strategic delegation with multiproduct firms

Strategic Delegation with MultiproductFirms

RAFAEL MONER-COLONQUES

Department of Economic AnalysisUniversity of [email protected]

JOSE J. SEMPERE-MONERRIS

Department of Economic AnalysisUniversity of Valencia

[email protected] and IRES

AMPARO URBANO

Department of Economic AnalysisUniversity of [email protected]

This paper shows that a multiproduct firm may find it optimal not to delegatethe sales of all products and therefore to employ different distribution channelsfor different products. It faces the following trade-off: There is a strategic effectassociated with delegation, but if both products’ sales are delegated, intrafirmcompetition is not internalized. By delegating the sales of just one of the productswhile selling the other product directly—partial delegation—the multiproductmanufacturer strikes just the right compromise: The externalities between itsowns products are internalized partially while a strategic advantage is achievedagainst its rival single-product manufacturer. Partial delegation also holds ifboth products are sold by a common retailer; it dominates full delegation whenboth manufacturers are multiproduct firms.

We thank seminar participants at the Departament d’Economia i d’Historia Economica,Universitat Autonoma de Barcelona. We also are grateful to a coeditor and two refereesfor their useful comments and suggestions, which improved the presentation. Financialsupport from Ministerio de Educacion y Ciencia DGCYT under project PB-95-1074and from the Instituto Valenciano de Investigaciones Economicas, IVIE, gratefully isacknowledged. A previous version appears as IVIE WP 2000-19 under the title “ProductQuality and Distribution Channels.” Sempere-Monerris acknowledges the support ofthe Belgian research programs “Poles di Attraction Inter-universitaires” PAI P5/21 and“Action de Recherches Concertee” ARC 03/08-302.

c© 2004 Blackwell Publishing, 350 Main Street, Malden, MA 02148, USA, and 9600 Garsington Road,Oxford OX4 2DQ, UK.Journal of Economics & Management Strategy, Volume 13, Number 3, Fall 2004, 405–427

406 Journal of Economics & Management Strategy

1. Introduction

A question much analyzed in the literature on vertical relations hasbeen whether manufacturers wish to distribute their products throughindependent retailers (vertical separation) or wish to be the retailersof their own products (vertical integration). It typically is argued thatintegration serves agents inside the firm to coordinate their interests.However, in an oligopolistic setting, delegation of sales to independentretailers may be advantageous strategically for manufacturers. Thesetwo apparently conflicting views reflect a trade-off between verticalcontrol and strategic delegation, yet they can be reconciled by assumingmultiproduct firms. The present paper shows that multiproduct firmsmay find it optimal not to delegate the sales of all products and thereforeto employ different distribution channels for different products.

Lladro, a Spanish handcraft porcelain producer, has its own distri-bution network in Europe, the United States, and Asia, where porcelainwith their “flower and an ancient chemical symbol” tag is sold. Otherporcelain works of art, without the tag, also are manufactured by Lladro,but these are sold through privately owned dealerships. Burberry, theoriginal British luxury brand, sells in directly operated Burberry storesas well as through leading specialty retailers worldwide. In August 2003,Thomas Burberry brand will be available only in leading departmentstores such as Harrods, Selfridges, and Harvey Nichols. As for Spain,Adolfo Dominguez sells its collections in its own shops and franchises.However, its recently launched U collection for young people is soldonly at other franchised shops.

Earlier work on delegation has shown that single-product firmshave a unilateral incentive to delegate tasks to independent agents.Representative papers, where the final stage choices are in quantities,include Vickers (1985), Fershtman and Judd (1987), Sklivas (1987), andFershtman, Judd, and Kalai (1991). By delegating output choices, up-stream firms instruct their retailers or managers to choose an equilibriumoutput that is greater than under the standard (vertical integration)Cournot equilibrium. In fact, the vertically separated firm, if the rivalis integrated, achieves the outcome of a Stackelberg leader when thechoice variables are strategic substitutes. Hence, the previous litera-ture suggests that firms always should delegate for strategic reasons.1

Coughlan and Wernerfelt (1989) and Katz (1991) have shown thatresults change significantly if contracts are unobservable and/or can

1. The strategic advantage of delegation remains when the final stage variables arestrategic complements, as in McGuire and Staelin (1983) and Bonanno and Vickers (1988).However, in contrast with the Cournot case, delegation is both in the individual andcollective interest of manufacturers. Contracting with independent retailers is a means toalleviate the intensity of price competition.

Strategic Delegation with Multiproduct Firms 407

be renegotiated since the precommitment value of delegation is lost—see Irmen (1998) for a survey. We wish to show that the aforementionedphenomena can be found consistent with the Vickers-type model in astrategic setting when upstream firms are allowed to be multiproduct.2

We also will consider the possibility that the downstream firms sellmultiple products, and hence our analysis contributes to the literatureon the convenience of having separate or common retailers, as doneby Bernheim and Whinston (1998), Lin (1990), and O’Brien and Shaffer(1993).

The analysis of firms that sell multiple products makes the problemof delegation a more complex one, and the received intuition fromthe single-product case fails. Note that, although the strategic role ofdelegation remains, a multiproduct manufacturer cannot internalizecompetition among its own products if sales are given out to indepen-dent retailers. Additionally, it can opt for not delegating the sales of thefull product range, i.e., partial delegation. It will be shown that it cancoordinate better the intensity of intrafirm and interbrand competitionby delegating the sales of just one of the products while selling the otherproduct directly. In doing so, the multiproduct manufacturer strikes justthe right compromise: The externalities between its owns products areinternalized partially while a strategic advantage is achieved against itsrival single-product manufacturer. The previous examples indicate thatpartial delegation is a way of balancing such trade-off. Furthermore, par-tial delegation also holds if both products are sold by a common retailer;partial delegation dominates full delegation when both manufacturersare multiproduct firms.

In this paper we focus exclusively on studying whether previousresults on delegation with single-product firms generalize to the multi-product firm case. As such the model will be stylized to exclude channelcoordination issues, retailer buyer power, cost-related justifications, andso on. With this aim in mind we propose a noncooperative three-stagegame with observed actions. In the first stage, the multiproduct andthe single-product manufacturers simultaneously decide whether todelegate sales or wish to market the products themselves—i.e., theychoose their delegation policy. In the second stage, and dependingon their earlier choice, manufacturers select the terms of payment,a two-part tariff, for their contract with their respective retailers.

2. There exist few theoretical atempts to examine multiproduct firms’ competition inthe context of manufacturer–retailer relationships. Among these, Shaffer (1991) and Villas-Boas (1998) are worth a mention. These authors concentrate on channel coordinationproblems, i.e., what contractual arrangements or strategies allow a manufacturer whosells through independent retailers to obtain the profits that otherwise could get undervertical integration.


There is Cournot competition in the last stage of the game. The nextsection sets out the model and specifies the equilibria of the third-and second-stage subgames. In section 3 we examine whether themultiproduct manufacturer prefers delegation of both products’ salesto separate retailers rather than to a common retailer. Section 4 char-acterizes the equilibria of the full game. The case of two multiproductmanufacturers is taken up in section 5. Some extensions and remarksconclude.

2. The Model

Consider a market with two differentiated products, A and B. Their in-verse demand functions are assumed to be represented by the followinglinear system:3

pA = DA − QA − dQB ; pB = DB − QB − d QA, (1)

where pi and Qi are the price and the aggregate quantity of producti, i = A, B. Parameter d is strictly positive and smaller than 1, as owneffects on prices are greater than cross effects. It measures the degree ofinterbrand competition between products A and B, which are imperfectsubstitutes. When d approaches 1 products become closer substitutes,and interbrand competition increases.

Both A and B are produced under constant returns to scale withunit costs equal to cA and cB, respectively. There are two manufacturers,M1 and M2, and a competitive supply of retailers. We assume that M1 is amultiproduct firm and that M2 is a firm that produces just one product,say product B. We denote by qA, and qB1 the quantity of product Aand product B produced by M1, and by qB2 the quantity of product Bproduced by M2. Then QA = qA and QB = qB1 + qB2. Also, it is assumedthat DA > cA and DB > cB. No further assumption is made about whichmarket is relatively more profitable.

The two manufacturers and the retailers play the following mul-tistage game G with observed actions. In the first stage, manufacturersdecide simultaneously and independently whether to delegate sales toretailers or to sell the product directly to consumers; that is, they choosetheir delegation policy. Then, M1 may decide either (1) to produce and tosell both products himself (action N); (2) to hire a retailer for distributingproduct A and to sell product B himself (action A); (3) the reverse of(2) (action B); or (4) to delegate the sales of both products to independentand different retailers (action AB). Actions N and AB involve the use ofthe same types of channel for both products; action N corresponds to

3. A more general inverse demand structure is analyzed in a companion paperavailable at http://www.uv.es/sempere/jems-general.pdf. It covers the most widelyemployed models for differentiated products either of the address or nonaddress type.


vertical integration of the multiproduct manufacturer, whereas action ABwill be referred to as full delegation. On the other hand, actions A and Bsuppose a different type of distribution channel for each of the products,called partial delegation by the multiproduct manufacturer. ManufacturerM2 also chooses the way its product will be distributed: either sold by themanufacturer itself (action N) or through a retailer (action B). Any pairof delegation policies chosen by the manufacturers defines a delegationpattern.

In the second stage, each manufacturer decides on the terms ofthe two-part tariff contract to be signed with each of the correspondingretailers, if appropriate. Otherwise, no action is taken at this stage. Eachretailer, h, is supplied by its corresponding manufacturer at a constantunit price, wh, the transfer price, and has to pay an up-front fixed fee,Fh, for h = A, B1, B2. We assume that retailers are not differentiated inthe sense that consumers of product B receive the same utility no matterwhich retailer, either B1 or B2, is selling product B to them. Finally, allthe active agents (manufacturers and/or retailers) play simultaneouslyand independently a quantity game.

We look for the subgame perfect equilibria of this three-stage game.As usual, by backward induction, we begin by characterizing the third-stage Nash equilibrium quantities for each possible delegation patternand two-part tariff set of contracts chosen in the previous stages. Thenwe compute the subgame perfect Nash equilibrium choice of transferprices and up-front fixed fees by manufacturers, when appropriate.Finally, in the first stage, we find the subgame perfect Nash equilibriumdelegation pattern.

2.1 The Third-Stage Quantity Subgame

There appear to be three outputs in the market, qA, qB1, and qB2. Thus,we identify interbrand competition (between qA and both qB1 and qB2),intrabrand competition (between qB1 and qB2), and intrafirm competition(between qA and qB1). In our setting, the multiproduct manufacturerhas two decision variables to control for the intensity of intrafirmcompetition: (1) the transfer price(s), which is also a means to gain salesvis a vis its rival; and (2) the choice of the delegation policy. Hence, whenthe multiproduct firm decides not to delegate sales at all, it directlyinternalizes intrafirm competition, whereas under full delegation, theintensity of intrafirm competition is maximal.

The different subgames can be classified into two types: thosesubgames with two active sellers, one of which is a multiproduct seller,the (N, N) and (N, B) subgames; and the remaining six subgames withthree single-product active sellers. We begin by considering the latter


case. For any given triplet of marginal costs of the sellers, (rA, rB1, rB2),where rA ∈ {cA, wA}, rB1 ∈ {cB, wB1}, and rB2 ∈ {cB, wB2}, the followingequilibrium quantities are obtained:

q A(rA, rB1, rB2) = 3DA − 2d DB − 3rA + d(rB1 + rB2)2(3 − d2)

qB1(rA, rB1, rB2) = 2DB − d DA − (4 − d2)rB1 + (2 − d2)rB2 + drA

2(3 − d2)

qB2(rA, rB1, rB2) = 2DB − d DA − (4 − d2)rB2 + (2 − d2)rB1 + drA

2(3 − d2).

(2)

It is easy to check that the choice variables are strategic substi-tutes. The comparative statics yield the standard signs under oligopoly.Second-order conditions for a maximum and the required stabilityconditions are satisfied, given the assumptions on the demand schedule.

We next examine the subgames with two sellers, one of which is amultiproduct seller. The following equilibrium quantities are obtained:

q ′A(rA, rB1, rB2) = DA − d DB − rA + drB1

2(1 − d2)

q ′B1(rA, rB1, rB2) = (2+d2)DB − 3d DA − (4 − d2)rB1 +2(1 − d2)rB2 +3drA

6(1 − d2)

q ′B2(rA, rB1, rB2) = DB − 2rB2 + rB1

2(1 − d2).

.

(3)

Although the game under consideration only allows one of themanufacturers to sell multiple products, we have kept the above pre-sentation since the possibility that a retailer sells multiple productswill be contemplated in the next section. There are some differencescommenting on as compared with the case of three single-productsellers. In particular, ∂q ′

A∂rB2

= 0, which means that the equilibrium outputq′

A(rA, rB1, rB2) does not depend on the delegation policy of the single-product firm. Besides, ∂q ′

B2∂ DA

= ∂q ′B2

∂rA= 0; that is, the equilibrium output of

the single-product firm is not related with parameters that characterizethe demand and technology for product A. Therefore, decisions regard-ing the delegation policy of product A have no direct effect on M2’s salesof product B.

2.2 The Second-Stage Terms of Payment Subgame

In the second stage, manufacturers decide simultaneously and indepen-dently on the terms of payment, the transfer price, and the up-front fixedfee, when appropriate. As there is a competitive supply of retailers, theterms of payment are dictated by manufacturers. Given that sales are


Table I.

Equilibrium Manufacturer Margins

wABBA − c A = −d(wABB

B1 − cB ) = d[−d(5−2d2)a+(2+2d2−d4)b]2(5−2d2)(1−d2)

wABBB2 − cB = −b

(5−2d2)

wBBB1 − cB = −2[−d(5−2d2)a+(2+2d2−d4)b]

(2−d2)(10−8d2+d4)wBB

B2 − cB = −[d3a+4(1−d2)b](2−d2)(10−8d2+d4)

wABA − c A = 2d[−d(5−2d2)a+(2+2d2−d4)b]

(24−28d2+7d4)wAB

B2 − cB = −[−3da+2(3−d2)b](24−28d2+7d4)

wNBB2 − cB = −b

4

delegated to separate retailers, the equilibrium up-front fixed fee set bymanufacturers will be a fully extracting fee. We further assume, and forthe sake of the exposition, that the single-product manufacturer alwaysdelegates sales.4 For illustrative purposes consider the (AB, B) subgame.Manufacturers’ payoffs are given by

�1(wA, wB1, wB2) = (wA − c A)q A(wA, wB1, wB2) + FA

+ (wB1 − cB)qB1(wA, wB1, wB2) + FB1

�2(wA, wB1, wB2) = (wB2 − cB)qB2(wA, wB1, wB2) + FB2,

(4)

where Fh is set equal to the variable profit of each retailer as follows:FA = (pA − wA)qA(wA, wB1, wB2), FB1 = (pB − wB1)qB1(wA, wB1, wB2), andFB2 = (pB − wB2)qB2(wA, wB1, wB2). The equilibrium transfer prices, ob-tained by solving ∂�1

∂wA= ∂�1

∂wB1= ∂�2

∂wB2= 0, are denoted by wABB

h , wheresuperscripts will be used to identify the subgame. The remainingtransfer prices for subgames (A, B), (B, B), and (N, B) follow by proceed-ing in the same manner. All the equilibrium manufacturers’ marginsare summarized in Table I where the following simplifying notation,a ≡ DA − cA and b ≡ DB − cB has been used.

The third-stage equilibrium quantities, once the equilibrium trans-fer prices have been substituted for, are positive as long as ( a

b ), the rela-tive per-unit profitability of products for the manufacturers, belongs tothe following interval: d < a

b < 1+d2

2d (see the Appendix). Note that whenab equals 1 all outputs are positive for the full range of d; as it divergesfrom 1, positiveness for all outputs only happens for sufficiently smalld values. The interval widens as d tends to zero, this meaning that thedegree of product asymmetry admitted by the restriction for positivequantities increases with the degree of product differentiation.

Whenever the equilibrium transfer prices differ from unit produc-tion costs, manufacturers and retailers do not face the same incentive

4. Delegation is a dominant strategy for the single-product manufacturer. Please referto http://www.uv.es/∼sempere/jems-general.pdf.


structure. This is a standard result in the literature of strategic delegationwith single-product firms. When the choice variables are strategic substi-tutes and other things are equal, a transfer price below unit productioncosts produces a shift-out of the firm’s reaction function. In this waythe manufacturer induces the retailer to sell more than under verticalintegration; i.e., the retailer is given stronger sales incentives to gainmarket share strategically against its rival.

From Table I, we see that the equilibrium transfer prices set bythe single-product manufacturer are below unit production cost. Notethat wAB

B2 < cB as long as ab <

2(3−d2)3d , and this always holds given the

constraint for positive equilibrium outputs. Concerning the multiprod-uct manufacturer, the above standard result fails. Whether transferequilibrium prices are below unit production costs depends on thesize of the ratio a

b . In particular, if d < ab < 2+2d2−d4

d(5−2d2) , then wABBA >

cA, wABBB1 < cB, wBB

B1 < cB, and wABA > cA. The direction of the inequalities

is reversed for 2+2d2−d4

d(5−2d2) ≤ ab < 1+d2

2d . In view of this argument, we statethe following.5

Proposition 1: Strategic delegation is not sales enhancing necessarily forthe multiproduct manufacturer, while the single-product manufacturer alwaysgives the retailer stronger sales incentives.

When setting the transfer prices, the multiproduct firm takes intoaccount the effect of each of its products on the profits generatedby the other products it produces and consequently on the amountof the equilibrium up-front fixed fees. We just have shown that themultiproduct manufacturer never sets both transfer prices below orabove the corresponding unit production costs when each product’ssales are delegated. This is regardless of the delegation policy choicemade by the rival single-product manufacturer. Which product’s trans-fer price is set below unit production costs depends on the size of a

band the degree of product differentiation. Thus, for a given degree ofproduct differentiation, the multiproduct manufacturer better had givean incentive for greater sales of product A for higher levels of the ratio a

b .By setting wABB

A < cA and wABBB1 > cB, product A’s market share increases

relative to product B.An intuition can be understood by supposing that the rival single-

product manufacturer sells directly to consumers. Then, it is straightfor-ward to check that the payoffs corresponding to the (AB, N) subgame

5. Just note that when ab < 0.964, then a

b < 2+2d2−d4

d(5−2d2)for all d ensuring positive

quantities, where 0.964 is the value of 2+2d2−d4

d(5−2d2)evaluated at its minimum, i.e., when

d = 0.794.


coincide with those in the (N, N) subgame when quantities qA andqB1 are chosen before the single-product firm sets qB2; this would bea Stackelberg type of equilibrium with multiproduction. If the multi-product manufacturer wishes to realize such payoffs and employs twoseparate retailers, then a kind of first-mover advantage can be exploitedby setting one of the transfer prices above and the other one belowunit costs. A similar argument applies when only the sales of one ofthe products are delegated; it sets the transfer price in a way that it canattain the payoffs were its retailer allowed to choose output before theother two sellers. To sum up, the multiproduct manufacturer employstransfer prices to control both for the intensity of competition betweenits own products and also for the intensity of interbrand competition.The decision to delegate together with the choice of transfer prices givesthe multiproduct manufacturer the flexibility to attain some sort of first-mover advantage, either in market A or B or both.

3. Full Delegation: Separate versusCommon Dealership

Before characterizing the delegation pattern equilibria, it may be usefulto discuss the different implications of full delegation either to separateretailers or to a common retailer. The former case, as we just haveseen, involves fully extracting fees and transfer prices, which are notsimultaneously sales enhancing. This is not the case when both products’sales are delegated to the same retailer. There are two opposing effectsat work. On the one hand, intrafirm competition is internalized fully bythe common retailer. On the other hand, the retailer has the power tothreaten the multiproduct manufacturer credibly with dropping one ofthe products. Such power stems from its discretion over product choicebecause both products are substitutes. Therefore, the retailer will notaccept a contract with product specific fully extracting fees. In otherwords, the most up-front fixed fee the multiproduct manufacturer canelicit is each product’s marginal contribution to the retailer’s profit.Therefore, the retailer earns strategic rent, i.e., the foregone profit fromthe reduced sales of substitute products. Our analysis in this sectioncan be viewed as an extension of Shaffer (1991) and O’Brien andShaffer (1993).6 Result 1 in the Appendix proves that the multiproductmanufacturer cannot extract fully the retailer payoffs by using a productspecific two-part tariff contract. Therefore, it is a priori unclear which of

6. A positive strategic rent occurs regardless of the fact that the two products belong tothe same manufacturer, as in Shaffer (1991), or to different single-product manufacturers,as in O’Brien and Shaffer (1993).


these full delegation policies entails higher payoffs to the multiproductmanufacturer.

Let us now examine this question in more detail. The multiproductmanufacturer offers the common retailer the following product specifictwo-part tariff contract {wA, FA, wB1, FB1}, while the single-product oneoffers {wB2, FB2}. The fixed fee, FB2, is a fully extracting fee, yet the fixedfees, FA and FB1, now are equal to the corresponding product’s marginalcontribution to the retailer’s profits. Specifically,

FA = (pA − wA)q ′A(wA, wB1, wB2) + (pB − wB1)q ′

B1(wA, wB1, wB2)

− ( pB − wB1)qB1(wB1, wB2)

FB1 = (pA − wA)q ′A(wA, wB1, wB2) + (pB − wB1)q ′

B1(wA, wB1, wB2)

− ( pA − wA)q A(wA, wB2).

The notation is the following: pA is product A’s inverse demandfunction, and q A the equilibrium output when the common retailer dropsproduct B from M1. Similarly for pB and qB1. The first two terms in FA

and FB1 are the retailer’s profits of carrying both products, while thethird term is the profits when one of the products is dropped. Notethat the transfer prices serve an additional purpose, i.e., to controlfor the amount of the strategic rent that can be captured from the retailer.If the retailer threatens with dropping product A, then the multiproductmanufacturer may use wB1 to reduce the magnitude of the third term sothat it has an incentive to increase wB1. The equilibrium manufacturers’margins turn out to be

wCA − c A = −d[d(52 + 5d2)a − (32 + 20d2 + 5d4)b]

80(1 − d2)

wCB1 − cB = [6da − (1 + 5d2)b]

5(1 − d2)

wCB2 − cB = −[3da + (2 − 5d2)b]

10(1 − d2),

(5)

where the superscript C stands for common retailer. It is easy to checkthat regardless of the size of a

b , wCA > c A, and wC

B2 < cB. Furthermore, itis also the case that wC

B1 > cB either when product differentiation is weak,when 0.606 < d < 1, or when it is strong and a

b is sufficiently large (seeResult 2 in the Appendix). As usual, the single-product manufactureralways creates an incentive for greater sales than would occur otherwise.Matters change for the multiproduct manufacturer. In particular, thetransfer price for product A always is set above unit production cost, thismeaning that the incentive to reduce the strategic rent that the common


retailer can keep from the sales of product B dominates the incentive togreater sales. Concerning product A, in which M1 is the sole producer,the opposite might happen.

Proposition 2: The multiproduct manufacturer obtains higher payoffs ifit employs two separate retailers rather than one common retailer when productspecific two-part tariff contracts are used.

For the proof, see the Appendix.One possibility for the above conclusion to be reversed is that

the multiproduct manufacturer can implement fully extracting feeseffectively. This would require the introduction of a vertical restraintin the contract, such as full-line forcing, a clause that does eliminate theretailer’s ability to make its threat a credible one. Such a contract willconsist of a fully extracting fee and transfer prices, wFLF

A = c A − db5 and

wFLFB1 = cB − b

5 , which imply that transfer prices are set below unit pro-duction costs. The strategic effect supposes an output increase that goesin the opposite direction to the output decrease due to the internalizationof intrafirm competition. Since the former effect dominates the latter,delegation to a common retailer plus full-line forcing will be preferredto delegation to separate retailers by the multiproduct manufacturer(see the Appendix).

Thus, whether a multiproduct manufacturer is better off by del-egating both products to a common retailer depends on the kind ofcontracts competition authorities may allow. Full-line forcing is a formof tying where distributors are required contractually to take severalproducts or the entire product range. Legally tying is treated differentlyin the United States and the European Union. In the United States itmay be found to be a violation of section 3 of the Clayton Act. It issubject to per se prohibition if it applies to different commodities, ifthe seller has sufficient power for the tying product to enable it tocoerce the buyer into accepting the tied product, and if there is proof ofanticompetitive effects in the tied market. Even if a tying arrangementdoes not meet these criteria, it still may be considered an unreasonablerestraint of trade under the U.S.’s rule-of-reason approach. The FederalTrade Commission and the Antitrust Division of the Department ofJustice have new enforcement initiatives under way as the VerticalRestraints Guidelines were rescinded in 1998. Tying also can be foundcontrary to competition in the European Union. The Guidelines onVertical Restraints (2000/C291/01) take an economic based approachto assess the likely damaging effects of vertical restraints. Some clausesare considered hard-core restrictions; tying is not per se prohibited andcan be exempted when the market share of the tied and the tying productdoes not exceed 30%. Since the current legislative approach to vertical


restraints particularly emphasizes their effects on inter- and intrabrandcompetition, full-line forcing should be looked at carefully in a settinglike the one presented herein.

Our analysis has assumed that a retailer never sells the prod-ucts of more than one manufacturer. However, in the real world weobserve retailers carrying substitutable products of several manufac-turers. To conclude this section we elaborate on the case where acommon retailer may carry products A and B, the latter suppliedby the two manufacturers. The equilibrium set of contracts is givenby wCA

A = c A, F CAA = (a−db)2

4(1−d2) , wCAB1 = wCA

B2 = cB, F CAB1 = F CA

B2 = 0, where thesuperscript CA stands for common agent. Since the two manufacturersengage in Bertrand competition in setting the terms of the contractsfor product B, the equilibrium outcome is that both transfer prices beset equal to unit costs and that the corresponding up-front fixed feesequal to zero. On the other hand, the multiproduct manufacturer willset an up-front fixed fee for product A equal to that product’s marginalcontribution to the retailer’s profits. Given that M2 is not able to extractany rents from the common retailer, it never would choose delegation if itknew that M1 was to delegate both products’ sales to the common agent.It is shown in the Appendix that under full delegation, the multiproductmanufacturer earns higher payoffs by using two separate retailers. Thishappens regardless of the degree of product differentiation and the sizeof a

b .

4. Derivation of the Equilibriaof the Three-Stage Game

In this section we solve for the first-stage of the game where bothmanufacturers decide on their delegation policies. The equilibria canbe described in terms of the ratio a

b and the degree of product differen-tiation. The complete characterization is relegated to the Appendix andthe next proposition, which establishes sufficient conditions, is stated toillustrate the following nonoccurrence result on delegation.

Proposition 3: At the first-stage equilibrium of game G, the multiprod-uct manufacturer chooses partial delegation when interbrand competition issufficiently strong and the relative per-unit profitability of products for themanufacturers is sufficiently large.

For the proof, see the Appendix.This finding states that the delegation equilibrium does not always

hold, thus suggesting that previous results are not robust when a firmsells multiple products. As already indicated, the multiproduct firm


faces the following trade-off. On the one hand, there is the strategiceffect of delegation. On the other, competition between its own prod-ucts cannot be internalized when both products’ sales are delegated.However, it has two instruments to control directly for the intensityof interbrand and intrafirm competition—and indirectly for intrabrandcompetition. These instruments are the two transfer prices. Note thatpartial delegation is in fact a commitment to setting one of the transferprices equal to unit costs. Also, the delegation policy choice is a wayto choosing in which product the multiproduct manufacturer wishes toexploit a first-mover advantage.

To see the intuition, fix a = b so that, for d ∈ (0, 0.616)wABB

A > cA, wABBB1 < cB, wBB

B1 < cB, and wABA > cA. In this range of d values

M1 optimally chooses either action AB or action B. The equilibriumpayoffs can be written in the following useful way, with some obvi-ous saving on notation: MABB

1 = (qABBA )2 + (qABB

B1 )2 + (wABBA − cA)qABB

A+(wABB

B1 − cB)qABBB1 and MBB

1 = (qBBA )2 + (qBB

B1 )2 + (wBBB1 − cB)qBB

B1 . The firsttwo terms in MABB

1 and the second term in MBB1 are the corresponding

up-front fixed fees, while the first term in MBB1 is the profit from direct

sales of product A. The remaining terms reflect the benefits from notgiving sales incentives or the costs from giving sales incentives. Thedifference, MABB

1 − MBB1 , can be decomposed into the aggregate net

effect on profits from each product. For product B, since wABBA > cA and

by strategic substitution it happens that wABBB1 < wBB

B1 < cB (this meansthat the multiproduct manufacturer becomes less aggressive in sales inmarket for product B) and wABB

B2 < wBBB2 < cB. Therefore, there is a cost

increase from sales incentives but a gain in the payoffs extracted viathe fee. The net effect is positive. For product A, there is a benefitfrom not giving sales incentives and a loss in the payoffs extractedvia the fee. The net effect is negative. The multiproduct manufactureris better off with full delegation when interbrand competition is low,d ∈ (0, 0.302), whereas partial delegation of product B is chosen ford ∈ (0.302, 0.616). A similar reasoning can be applied to the differ-ence MABB

1 − MAB1 so that partial delegation of product A is chosen

for d ∈ (0.616, 1), where wABBA < cA, wABB

B1 > cB, wBBB1 > cB, and wAB

A < cA.Furthermore, when product differentiation is very weak, d ∈ (0.919, 1),the multiproduct manufacturer will sell both products directly becauseinterbrand competition is so intense that setting a transfer price belowunit costs does not give it any profitable strategic advantage, and theinternalization of competition effect is stronger. See Lemma 2 in theAppendix.7

7. It is worth remarking that partial delegation is the only equilibrium delegation pat-tern in a linear demand system taking after the Gabszewicz and Thisse (1979) specification(see Moner-Colonques, Sempere-Monerris, and Urbano 2000).


In summary, when delegating the sales of just one of the products,the multiproduct manufacturer commits to a greater output, firstlyby dispensing with the full internalization of intrafirm competitionand secondly by not setting any equilibrium transfer price strictlyabove unit production cost. Partial delegation enables the multiproductmanufacturer to exploit the strategic advantage in one of the mar-kets while partially internalizing the externalities between its ownproducts.

5. The Case of Two Multiproduct Manufacturers

The existing research on delegation with single-product firms has ex-amined the delegation problem of symmetric manufacturers competingin a channel. It therefore seems natural to consider symmetric upstreammanufacturers who can sell the two products. It must be noted thatthis introduces considerable extra complexity into the model sincethere are many asymmetric delegation patterns possible. Thus, werestrict the analysis and compare three of them. In the first one, whereboth multiproduct manufacturers sell directly to consumers, there isintrabrand competition in both products, but intrafirm competition isinternalized (vertical integration). In the second, each manufacturerdelegates both products’ sales to separate retailers; that is, there arefour retailers that compete in quantities (full delegation). Finally, eachmanufacturer delegates the sales of just one of the products—either thesame product as the rival or a different one—whereas the other productis sold directly to consumers (partial delegation).

It is easy to prove that full delegation is not sales enhancing onboth products when market profitability for each product differs much.However, transfer prices are below unit costs for a

b close to 1. Similarly,partial delegation is not always sales enhancing.8 Overall, delegationleads to a more competitive outcome than under vertical integration.The internalization of intrafirm competition effect dominates the strate-gic benefits of partial and full delegation. The following propositionaddresses the comparison of these delegation patterns when a = b.

Proposition 4: Each of the two multiproduct manufacturers achievesthe highest payoffs when selling themselves directly to consumers. Partialdelegation yields higher payoffs than full delegation. Finally, within partialdelegation, manufacturers prefer to delegate the same product rather than adifferent product each.

For the proof, see the Appendix.

8. These computations are available from the authors upon request.


Not surprisingly, and in such a symmetric setting, payoffs arehighest under vertical integration, noting that they correspond to thoseof a Cournot differentiated multiproduct duopoly. Still, partial delega-tion dominates full delegation, which means that each multiproductmanufacturer prefers having one of the product’s sales delegated to aretailer. Partial delegation may be reinterpreted as the use of differentcontracts to different retailers, which bears a link with the literature onchannel coordination. Bolton and Bonanno (1988), Winter (1993), andIyer (1998) have examined channel coordination issues in single-productenvironments when firms compete on price and nonprice dimensions.Winter (1993) only considers symmetric contracts. However, Iyer (1998),by questioning the symmetric contracting assumption, shows thatoffering contracts that are uniform across ex ante identical retailersis not always the optimum policy for the manufacturer. Therefore,a kind of partial delegation may arise when there is just intrabrandcompetition. He also analyzes the case of both inter- and intrabrandcompetition establishing conditions under which two single-productmanufacturers use different coordination strategies. In Iyer’s (1998)model, consumers are heterogeneous in their locations and in theirwillingness to pay for retail services. His results do not carry overto our setting with two multiproduct manufacturers. This is becausethere is neither consumer heterogeneity nor retailer differentiation,and, most importantly, both manufacturers are symmetric. One wayto build up some asymmetry into our model is to assume that each ofthe manufacturers produces two products that are differentiated fromthe rival’s. It then might be possible to have an asymmetric delegationpattern.

6. Extensions and Concluding Remarks

This paper has investigated whether a multiproduct manufacturerstrategically will employ a different distribution channel for each ofits products. By letting it delegate the sales of only one of the products,it has been proven that previous results on delegation are not robustwhen a firm sells multiple products. This nonoccurrence result holdseven if both products were sold by a common retailer rather thanseparate retailers. By appropriately setting the transfer price levelstogether with the decision of not delegating the sales of both prod-ucts, the multiproduct manufacturer strikes just the right compromise:The externalities among its own products are internalized partiallywhile a strategic advantage is achieved against its rival single-productmanufacturer. Furthermore, with two multiproduct manufacturers andwhen considering only symmetric delegation patterns, full delegation


is dominated vis a vis partial delegation and vertical integration. A fewcomments on extensions are in order.

It has been shown elsewhere that single-product manufacturershave a unilateral incentive to delegate sales with price competition.Suppose that there is competition in prices in the last stage of gameG and assume some differentiation among all three products. Thetrade-off pointed out under quantity competition does show up. How-ever, the decision variables, prices, and transfer prices are now strategiccomplements—reaction functions are upward sloping. In equilibrium,transfer prices exceed unit production costs, which reduces competition,and thus full delegation leads to a more collusive outcome. If one ofthe products were sold directly to consumers, then the multiproductmanufacturer would compete with a lower cost for that product, thusresulting in a more competitive outcome. Since variables are strategiccomplements, the outcome under partial delegation is more competitivethan under full delegation, given that a decrease in a transfer priceinduces a decrease in the other two. We conjecture that it is in the interestof both manufacturers to coordinate in the most collusive outcome, andthen, at equilibrium, the sales of the three products will be delegated toretailers.9 With quantity competition, a multiproduct manufacturer doesnot want to delegate both products’ sales because this will exacerbatethe competition between the two products. With price competition, itwishes to delegate both products’ sales because then competition isless intense. Confronting the results under strategic substitutability andcomplementarity leads us to a testable implication.

Another interesting extension is to examine the case with twomultiproduct manufacturers where each delegates both products toa common retailer. As argued already, fully extracting fees can beimplemented if full-line forcing is included in the contract. This del-egation pattern entails the internalization of intrafirm competition byeach manufacturer. It then is easy to check that the strategic effect ofdelegation leads to transfer prices below unit costs. Consequently, two-part tariff contracts plus full-line forcing are not sufficient to recover theprofits of a Cournot differentiated multiproduct duopoly. This findingbrings us to the problem of channel coordination and is an invitation forfurther research in multiproduction and vertical relationships.

Appendix

It is important to begin by establishing which are the restrictions soas to ensure that all equilibrium quantities in game G are positive.

9. This point was raised to us by a referee.


Substituting the equilibrium quantities in (2)–(3) for the correspondingequilibrium transfer prices, we obtain the following table of equilibriumoutputs:

q ABBA = a−db

2(1−d2)q ABB

B1 = −d(5−2d2)a+(4−d2)b2(1−d2)(5−2d2)

q ABBB2 = (4−d2)b

2(5−2d2)

q BBA = (10−3d2)a−2d(4−d2)b

2(10−8d2+d4)q BB

B1 = −d(20−10d2+d4)a+4(4−d2)b2(2−d2)(10−8d2+d4)

q BBB2 = (4−d2)[d3a+4(1−d2)b]

2(2−d2)(10−8d2+d4)

q ABA = (24−9d2)a−2d(11−4d2)b

2(24−28d2+7d4)q AB

B1 = (2−d2)[−3da+(6−2d2)b]2(24−28d2+7d4)

q ABB2 = (4−d2)[−3da+(6−2d2)b]

2(24−28d2+7d4)

q NBA = a−db

2(1−d2)q NB

B1 = −2da+(1+d2)b4(1−d2)

q NBB2 = b

2

All equilibrium quantities are positive as long as the ratio ( ab )

belongs to the following interval: d < ab < 1+d2

2d .

Result 1: The common retailer selling both multiproduct manufacturers’products earns a positive strategic rent. Therefore the multiproduct manu-facturer cannot extract fully the retailer’s payoffs by using a product specifictwo-part tariff contract.

Proof of Result 1. If M1 wants the common retailer to distribute itstwo products it must offer a two-part tariff contract that pays thecommon retailer. Assume that a retailer accepts the following contract{wA, FA, wB1, FB1} to distribute M1’s products, and another retailer ac-cepts the contract {wB2, FB2} to distribute M2’s product. Denote by q′

A, q′B1,

and q′B2 the equilibrium quantities. Let pB(0, qB1 + qB2) denote product

B′s inverse demand function when the common retailer does not sellproduct A, and let pA(q A, qB2) be product A′s inverse demand functionwhen the common retailer does not sell product B from M1. Then thecommon retailer accepts the contract if and only if

(pA(q ′A, q ′

B1 + q ′B2) − wA)q ′

A + (pB(q ′A, q ′

B1 + q ′B2) − wB1)q ′

B1 − FA − FB1

≥ max{maxq A

( pA(q A, qB2) − wA)q A − FA), maxqB1

( pB(0, qB1 + qB2)

− wB1)qB1 − FB1)}.If FA and FB1 are fully rent extracting then the left-hand side of the aboveexpression will be zero, while the right-hand side is not zero, giventhat

maxq A( pA(q A, qB2) − wA)q A − FA ≥ ( pA(q ′A, qB2) − wA)q ′

A − FA

= ( pA(q ′A, qB2) − wA)q ′

A − (pA(q ′A, q ′

B1 + q ′B2) − wA)q ′

A

= ( pA(q ′A, qB2) − pA(q ′

A, q ′B1 + qB2))q ′

A > 0.


The case is similar for product B. In fact, the retailer is able to earna positive rent on each product, since if M1 offered a contract with fullrent extraction on only one product, then the common retailer still couldthreaten with dropping precisely this product.

Result 2:

(1) wCA > c A∀ a

b that ensures positive quantities;(2) wC

B1 > cB either if 0 < d < 0.606 and 1+5d2

6d < ab < 16+40d2−5d4

d(56−5d2) or if0.606 ≤ d < 1 and ∀ a

b that ensures positive quantities, while wCB1 ≤ cB

otherwise;(3) wC

B2 < cB∀ ab that ensures positive quantities.

Proof of Result 2. We first report the equilibrium quantities when acommon retailer carries the two products of M1,

q CA = (80 + 68d2 + 5d4)a − d(128 + 20d2 + 5d4)b

160(1 − d2)

q CB1 = (4 − d2)[−d(56 − 5d2)a + (16 + 40d2 − 5d4)b]

5(1 − d2)

q CB2 = 3da + (2 − 5d2)b

5(1 − d2),

which are all positive as long as d(128+20d2+5d4)80+68d2+5d4 < a

b < 16+40d2−5d4

d(56−5d2) .

From (5) in the text note that wCA > cA iff a

b < 32+20d2+5d4

d(52+5d2) andsince 16+40d2−5d4

d(56−5d2) < 32+20d2+5d4

d(52+5d2) ∀d ∈ (0, 1) then part (1) is proven. Con-cerning part (2), wC

B1 > cB iff ab > 1+5d2

6d and since d(128+20d2+5d4)80+68d2+5d4 < 1+5d2

6dfor 0 < d < 0.606, while the opposite holds for 0.606 ≤ d < 1, part (2) isproven. Finally, wC

B2 < cB iff qCB2 > 0. �

Proof of Proposition 2. It follows by comparing the following expres-sions that refer to M1’s payoffs for the case of full delegation to separateretailers and full delegation to a common retailer, respectively:

MABB1 = (5 − 2d2)2a2 − 2d(5 − 2d2)2ab + (8 + 9d2 − 10d4 + 2d6)b2

4(5 − 2d2)2(1 − d2)

MC1 =

(400 + 448d2 + 25d4)a2 − 2d(688 + 160d2

+ 25d4)ab + (128 + 720d2 + 25d6)b2

1600(1 − d2)2 .

The sign [MC1 − MABB

1 ] is given by the sign of the following convexquadratic polynomial on a

b : �( ab ) = d(5 − 2d2)2(848 + 25d2)( a

b )2 − 2d(5 − 2d2)2(288 + 560d2 + 25d4) + d(15040 − 6288d2 − 1295d4 + 300d6 +


100d8). Denote by σs(d) and σg(d) the smallest and the largest roots of �,which are a function of d. Note that �( a

b ) < 0 for all σs(d) < ab < σg(d).

It is checked easily that σs(d) <d(128+20d2+5d4)

80+68d2+5d4 < 16+40d2−5d4

d(56−5d2) < σg(d).

Therefore, MC1 < MABB

1 for all d(128+20d2+5d4)80+68d2+5d4 < a

b < 16+40d2−5d4

d(56−5d2) . �

We now prove that a multiproduct manufacturer that can im-plement full-line forcing effectively obtains greater payoffs than un-der full delegation to separate retailers. First note that MFLF

1 =25a2−50dab+(8+17d2)b2

100(1−d2) . Then, MFLF1 − MABB

1 = d2(20−9d2)b2

50(5−2d2)2 , which is positive.Finally, under delegation to a common agent, both manufacturers

engage in Bertrand competition when setting the terms of paymentsof product B. This together with the fact the common agent earnsstrategic rent on product A results in wCA

A = cA and a fee equal to (a−db)2

4(1−d2) .Subtracting the payoffs with a common agent from those with separateretailers yields b2(2−d2)2

2(5−2d2)2 , which is positive.

Proof of Proposition 3. We first present M1’s payoffs for the foursubgames under consideration.

MNB1 = 4a2 + 8dab + (1 + 3d2)b2

16(1 − d2);

MAB B1 = (5 − 2d2)2a2 − 2d(5 − 2d2)2ab + (8 + 9d2 − 10d4 + 2d6)b2

4(5 − 2d2)2(1 − d2)

MB B1 =

(200 − 220d2 + 58d4 + d6 − d8)a2 − 8d(50 − 59d2

+ 20d4 − 2d6)ab + 4(2 − d)(2 + d)(4 + 4d2 − 6d4 + d6)b2

4(2 − d2)(10 − 8d2 + d4)2

MAB1 =

3(3 − d2)(4 − 3d2)(16 − 7d2)a2 − 4d(252 − 383d2

+ 183d4 − 18d6)ab + 4(36 + 17d2 − 79d4 + 44d6 − 7d8)b2

4(24 − 28d2 + 7d4)2

(1) First, MBB1 > MABB

1 iff [d(5 − 2d2)a − (2 + 2d2 − d4)b][−d(5 − 2d2)(50 − 59d2 + 20d4 − 2d6)a + (60 + 74d2 − 200d4 + 117d6 − 26d8

+ 2d10)b] > 0, which happens either when both terms in bracketsare positive or when both are negative. The first and secondterms in brackets are positive as long as a

b < 2+2d2−d4

d(5−2d2) andab >

(60+74d2−200d4+117d6−26d8+2d10)d(5−2d2)(50−59d2+20d4−2d6) , respectively. Let 1(d) =

(60+74d2−200d4+117d6−26d8+2d10)d(5−2d2)(50−59d2+20d4−2d6) and 2 (d) = 2+2d2−d4

d(5−2d2) . Further notice

that the following ranking holds: d < 1(d) < 2 (d) < 1+d2

2d . Also 1(d)and 2(d) reach a minimum at d = 0.569 and d = 0.794, respectively,where 1(d = 0.569) = 0, 818 and 2(d = 0.794) = 0, 964. See Figure 1.


FIGURE 1. PARTIAL DELEGATION OF PRODUCT B VERSUS FULLDELEGATION

Then, we conclude that MBB1 > MABB

1 iff 1(d) < ab < 2(d) and

MBB1 ≤ MABB

1 otherwise. It is important to remark that for ab ≤ 0.818

then MBB1 ≤ MABB

1 for all d values that ensure positive outputs.(2) Second, MAB

1 > MABB1 iff [d(5 − 2d2)a − (2 + 2d2 − d4)b]

[−d(5 − 2d2)(108 − 173d2 + 88d4 − 14d6)a + (504 − 898d2 + 454d4 + 27d6

−74d8 + 14d10)b] > 0.Proceeding as above, the following ranking holds:d < 2(d) < 1+d2

2d < 504−898d2+454d4+27d6−74d8+14d10

d(5−2d2)(108−173d2+88d4−14d6) . We conclude that

MAB1 > MABB

1 iff 2(d) < ab < 1+d2

2d ; MAB1 ≤ MAB B

1 otherwise. Now notethat for a

b ≤ 0.964 then MAB1 ≤ MABB

1 for all d values that ensure positiveoutputs.


(3) Third, MBB1 > MAB

1 iff [d(5 − 2d2)a − (2 + 2d2 − d4)b][d(1080−3926d2 +5415d4 −3641d6 + 1260d8 −214d10 +14d12)a − (1008 − 2444d2

+1346d4 + 1022d6 − 1439d8 + 601d10 − 107d12 + 7d14)b] > 0. The follo-wing ordering verifies

d < 2(d) < 1+d2

2d < 1008−2444d2+1346d4+1022d6−1439d8+601d10−107d12+7d14

d(1080−3926d2+5415d4−3641d6+1260d8−214d10+14d12) .Therefore, MBB

1 > MAB1 iff d < a

b < 2(d); MB B1 ≤ MAB

1 otherwise.Similarly, as in (2), for a

b ≤ 0.964 then MAB1 ≤ MBB

1 for all d values thatensure positive outputs.

Lemma 1, which compares full delegation with partial delegation,summarizes this reasoning.

Lemma 1: Full delegation dominates if d < ab < 1(d) < 2(d) < 1+d2

2d .Partial delegation of product B dominates if d < 1(d) < a

b < 2(d) < 1+d2

2d .Finally, partial delegation of product A dominates if d < 1(d) < 2(d) <ab < 1+d2

2d .

We finally compare with action N.(4) Fourth, MABB

1 > MNB1 iff 7 − 12d2 + 4d4 > 0, that is for

d ∈ (0, 0.890)(5) Fifth, MAB

1 > MNB1 if (a2d(432 − 692d2 + 352d4 − 56d6)

+ 8ab(− 72 + 74d2 + 12d4 − 30d6 + 7d8) + b2d(688 − 1376d2 + 1000d2

−311d6 + 35d8)) < 0. The smallest root of the this convex quadraticpolynomial on a

b is given by

3(d) =−144 + 148d2 + 24d4 − 60d6 + 14d8 + (24 − 28d2 + 7d4)

×√

(1 − d2)(36 − 83d2 + 60d4 − 14d6)−216d + 346d3 − 176d5 + 28d7 ,

where 3(d) takes nonreal values for d ∈ (0.932, 1) and thereforethe above inequality is positive regardless of the value of a

b (i.e.,MAB

1 < MNB1 ). It happens that for d ∈ (0, 0.922), d < 3(d) < 1+d2

2d , whilefor d ∈ (0.922, 0.932), d < 1+d2

2d < 3(d). We conclude that for d < ab <

3(d) < 1+d2

2d then MAB1 < MNB

1 , whereas for d < 3(d) < ab < 1+d2

2d thenMAB

1 > MNB1 ; and for d < a

b < 1+d2

2d < 3(d) then MAB1 < MNB

1 . To sum up,action N dominates action A if, for a given d, a

b is small enough or, for agiven a

b , d is large enough.When a

b > 1 and d is close to one action A dominates actions B andAB. For this parameter range the first-stage equilibrium is either partialdelegation of product A or vertical integration. In particular, MAB

1 ≥ MNB1

for ab ≥ 1.003 and for every d that ensures positive outputs.

When ab < 1 and d is sufficiently large action AB dominates actions

A and B, but from (4) full delegation dominates vertical integration when


d < 0.890. In particular, it suffices that ab ≤ 0.890 to have MABB

1 ≥ MNB1

for every d that ensures positive outputs.

Lemma 2: In the particular case of a = b, at the first-stage equilibrium ofgame G, M1 chooses either

(a) Full delegation for d ∈ (0, 0.302), or(b) Partial delegation of product B for d ∈ [0.302, 0.618), or(c) Partial delegation of product A for d ∈ [0.618, 0.919), or(d) Vertical integration for d ∈ [0.919, 1),

where 0,302, 0.618, and 0.919 are obtained from equating 1(d), 2(d)and 3(d) to one.

Proof of Proposition 4. The payoffs when both manufacturers produceA and B are given by

MVI = a2 − 2dab + b2

9(1 − d2);

MFD = (50 + 3d2 − 4d4)(a2 + b2) − d(130 − 32d2)ab(25 − 4d2)2(1 − d2)

MPD(A) =(3 − d2)(3 − 2d2)(9 + 4d2)a2 − 2d(261 − 262d2 + 56d4)ab

+ (225 − 152d2 − 28d4 + 16d6)b2

(45 − 46d2 + 8d4)2

MPD(AB)

=(3 − 2d2)[(3 − 2d2)(288 − 183d2 − 24d4 + 16d6)a2 − 2d(45 − 28d2)(20

− 19d2 + 4d4)ab + 2(216 − 201d2 − 33d4 + 72d6 − 16d8)b2

(144 − 241d2 + 120d4 − 16d6)2 ,

where MVI denote the payoffs under vertical integration; MFDare thepayoffs under full delegation; MPD(A) denote the payoffs under partialdelegation when both manufacturers delegate the sales of product A(the case for product B is straightforward by exchanging the coefficientsof a2 and b2); and MPD(AB) are the payoffs under partial delegationwhen one of the manufacturers delegates the sales of product A whilethe other delegates the sales of product B. In the particular case of a = bthe payoffs are ranked as follows: MVI > MPD(A) > MPD(AB) > MFD.

References

Bernheim, B. and M. Whinston, 1998, “Exclusive Dealing,” Journal of Political Economy,106, 64–103.

Bolton, P. and G. Bonanno, 1988, “Vertical Restraints in a Model of Vertical Differentiation,”Quarterly Journal of Economics, 103, 555–570.


Bonanno, G. and J. Vickers, 1988, “Vertical Separation,” Journal of Industrial Economics, 36,257–265.

Coughlan, A. and B. Wernerfelt, 1989, “On Credible Delegation by Oligopolists: ADiscussion of Distribution Channel Management,” Management Science, 35, 226–239.

Fershtman, C. and K. Judd, 1987, “Equilibrium Incentives in Oligopoly,” American Eco-nomic Review, 77, 927–940.

——,—— and E. Kalai, 1991, “Observable Contracts: Strategic Delegation and Coopera-tion,” International Economic Review, 32, 551–559.

Gabszewicz, J.J. and J.F. Thisse, 1979, “Price Competition, Quality, and Income Dispari-ties,” Journal of Economic Theory, 20, 340–354.

Irmen, A., 1998, “Precommitment in Competing Vertical Chains,” Journal of EconomicSurveys, 12, 333–359.

Iyer, G., 1998, “Coordinating Channels under Price and Nonprice Competition,” MarketingScience, 17, 338–355.

Katz, M., 1991, “Game-Playing Agents: Contracts as Precommitments,” Rand Journal ofEconomics, 22, 307–328.

Lin, Y.L., 1990, “The Dampening-of-Competition Effect of Exclusive Dealing,” Journal ofIndustrial Economics, 39, 209–223.

McGuire, T.W. and R. Staelin, 1983, “An Industry Equilibrium Analysis of DownstreamVertical Integration,” Marketing Science, 2, 161–191.

Moner-Colonques, R., J.J. Sempere-Monerris, and A. Urbano, 2000, “Product Quality andDistribution Channels,” Working paper, IVIE, 2000–19.

O’Brien, D. and G. Shaffer, 1993, “On the Dampening-of-Competition Effect of ExclusiveDealing,” Journal of Industrial Economics, 41, 215–221.

Shaffer, G., 1991, “Capturing Strategic Rent: Full-Line Forcing, Brand Discounts, Aggre-gate Rebates, and Maximum Resale Price Maintenance,” Journal of Industrial Economics,39, 557–575.

Sklivas, S., 1987, “Strategic Choice of Managerial Incentives,” Rand Journal of Economics,18, 452–458.

Vickers, J., 1985, “Delegation and the Theory of the Firm,” Economic Journal, 95, Supple-ment, 138–147.

Villas-Boas, J.M., 1998, “Product Line Design for a Distribution Channel,” MarketingScience, 17, 156–169.

Winter, R., 1993, “Vertical Control and Price versus Nonprice Competition,” QuarterlyJournal of Economics, 108, 61–76.

strategic delegation with multiproduct firms

Documents