fi lowers the fixed fee in tarifft*i by $1. In Zone Zi

Menu* an increase in fi

decreases the profit of the ith (now the less profitable) retailer—requiring a fixed feereduction for both retailers in order to preclude (1) the ith retailer from not participatingin the channel and (2) the jth retailer from defecting; accordingly]PM

Menu*/]fi 5 2$2. Theeffect of a change in fj is precisely symmetric.

COORDINATION VERSUS NON-COORDINATION: THEORETICAL BASES

The preceding Sections described three pricing mechanisms: (1) a channel-coordinatingquantity-discount schedule, (2) a sophisticated Stackelberg two-part tariff that cannot

Legend: P*C [ Total profit of a coordinated channel; PMMenu* [ man-

ufacturer profit with a channel-coordinating menu. LMenu* [Lower bound on fixed cost fi below which the ith retailerobtains a positive profit. UMenu* [ Upper bound on fixedcost fi above which the jth retailer obtains a positive profit.

FIGURE 3

Manufacturer Profit with a Channel-CoordinatingMenu of Two-Part Tariffs


coordinate the channel but that manufacturer-profit dominates any single two-part tariff,and (3) a channel-coordinating menu of two-part tariffs. Will the manufacturer ever preferthe non-coordinating two-part tariff? If channel coordination is preferable, which methodof coordination will yield greater manufacturer profit—a quantity-discount schedule or amenu of two-part tariffs? This Section answers these questions theoretically. Additionally,in the final sub-Section we link our results to three classic articles from the marketingscience literature; Jeuland and Shugan (1983), McGuire and Staelin (1983), and Moorthy(1987).

Dimensionality: a Basis for Comparison

We re-parameterize manufacturer profit for each of our wholesale-price policies interms of two dimensions. They are (1) the degree of competition between the retailers and(2) the relative size of the retailers. We formalize these dimensions in the followingmanner. First, the degree of competition between the retailers is expressed as the ratio ofcross-price to own-price sensitivity (x [ (u/b)). The ratio ofu and b is a pure number.Further, becauseu . 0 is required for the retailers to be in competition, and becauseu ,b is necessary for the aggregate demand curve to be negatively sloped,x lies in the unitinterval. Second, the relative size of the retailers is expressed as their output ratio in thebaseline, vertically integrated state (Q*[ (Q*j/Q*i)). Because we have assumed Q*i . Q*j,Q* is a pure number that lies in the unit interval.

The parametersx and Q* form a two-dimensional unit-square within which we maymap manufacturer profit for each wholesale-price policy (see Table 1). Although the profitlevels are presented in terms of b,u, Q*i, and Q*j, they are actually combinations of theprimitive values Ai, Aj, ci, cj, C, b, andu as can be seen by writing out Q*:

Q* ;Q*jQ*i

; SAj 2 b~cj 1 C! 1 u ~ci 1 C!

Ai 2 b~ci 1 C! 1 u ~cj 1 C!D . (28)

Thus, manufacturer profit may be expressed in the abbreviated notation of Q* andx.It is important to recognize that although Q* represents the ratio of optimal retailer

outputs for a profit-maximizing, vertically-integrated firm, the parametric value Q* isexogenous from the perspective of the wholesale-price policy chosen by the decentralizedmanufacturer. It is exogenous because a decentralized price strategy has no effect on thevalues of either Q*i or Q*j. Thus, the manufacturer’s profit from a wholesale-price policy isendogenouswhereas the basic parametric values Q* andx areexogenous.

Manufacturer profit is also affected by fixed costs at retail and at manufacturer. Thelatter value (F) affects each price-policy equally, shifting manufacturer profit uniformlyupward or downward in response to changes in F; hence we ignore F in what follows. Incontrast, the difference in retailers’ fixed costs, which may be of any sign, has a compleximpact on profitability because it affects what Zone is relevant for each wholesale-pricepolicy.


The Manufacturer-Optimality Conditions

In this sub-Section we investigate how the manufacturer’s choice of a wholesale-pricepolicy varies with the parameters that are encapsulated within the dimensions Q* andx.For ease of exposition we begin with Figure 4, drawn for the special case of equalretailers’ fixed costs; this provides a starting point for our analysis. Also, because themarketing science literature on distribution channels largely ignores fixed costs, this caseprovides the basis for a comparison with extant knowledge.15,16

Within the unit-square of Figure 4 we identify four Regions, each of which has its owndistinctive, manufacturer-optimal wholesale-price strategy. These Regions are defined interms of the following three Conditions, which we phrase as questions:

Condition 1: Does coordination with a channel-coordinating menu of two-part tar-iffs alwaysmanufacturer-profit dominate a sophisticated Stackelbergtariff?

TABLE 1A

The Sophisticated Stackelberg Two-Part Tariff:Channel Members’ Profits by Zone

Channel Member’s Profits in Zone ZjSS*

The jth retailer: pjSS*~j! 5 0

The ith retailer: piSS*~j! 5 2F3u~Q*i !

2 1 4~b 2 u!Q*iQ*j 2 ~4b2 3u!~Q*j !2

~2b1 u!2 G2 ~fi 2 fj!

Manufacturer: PMSS*~j! 5 F~4b2 2 4bu 1 9u2!~Q*i !

2 2 2~4b2 2 16bu 1 3u2!Q*iQ*j1 ~20b2 2 12bu 1 u2!~Q*j !

2

2~b 2 u!~2b1 u!2G2 2fj 2 F

Channel: PCSS*~j! 5 F~4b2 1 8bu 2 3u2!@~Q*i!

2 1 ~Q*j !2# 1 2~4b2 1 5u2!Q*iQ*j

2~b 2 u!~2b1 u!2 G2 fi 2 fj 2 F

Channel Member’s Profits in Zone ZijSS*

The ith & jth retailers: piSS*~a! 5 0 5 pj

SS*~a!

Manufacturer: PMSS*~a! 5 HPC

SS*~j! 18a~1 2 a!~b 2 u!~Q*i 2 Q*j !

2

~2b1 u!2 JChannel: PC

SS*~a! 5 PMSS*~a!

Value of a:1 $ a ;

~~fi 2 fj! 2 LSS*!

~USS* 2 LSS*!$ 0

Channel Member’s Profits in Zone ZiSS*

Reversing i’s and j’s in the profit expressions for Zone ZjSS* yields the results for Zone Zi

SS*.


Condition 2: Does sophisticated Stackelberg maximizationalways manufacturer-profit dominate a channel-coordinating menu when differences inretailers’ fixed costs are “large”? (By “large” we mean (fi 2 fj) .USS*.)

Condition 3: Does coordination with a quantity-discount scheduleevermanufactur-er-profit-dominate coordination with a menu of two-part tariffs?

The following Table illustrates that the answer to each question is “yes” in only a singleRegion:

TABLE 1B

The Channel-Coordinating Menu of Two-Part Tariffs:Channel Members’ Profits by Zone

Channel Member’s Profits in Zone ZjMenu*

The jth retailer: pjMenu* 5 0

The ith retailer: piMenu* 5 ~R*i 2 fi! 2 ~R*j 2 fj 1 Bj!

Manufacturer: PMMenu* 5 v* 1 @2R*j 2 2fj 1 Bj# 2 F

Margin on units sold:v* ;

u2~Q*i !2 1 2buQ*iQ*j 1 u2~Q*j !

2

b~b2 2 u2!

Channel:PC

Menu* 5@b~Q*i!

2 1 2uQ*iQ*j 1 b~Q*j !2#

~b2 2 u2!2 fi 2 fj 2 F.

Channel Member’s Profits in Zone ZijMenu*

The ith & jth retailers: piMenu* 5 0 5 pj

Menu*

Manufacturer:PM

Menu* 5@b~Q*i !

2 1 2uQ*iQ*j 1 b~Q*j !2#

~b2 2 u2!2 fi 2 fj 2 F.

Channel: PCMenu* 5 PM

Menu*

Channel Member’s Profits in Zone ZiMenu*

Reversing i’s and j’s in the profit expressions for Zone ZjMenu* yields the results for Zone Zi

Menu*.

Condition 1:Is PM

Menu* $ PMSS*

for all (fi 2 fj)?

Condition 2:Is PM

Menu* , PMSS* for

all USS* , (fi 2 fj)?

Condition 3:Is PM

Menu* # PMQD* for some

dQD* , (fi 2 fj) , LMenu*?

Region 1 Yes No NoRegion 2 No Yes NoRegion 3 No No NoRegion 4 No No Yes


Condition 1 holdsonly in Region 1; Condition 2 holdsonly in Region 2; none of theConditions holds in Region 3; and Condition 3 holdsonly in Region 4. We nowdescribe the optimal wholesale pricing by Regions. (The intuition and formal proofsfor our results are sketched in the Technical Appendix, available from the authors.)

The Manufacturer-Optimal Wholesale-Price Strategy by Region

In this sub-Section we identify the optimal pricing strategy within each Region definedin the preceding Table. In each case we present a decision rule that summarizes themanufacturer’s optimal wholesale-price policy.

Legend: x [ degree of inter-retailer competition; Q* [ relative retailer size; 1 [Region 1; 2 [ Region 2; 3 [ Region 3; and 4 [ Region 4. See the fourthSection for a description of the Regions.

FIGURE 4

The Manufacturer-Optimal Wholesale-Price Policy across theFour Regions at fi 5 fj


Region 1

Region 1 occupies a small, parabolic space in the upper left-hand corner of Figure 4.From the Table immediately above we see that in Region 1 the manufacturer alwaysprefers the channel-coordinating menu (1) to the sophisticated Stackelberg tariff (Condi-tion 1 is met) and (2) to the channel-coordinating quantity-discount schedule (Condition3 is not met). Thus the manufacturer-optimal wholesale-price strategy in Region 1 is:

● Employ the channel-coordinating menu of tariffs.

Notice that this Region exists only for a Q* ratio close to zero and ax ratio close toone.17 To understand what this means consider the special case of equal costs acrossretailers (ci 5 cj). Under this assumption inter-retailer differences arise solely fromvariations in base levels of demand (Ai and Aj). Numerical analysis makes clear that theparametric values necessary to be in Region 1 yield a peculiar result: the ith retailer hasa slightly higher equilibrium price but sells a massively larger quantity—even though thetwo retailers are near perfect competitors! We find it difficult to conceive of a real-worldexample that is compatible with these characteristics so we are inclined to regard Region1 as a rare occurrence. This is important because Region 1 is the only Region of theunit-square within which channel coordination is optimal for all values of (fi 2 fj).

Region 2

Region 2 occupies the bottom of Figure 4. As in Region 1, the channel-coordinatingmenu always dominates the channel-coordinating quantity-discount schedule (Condition 3is not met). The manufacturer’s optimal wholesale-price strategy is a choice between thechannel-coordinating menu and the sophisticated Stackelberg tariff. But Region 2 isdefined by the condition that the sophisticated Stackelberg two-part tariff is manufacturer-preferred to the channel-coordinating menu for any (fi 2 fj) . USS*. Two other conditionsof Region 2 influence the optimal manufacturer price-strategy. First, it can be shown thatthe sophisticated Stackelberg two-part tariff is manufacturer-preferred to the channel-coordinating menu when (fi 2 fj) , LSS*. Second, we have already established that,because the channel-coordinating menu extracts all profits from both retailers, the menudominates the sophisticated Stackelberg tariff whenever LMenu* # (fi 2 fj) # UMenu*.

Taken together, these three conditions imply the existence of two distinct two (fi 2 fj)values at which the sophisticated Stackelberg and the menu policies are manufacturer-profit-equivalent. Let V1 denote the critical value in the interval LSS* , V1 , LMenu* thatsatisfies the condition:

~f i 2 f j! 5 V1f PMMenu* 5 PM

SS* (29)

Similarly, let V2 denote the critical value located in the interval UMenu* , V2 , USS* thatsatisfies:



SS* (30)

These critical values define the manufacturer-optimal wholesale-price strategy in Region2:

● If (f i 2 fj) , V1 employ the sophisticated Stackelberg tariff;● If V 1 # (fi 2 fj) # V2 use the channel-coordinating menu of tariffs; and● If (f i 2 fj) . V2 employ the sophisticated Stackelberg tariff.

We numerically illustrate this rule in the next Section.

Region 3

Region 3 occupies the middle of Figure 4. This Region is defined by the fact thatnoneof the three Conditions hold. This is consistent with using the sophisticated Stackelbergtariff at “small” values of (fi 2 fj) and the channel-coordinating menu at higher values.(Because Condition 3 is not met the quantity-discount schedule is not manufacturer-preferred anywhere in this Region.)

We can show that this strategy is indeed manufacturer-optimal by use of the followingresults. (We stress that these results hold in Region 3, not throughout the unit-square.)First, for (fi 2 fj) , LSS* the manufacturer prefers the sophisticated Stackelberg two-parttariff to the channel-coordinating menu. (This statement also held in Region 2.) Second,for any (fi 2 fj) $ LMenu* the manufacturer prefers the channel-coordinating menu oftariffs. Taken together, these two conditions imply the existence of a (fi 2 fj) value atwhich the sophisticated Stackelberg tariff and the menu are manufacturer profit-equiva-lent. This critical value, which we denote as V3, is located in the interval LSS* , V3 ,LMenu*. It satisfies the condition:


SS* (31)

Because this Region is characterized by a single critical value, we summarize themanufacturer-optimal wholesale-price strategy in Region 3 as:

● If (f i 2 fj) , V3 employ the sophisticated Stackelberg tariff; and● If (f i 2 fj) $ V3 utilize the channel-coordinating menu of tariffs.

Region 4

Region 4 occupies the upper right-hand portion of Figure 4. This Region is defined bythe fact that, for some values of (fi 2 fj), the manufacturer prefers the channel-coordi-nating quantity-discount schedule to the channel-coordinating menu of tariffs. It can beshown that this occursonly whendQD* , LMenu*.


Three results are needed to describe fully this Region’s optimal pricing strategy. First,because the manufacturer obtains all channel profit at (fi 2 fj) 5 dQD*, the channel-coordinatedquantity-discount schedule is manufacturer-preferred at this point (and in its vicinity). Second,for any (fi 2 fj) , LSS* the manufacturer prefers the sophisticated Stackelberg two-part tariffto the channel-coordinating quantity-discount schedule. Third, for any (fi 2 fj) . LMenu* themanufacturer prefers the channel-coordinating menu of tariffs.

Taken together, these three conditions imply the existence of two distinct (fi 2 fj)values at which the sophisticated Stackelberg and the menu policies are manufacturer-profit-equivalent. In particular, let V4 denote the critical value that is located in the intervalLSS* , V4 , dQD* and that satisfies the following condition:

~f i 2 f j! 5 V4f PMQD* 5 PM

SS* (32)

Similarly, let V5, located indQD* , V5 , LMenu*, denote the critical value satisfying thecondition:


QD* (33)

Then the manufacturer-optimal wholesale-price strategy in Region 4 may be defined as:

● If (f i 2 fj) , V4 employ the sophisticated Stackelberg tariff;● If V 4 # (fi 2 fj) # V5 use the channel-coordinating quantity-discount schedule; and● If (f i 2 fj) . V5 apply the channel-coordinating menu of tariffs.

Regional Summary

We opened this Section by asking (in effect) “Is it more profitable for the manufacturerto coordinate or not to coordinate the channel?” The following Table summarizes ourobservations. In brief, the answer to our question is dependent on the difference inretailers’ fixed costs (fi 2 fj) and on both the relative size of the retailers (Q*) and thedegree of competition between them (x). The pair^Q*, x& determines which Region isrelevant and the absolute difference in retailers’ fixed costs determines which wholesale-price policy is manufacturer profit-preferred.

Manufacturer-Optimal Wholesale-Price Strategies

(fi 2 fj) # LSS* LSS* , (fi 2 fj) , max(USS*, UMenu*) max(USS*, UMenu*) # (fi 2 fj)

Region 1 Menu Menu MenuRegion 2 SS SS, Menu, SS SSRegion 3 SS SS, Menu MenuRegion 4 SS SS, QD, Menu Menu

Legend: Menu [ Channel-Coordinating Menu of Two-Part TariffsSS [ Channel Non-Coordinating Sophisticated Stackelberg Two-Part TariffQD [ Channel-Coordinating Quantity-Discount Schedule


Relationship to the Literature

In this sub-Section we frame our results relative to three classic articles in the analyticalchannels literature. We shall show that two of them are special cases of our model whereasthe third may be interpreted within the context of Figure 4.

First, Jeuland and Shugan (1983) examined a bilateral monopoly channel. The parallelwithin our context is given by assuming that the optimal output ratio Q* is zero. Of course,this effectively means that the degree of substitutability is also zero since there is only oneretailer! In our language, the Jeuland and Shugan bilateral monopoly model is an analysisof the manufacturer-optimal wholesale-price strategy under the assumptions Q*5 0 andx 5 0. (Graphically this is a solitary point—the lower left-hand corner—of Figure 4.)They showed that a properly specified quantity-discount schedule permits the manufac-turer to maximize channel profit, all to its own benefit.18 Second, Moorthy (1987) used thesame bilateral monopoly model to demonstrate that the manufacturer can obtain allchannel profit with a channel-coordinating two-part tariff.

In the context of our model it can be proven that, under the restricted parametric valuesQ* 5 0 andx 5 0, the sophisticated Stackelberg tariff also maximizes channel profit andpermits the manufacturer to extract all profit from the retailer. This is a critical feature ofthe sophisticated Stackelberg tariff. Unlike the naı¨ve Stackelberg tariff that can nevercoordinate any channel, the sophisticated version of the Stackelberg approach doescoordinate a bilateral monopoly channel. Thus, our menu, our quantity-discount scheduleand our sophisticated Stackelberg tariff yield the same behavioral results in the specialcase of a bilateral monopoly. However, our approach of adding a second, competingretailer expands our analytical capability from a single corner point to the entire unit-square.

Third, McGuire and Staelin (1983) examined a market containingtwo channels withidentical parametric values—hence the channels have equal output.19 This is equivalent toassuming that the optimal output ratio is unity (Q*5 1); however, the degree ofcompetition may range from zero to one. It is easy to show that a properly specifiedquantity-discount schedule permits the manufacturer to maximize channel profit andextract all profit from both retailers—provided fixed costs at retail are identically zero [asis assumed by McGuire and Staelin (1983)]. Similarly, a channel-coordinating menu hasthe same properties, but due to the fact that the retailers are identical the menu has a singleelement—it is a degenerate menu. Finally, under this set of parametric values thesophisticated Stackelberg tariff also maximizes channel profit and extracts all profit fromboth retailers—again duplicating the menu and quantity-discount schedules. Thus, bypermitting our retailers to have different parametric values we are able to extend theanalysis of competing retailers from the special case of equal market shares—a singlevertical line in Figure 4—to all possible market shares as depicted by the unit square.Nonetheless, there is a fascinating parallelism between our work and their work. McGuireand Staelin (1983) were concerned with how the degree of competition affects a manu-facturer’s decision whether to vertically integrate or to utilize an independent retailer. Ourapproach deals with how the degree of competition affects a manufacturer’s decision to


coordinate or not to coordinate the channel. In the next Section we illustrate thisphenomenon.

We conclude by stating that for the models analyzed in this sub-Section neither achannel-coordinating menu nor a channel-coordinating quantity-discount schedule hasany inherentadvantage for a manufacturer relative to a channel non-coordinating sophis-ticated Stackelberg. The sophisticated tariff is a powerful tool for a manufacturer even forthe special cases of selling through a single retailer or serving a pair of competing, butidentical, retailers.

COORDINATION VERSUS NON-COORDINATION:PRACTICAL ILLUSTRATIONS

We have seen in Figure 4 that the manufacturer’s choice of a wholesale-price policy isdependent upon relative retailer size and the degree of inter-retailer competition when theretailers have equal fixed cost. In this Section we numerically illustrate the implicationsfor the manufacturer-optimal wholesale-price policy when the retailers’ fixed costs areunequal. In effect, by analyzing different fixed cost values we are converting ourunit-square to a “unit-cube” with the third dimension being the difference in retailer fixedcosts.20

As a practical matter it is clear that whilewecan write out the conditions under whichcoordination or non-coordination is manufacturer-profit-optimal, managers have a moredifficult time making the same calculations. At the heart of this differential capability isthe fact that as academics we may assume full information about parametric values, butmanagers must actually work with imperfect knowledge of the three fundamental para-metric values: relative retailer size (Q*), the difference in fixed cost (fi 2 fj), and thedegree of competition (x). In general we expect managers to have an excellent grasp ofQ*, but to have less accurate information about the other two dimensions. Therefore, toillustrate the impact of these two parameters we holdx fixed in the first sub-Section whilevarying (fi 2 fj). In the second sub-Section we hold (fi 2 fj) fixed and varyx.

Illustration of the Effects of Changes in (fi 2 fj)

To keep our presentation manageable we present an example based on a singlemagnitude of competition. We work with the parametric values Ai 5 150, Aj 5 100, ci 5cj 5 C 5 $10, and (b2 u) 5 0.5. Hence Q*i 5 70, Q*j 5 45 and relative retailer size isQ* [ (Q*j/Q*i) 5 0.643. Note that by focusing upon a single magnitude of competition—avertical slice of Figure 4—we can clearly delineate the effects of the other two factors:xand (fi 2 fj). This particular slice is compatible with Regions 2 through 4. Other possible“slices” would generate similar observations (except at very low magnitudes of compe-tition where Region 1 would appear at sufficiently highx values). To further simplifymatters we set F5 $1,000 and fj 5 $0 so that (fi 2 fj) 5 fi.


We examine the following three sets of (b,u) parametric values:

These three examples represent progressively higher degrees of competition. Variationsin the degree of competition may be interpreted in terms of geographic space or customerservice. In the spatial realm, close geographic proximity of the retailers implies a highdegree of competition. For example, two retailers located in the same shopping mall willbe stronger competitors than they would be if they were in different malls. In the servicerealm, comparable customer service across the two retailers implies stronger competitionthan differential service would imply. Other marketing mix factors could be considered inthe same vein.

A Low Degree of Competition

We start with our lowest degree of competition:u/b 5 0.286. Details are presented inTable 2. The first profitability column is based on an arbitrary value (fi 2 fj) 5 $0. As canbe inferred from the Table, the sophisticated Stackelberg strategy is manufacturer-profitdominant for all 0# fi , $3,567.61. Thus the critical value V1 is $3,567.61. The menuis strictly preferred for $3,567.62, fi , $3,792.90. Finally, at fi . $3,792.91 (the criticalvalue V2) the sophisticated Stackelberg strategy is again preferred. Profit columns 2 and

TABLE 2

Region 2: The Sophisticated Stackelberg Tariff and The Channel-CoordinatingMenu

Manufacturer Profit Rankings

PMSS* . PM

Menu* PMSS* 5 PM

Menu* PMMenu* . PM

SS* PMMenu* 5 PM

SS* PMSS* . PM

Menu*

fi $ 0.00 $3,567.61 $3,680.26 $3,792.91 $4,082.03P*C $12,572.22 $9,004.61 $8,891.96 $8,779.31 $8,490.19PM

SS* $ 9,217.19 $8,998.49 $8,878.87 $8,733.27 $8,240.63PM

Menu* $ 8,998.49 $8,998.49 $8,891.96 $8,733.27 $8,155.03PM

QD* $ 8,921.43 $8,921.43 $8,862.50 $8,637.20 $8,058.96

Legend: PMX* [ manufacturer profit with pricing policy “X.” P*M [ Vertically-Integrated Channel; PM

SS* [Sophisticated Stackelberg; PM

Menu* [ Channel-Coordinating Menu of two-part tariffs; PMQD* [

Channel-Coordinating Quantity-Discount Schedule; fi [ fixed cost of the ith retailer.Note: We set the following parametric values Ai 5 150, Aj 5 100, ci 5 cj 5 C 5 $10, fj 5 $0, and F 5

$1,000. We hold (b 2 u) 5 0.5 and specify b 5 0.7 and u 5 .2; USS* 5 $4,082.03, LSS* 5$3,105.47, UMenu* 5 $3,746.87, LMenu* 5 $3,573.73 and dQD* 5 $3,650.79. It follows that Q*i 570, Q*j 5 45, p*i 5 $148.89 and p*j 5 $121.11.

b u x [ u/b Region

0.7 0.2 0.286 20.9 0.4 0.444 32.0 1.5 0.750 4


4 provide information on the fi level at which the menu and the sophisticated Stackelbergstrategies generate manufacturer profit-equivalence, whereas profit column 5 (at whichfi 5 USS*) demonstrates that at a sufficiently high fixed cost level non-coordination isagain manufacturer-preferred. Note that the quantity-discount strategy can never bepreferred for this set of parametric values because the chosen values of b andu areincompatible with Condition 3. For completeness we note that the sum of the ith retailer’sfixed cost and fully coordinated channel profit (denoted asP*C) is a constant in this andsuccessive Tables.

A Moderate Degree of Competition

At a slightly higher degree of competition (x 5 .444) the sophisticated Stackelbergstrategy is manufacturer-profit-preferred only at “low” levels of the fixed cost difference;it does not again become manufacturer-preferred at high values. For the example at handwe have the sophisticated Stackelberg tariff being manufacturer-profit-optimal for 0#fi , $2,633.31 (the critical value V3) and the menu being preferred at all higher (fi 2 fj).To conserve space we do not provide a Table to demonstrate this result.

Our terminology “low difference in fixed costs” embraces the very real case of anegative difference (i.e., fi , fj). The notion that the smaller retailer may have a higherfixed cost is plausible once we recognize that fixed cost incorporates opportunity cost, landrent, and other factors. If the smaller quantity retailer is located in a mall, while the largeris not, the former may pay higher rent than does the latter. Similarly, if the smaller retaileris an upscale boutique, while the larger retailer is a mid-market purveyor, then the formermay have a greater opportunity cost than the latter. What is important from this briefargument is that a “low” (or even a negative) value of (fi 2 fj) is eminently plausible.

A High Degree of Competition

Finally, the impact of a very high degree of competition (x 5 .750) is characteristic ofRegion 4. In particular, the sophisticated Stackelberg tariff is manufacturer-preferred atlow levels of (fi 2 fj), the quantity-discount schedule is preferred at “moderate” levels of(fi 2 fj), and the menu is manufacturer-optimal at even higher fixed cost differences. Theprecise critical values are V4 5 $1,094.47 and V5 5 $1,131.69.

Channel-Coordinating Wholesale-Price Policies

Contrasting these results demonstrates that there is an inverse relationship betweenthe degree of competition and the (fi 2 f j) value at which the sophisticated Stackelbergand a channel-coordinating policy attain manufacturer profit-equality. The result witha high degree of competition illustrates the benefit of a quantity-discount schedule tothe manufacturer: it allows channel-coordination without the potential defection costassociated with offering a menu of tariffs. A menu enables the manufacturer to


maximize total channel profit over a wide range of (fi 2 f j) values, but coordinationcomes at the price of sacrificing potential manufacturer profit to prevent defection.Defection is more likely when the elements of the menu are similar—which is the casewhen there is a high degree of competition between the retailers. Thus it should notbe surprising that the quantity-discount strategy is a viable supplement to the channel-coordinating menu only at high degrees of competition when the retailers are ofroughly comparable size.

Illustration of the Effects of Changes in the Degree of Competition

In this sub-Section we utilize a graphical presentation—based on numerical analy-sis—to illustrate the effect on manufacturer’s profit of changes in the degree of compe-tition (x) under each of our wholesale-price policies. We evaluate the impact ofx overfour combinations of Q* and (fi 2 fj):

These four examples represent sharply different relative retailer size and fixed costs.In Figures 5 and 6 we examine the case of the smaller retailer having a 40% marketshare (Q* 5 .67 5 40%/60%). This relative retailer size roughly reprises ourparametric value from Tables 2 through 4 (where we had Q*5 .643). Moreimportantly, it is representative of competition between rivals who are of approxi-mately equal size—metaphorically Macy’sversusMay Company. We shall refer tothis as “competition between equals.” In Figures 7 and 8 we examine the case of thesmaller retailer having a 10% market share (Q*5 .11 5 10%/90%). This isrepresentative of any “Davidversus Goliath” competition (e.g., a small apparelboutiqueversusa department store or a convenience food storeversusa supermarket).We shall refer to this as “competition between unequals.” In both cases we show themanufacturer-profit relationship between each of our wholesale-price policies at everypossible degree of competition. In Figures 5 and 7 we focus on an absence of fixedcosts (again reflecting general usage in the literature) whereas in Figures 6 and 8 weset fixed cost at a substantial level, 65% of a retailer’s net revenue in a coordinatedchannel.

Competition Between Equals

When the rival retailers are of roughly comparable size we find that the sophisticatedStackelberg tariff manufacturer-profit dominates both of the channel-coordinating whole-

Q* 5 .67 Q* 5 .11

fi 5 0 5 fj Figure 5 Figure 7fi 5 .65 5 fj Figure 6 Figure 8


sale-price policies atall degrees of competition. This dominance is relatively small at highvalues ofx, but is distinctly noticeable at lower levels of competition, especially whenfixed costs are substantial. The general inference, that the sophisticated Stackelberg tariffis a powerful managerial tool, holds as long as fixed costs are less than two-thirds thechannel coordinated net revenue. At larger fixed cost levels the menu and (sometimes) thequantity-discount schedule are manufacturer-profit-optimal athigh levels ofx. However,even when this is the case the sophisticated Stackelberg tariff generates a minimum of96% of the manufacturer-profit obtained with the optimal coordinating policy. Interest-ingly, at lower levels ofx the sophisticated Stackelberg tariff outperforms the channel-

Legend: The labels signify the following values:x [ (u/b) 5 Degree of Inter-Retailer CompetitionSS [ PM

SS*/P*C 5 Manufacturer Profit under the Non-Coordinating Sophisticated Stack-elberg Tariff relative to Integrated Channel Profit (dashed line)M [ PM

Menu*/P*C 5 Manufacturer Profit under Channel-Coordinating Menu of Tariffsrelative to Integrated Channel Profit (solid line)QD [ PM

QD*/P*C 5 Manufacturer Profit under Channel-Coordinating Quantity-Dis-count Schedule relative to Integrated Channel Profit (dotted line)

Note: In this Figure the dotted QD line is obscured by the solid M line.

FIGURE 5

Optimal Manufacturer Pricing Strategy When fi 5 0 5 fj and Q* 5 .67


coordinating policies by well in excess of 10%. We conclude that, when retail competitionis between “equals” who have roughly equal fixed costs, the sophisticated Stackelbergwholesale-price policy—which does not coordinate the channel—is apt to be the singlebest price-policy for the manufacturer to adopt when the precise degree of competition isunknown. It is the actual optimal policy over most of parametric space, and when it isnot-optimal it is an excellent alternative choice because it does not fall much short of theoptimum. In contrast, when the coordinating schedules are not optimal they are often poorperformers relative to the sophisticated Stackelberg tariff.21


SS*/P*C 5 Manufacturer Profit under the Non-Coordinating SophisticatedStackelberg Tariff relative to Integrated Channel Profit (dashed line)M [ PM



Note: In this Figure the dotted QD line is partially obscured by the solid M line.

FIGURE 6



Competition Between Unequals

When the rival retailers are of dramatically different size we again find that the sophis-ticated Stackelberg tariff manufacturer-profit dominates both channel-coordinating whole-sale-price policies atall degrees of competition. The basic message from Figures 7 and 8is the same as above: the sophisticated Stackelberg two-part tariff is manufacturer-profit-optimal; channel-coordination isnot in the manufacturer’s best interest. What is surprisingis that with a substantial disparity in competitor size the sophisticated Stackelberg tariffmassively out-performs the two channel-coordinating wholesale-price policies at lowdegrees of competition. Of course, asx goes to one the manufacturer’s advantage fromnon-coordination shrinks. Our comments in the previous paragraph (on the relative merits


SS*/P*C 5 Manufacturer Profit under the Non-Coordinating Sophisticated Stack-elberg Tariff relative to Integrated Channel Profit (dashed line)M [ PM



FIGURE 7



of the price policies when fixed costs exceed two-thirds of the net revenue undercoordination) continue to hold. At a substantial disparity in retailer size (i.e., at all Q*,.07), combined with a very high degree of competition, Region 1 becomes relevant (seefootnote 17 for details). Within this Region the menu is manufacturer-optimal. However,our numerical analysis shows that the menu only generates about 1% higher profit for themanufacturer than does the sophisticated Stackelberg tariff. (At degrees of competitionless than “very high” the sophisticated Stackelberg policy is optimal.)

The Practicality of Coordination versus Non-Coordination

We have seen that any of the three wholesale-price policies may be manufacturer-profit-optimal, depending on the specific parametric values of (1) the degree of compe-


SS*/P*C 5 Manufacturer Profit under the Non-Coordinating SophisticatedStackelberg Tariff relative to Integrated Channel Profit (dashed line)M [ PM



FIGURE 8

Optimal Manufacturer Pricing Strategy When fi 5 .65 5 fj and Q* 5 .11


tition (x), (2) the relative size of the competitors (Q*), and (3) the absolute difference inthe retailers’ fixed costs (fi 2 fj). However, extensive numerical analysis strongly suggeststhat the noncoordinating sophisticated Stackelberg strategy is the manufacturer-preferredwholesale-price policyunlesscompetition is very unequal or the level of fixed costs isvery high. Even then, competition must bevery intense for coordination to be manufac-turer-optimal. Further, when a coordinating wholesale-price policy does outperform thesophisticated Stackelberg policy it does so only by a small margin. The reverse is not true.When non-coordination is manufacturer-preferred the sophisticated Stackelberg policysubstantially outperforms either coordinating policy over a wide-range of parametricvalues. From this we conclude that unless a supply-chain manager is quite certain of thespecific parametric values that its retailers face, and unless those values fall in narrowlycircumscribed ranges, use of a noncoordinating sophisticated Stackelberg wholesale-pricestrategy is the manufacturer’s best option. In short, the standard prescription from theliterature to coordinate the channel rarely generates a substantial improvement in manu-facturer-profit over what could be obtained without seeking coordination and often yieldssharply lower profit. Stated simply, channel coordination is not all that it is cracked up tobe.

DISCUSSION AND CONCLUSION

Should a manufacturer that sells through competing retailers seek to coordinate every oneof its retail relationships? The marketing science literature has long argued “Yes, becausecoordination maximizes total channel profit, profit that may be re-distributed betweenchannel members to the benefit of all.” This argument is clearly correct for asinglechannel dyad, but its extension to multiple dyads is logically predicated upon themanufacturer “cutting a separate deal” with each of its retailers. However, commonpracticalities such as administrative, bargaining and contract development costs (Lafon-taine, 1990) and legal restrictions such as the Robinson-Patman Act (Monroe, 1990)suggest that manufacturers do not regularly cut such deals with their retailers. Rather, theevidence is that manufacturers typically employ a wholesale-price policy that is commonto many retailers.

Recent analytical work by Ingene and Parry (1995b) raised the possibility that channel-coordination mightnot be in the manufacturer’s interest when competing retailers aretreated comparably. Their results were based on a limited number of specific wholesale-price policies. Thus it has not been clear if they glimpsed a broadly generalizable principle(that channel-coordination is not all that it has been cracked up to be) or if they found aset of special circumstances under which channel-coordination is non-optimal for themanufacturer.

In this paper we have clarified the issue of generalizable principle versus specialcircumstances by analyzing a basic channel situation: a single manufacturer sellingthrough a pair of differentiated, competing retailers facing their own unique demand andcost curves. We constrained the manufacturer to treat its retailers comparably and weassumed a fixed cost [denoted as fk, k [ (i, j)] for each retailer. The legal and practical


issues cited above justify our equal treatment constraint; the nontrivial opportunity cost ofchannel participation justifies our assumption of fixed costs.

Our analytical results make two critical contributions the theory of distribution channelstructure. First, we now know that a manufacturer’s choice of an optimal wholesale-pricepolicy is dependent upon three factors: the difference in retailers’ fixed costs (fi 2 fj), therelative size of the retailers (i.e., the magnitude of competition), and the degree ofinter-retailer competition. Second, we proved that over virtually all possible combinationsof degreesand magnitudesof inter-retailer competition there exist plausible (fi 2 fj)values for which channel-coordination isnot in the manufacturer’s best interest. Giventhat our analysis compared a sophisticated Stackelberg tariff (the best of all singletwo-part tariffs) with a channel-coordinating menu of two-part tariffs, our conclusion isdefinitive over the range of all possible two-part tariffs.22 (A two-part tariff, whichconsists of a per-unit fee and a fixed fee, is common in franchising. More importantly, itis a nonlinear quantity-discount schedule: the retailer’s average, per-unit acquisition costdeclines with increases in quantity purchased.) The sophisticated Stackelberg tariff wasalso shown to manufacturer profit-dominate a linear, channel-coordinating quantity-discount schedule. Therefore, we may safely conclude thatchannel coordination is not allthat it has been cracked up to be.

We believe that a well-intentioned “managerial implication” from the literature (“seekto coordinate your channel”) may have inadvertently misled managers. Such adviceencourages pursuit of a strategy that is manufacturerprofit-reducingover a wide range ofparametric values. Our analysis clearly demonstrates that there is no reason to expect it tobe in the best interests of different manufacturers to offer their retailers the samewholesale-price strategy; just as each dyad within a channel faces different demand andcost conditions, so does every manufacturer face diverse parametric values. It follows thatidentical advice(“seek channel coordination”)cannot be appropriate for all channels atall times. Finally, if one insists on looking for asinglerule-of-thumb to guide wholesale-price decisions, the appropriate advice to supply-chain managers is to “forget channelcoordination and use a sophisticated Stackelberg tariff.”

The larger and (we believe) more important message is that commonly held modelingassumptions are not always innocuous. Within the context explored here such keyassumptions from the extant literature are (1) fixed costs do not affect behavior within achannel and (2) conclusions drawn from bilateral monopoly models can be generalized tonon-bilateral situations. We have proven that the difference in retailers’ fixed costs mattersa great deal, for it affects which wholesale-price policy the manufacturer will offer andhow much economic profit each retailer will earn. Further, we have shown that conclu-sions valid in a single dyadic framework are generally inaccurate for multiple dyads. Thisindicates that other channel management recommendations may be similarly affected byassumptions that seem innocuous.

One candidate for future assessment involves channel breadth. In this paper we havefollowed the conventional modeling assumption of a fixed channel breadth. We did notpursue the question of manufacturer-optimal channel breadth because doing so wouldhave made an already intricate problem even more complicated. We speculate that, if oneretailer’s profit contribution is small relative to the other retailer’s, then the manufacturermay find it more profitable to deal solely with the larger retailer. The reason is that by


serving a single retailer, thus sacrificing market coverage, the manufacturer can extract allchannel profit through a properly specified two part-tariff. We believe our intuition isreasonable, but it should be confirmed in future research. Similarly, we believe thatassumptions in other realms of the broad field of supply-chain management should berigorously investigated to ascertain their robustness. This conviction follows directly fromthe results presented in this paper, which provide dramatic evidence that seeminglyinnocuous assumptions can dramatically affect the conclusions reached regarding optimalmarketing decisions.

Acknowledgment: Appreciation is expressed to the guest editors and anonymous reviewers forhelpful comments on earlier drafts.

NOTES

1. Channel coordination is often linked to the concept of Pareto-optimality. In the context ofchannels, a situation is Pareto-optimal if and only if no channel member can be made better offwithout harming another channel member.

2. An exception is wholly owned subsidiaries—such as Sherwin–Williams—where differentprices across outlets are allowed.

3. Jeuland and Shugan (1983) demonstrated that coordination could be achieved in a bilateralmonopoly context through the judicious application of an appropriately valued quantity-discountschedule. Ingene and Parry (1995b) extended this argument to the case of competing retailers. Theymentioned the possibility of a channel-coordinating menu of two-part tariffs.

4. In this paper we prove the existence of a feasible, channel-coordinating menu of two-parttariffs.

5. Ingene and Parry (1995b) sketched a channel-coordinating menu without providing muchdetail.

6. Ingene and Parry (1995a) examined optimal channel breadth for a manufacturer selling toany number ofnon-competing retailers. They used a general demand curve rather than the lineardemand curve of Equation (1). A general demand curve precludes obtaining closed-form solutions.The assumption of no competition between retailers dramatically simplifies the problem, althougha glance at that paper should make clear that the solution is still very complex.

7. Recall that retailer-specific transfer-prices are legally permissible for wholly owned sub-sidiaries.

8. Although this sub-Section follows Ingene and Parry (1995b), we develop several points thatdid not appear in that paper. In particular, only Equations (7)—(10) appeared (1995b) and did soonly for the special case of b5 1. The Zones that are developed in expressions (12) are original withthis paper.

9. A Technical Appendix, available from the authors, provides details on the wholesale marginand fixed fee by Zones.

10. Expressions (14)–(16) appeared in Ingene and Parry (1998) with substantially differentnotation. Our notational change facilitates the ensuing comparison across wholesale-price regimes.

11. It is typical to attribute an absence of “channel coordination” (PCSS* , P*C) to the “double

marginalization” that occurs with Stackelberg pricing—that is, to the fact that manufacturer andretailers obtain a positive margin (Gerstner and Hess 1995). Equivalence between lack of channel


coordination and double marginalization is strictly accurate only within a bilateral monopoly model.With competing retailers the channel-optimal wholesale-price entails apositiveper-unit manufac-turer margin, just as the optimal retail-price entails apositive margin for both retailers; withcompeting retailers “double marginalization” is required for channel coordination.

12. This Section substantially expands a two-page Section of Ingene and Parry (1995b). Thosepages included only two of the subsequent Equations—(18) and (19)—and then only for the specialcase of b5 1. The Zones discussed below were not developed, nor was the issue of possible“defection” accorded attention.

13. A fourth case appears to be theoretically possible: the ith retailer chooses tarifft*j and the jth

retailer chooses tarifft*i. However, it can be shown that this possibility canneverbe an equilibriumoutcome.

14. Incidentally, a comparison of the reduction in net revenues due to defection shows that Bj .Bi . 0.

15. Notice that fixed costs that are equal to each other is more general than fixed costs that areequal to zero.

16. Notice also that Figure 5 is drawn for fi 5 0 5 fj. A fundamentally similar Figure is obtainedfor all fi 5 fj , .66. At higher fixed costs values Region 1 becomes more prominent.

17. Specifically, the parabola can be characterized as .99$ x $ .85 and Q*# .07.18. Jeuland and Shugan used a general demand curve, while we assume a specific demand

curve. Our assumption is necessary to obtain analytical results when Q*Þ 0 andx Þ 0. They alsostressed that channel profit could be split between channel members by negotiation. Our textualcomment is that a greedy manufacturercan obtain all profit.

19. We are indebted to an anonymous reviewer for suggesting this comparison. We stress thatthe McGuire–Staelin analysis differs from the analysis presented in this paper in two key respects.(1) Each of their retailers is supplied by a different manufacturer. (2) They analyze an optimalone-part tariff—they do not consider a nonzero fixed fee; hence their results cannot entail channelcoordination except when a channel is vertically integrated.

20. Strictly speaking the unit-cube requires that each retailer’s fixed cost be normalized by itsnet revenue under coordination. This standardized fixed cost lies in the unit interval.

21. Inspiration for these Figures comes from McGuire and Staelin (1983); their Figure 2 showedthe manufacturer’s profit—at various levels of competition—for various channel structures.

22. If a noncoordinatingmenu of two-part tariffs were preferred to the tariffs explored here itwould reinforce our point: channel coordination is often manufacturer non-optimal. (We stress thatwe do not know if such a menu exists or if it could be calculated analytically.)

REFERENCES

Gerstner, E. and J. Hess. (1995). “Pull Promotions and Channel Coordination,”Marketing Science,14 (Winter) 43–60.

Ingene, C. and M. Parry. (1995a). “Coordination and Manufacturer Profit Maximization: TheMultiple Retailer Channel,”Journal of Retailing,71 (Summer) 129–151.

and . (1995b). “Channel Coordination When Retailers Compete,”Marketing Sci-ence,14 (Fall) 360–377.

and . (1998). “Manufacturer-Optimal Wholesale Pricing,”Marketing Letters,9 (1)65–77.


Jeuland, A. and S. Shugan. (1983). “Managing Channel Profits,”Marketing Science,2 (Summer)239–272.

Lafontaine, F. (1990). “An Empirical Look at Franchise Contracts as Signaling Devices,” Pitts-burgh: Graduate School of Industrial Administration, Carnegie-Mellon University.

McGuire, T. and R. Staelin. (1983). “An Industry Equilibrium Analysis of Downstream VerticalIntegration,”Marketing Science,2 (Spring), 161–192.

Monroe, K. (1990).Pricing: Making Profitable Decisions, New York: McGraw–Hill, Inc.Moorthy, S. (1987). “Managing Channel Profits: Comment,”Marketing Science,6 (Fall) 375–379.


is channel coordination all it is cracked up to be?

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