Supply contracts in manufacturer-retailer interactions with manufacturer-quality and retailer effort-induced demand

Supply Contracts in Manufacturer-Retailer Interactions withManufacturer-Quality and Retailer Effort-Induced Demand

Haresh Gurnani,1 Murat Erkoc2

1 Department of Management, University of Miami, Coral Gables, Florida 33124

2 Department of Industrial Engineering, University of Miami, Coral Gables, Florida 33124

Received 22 September 2005; revised 19 December 2007; accepted 27 December 2007DOI 10.1002/nav.20277

Published online 14 February 2008 in Wiley InterScience (www.interscience.wiley.com).

Abstract: We consider a decentralized distribution channel where demand depends on the manufacturer-chosen quality of theproduct and the selling effort chosen by the retailer. The cost of selling effort is private information for the retailer. We considerthree different types of supply contracts in this article: price-only contract where the manufacturer sets a wholesale price; fixed-feecontract where manufacturer sells at marginal cost but charges a fixed (transfer) fee; and, general franchise contract where manufac-turer sets a wholesale price and charges a fixed fee as well. The fixed-fee and general franchise contracts are referred to as two-parttariff contracts. For each contract type, we study different contract forms including individual, menu, and pooling contracts. In theanalysis of the different types and forms of contracts, we show that the price only contract is dominated by the general franchisemenu contract. However, the manufacturer may prefer to offer the fixed-fee individual contract as compared to the general franchisecontract when the retailer’s reservation utility and degree of information asymmetry in costs are high. © 2008 Wiley Periodicals, Inc.Naval Research Logistics 55: 200–217, 2008

Keywords: supply contracts; information asymmetry; retailer effort

1. INTRODUCTION AND RELATED LITERATURE

In decentralized distribution channels, the manufacturerrelies on marketing intermediaries (such as retailers) to takethe product to the end-user. However, each firm makes deci-sions to maximize their individual profits and hence, thepresence of disparate incentives for the firms requires carefulconsideration in the design of supply contracts. Two basicissues commonly identified in supply contract design are thedouble marginalization effect [28] and information asymme-try. In this article, we assume that the manufacturer designsthe contract but both the manufacturer and retailer engage inactivities that influence final demand.

We consider the case when the manufacturer invests inthe technology used to make the product; a larger invest-ment in technology improves the quality of the productwhich results in an increased market potential (demand)for the product. However, the technology-investment/quality-improvement costs are directly incurred by the manufactureronly. As such, the manufacturer may charge a higher whole-sale price/transfer fee in order to recuperate these costs.

Correspondence to: H. Gurnani ([email protected])

For instance, in the electricity markets, power generatingfirms can invest in different dimensions of power quality suchas environmentally-friendly green power or premium powerfor sensitive computing, as opposed to lower quality inter-ruptible power for flexible producers [27]. In the fast foodretail business, final demand is not only affected by the retailprice and the value added by the retailer, it also depends on theinvestments made by the franchisor in its brand name [21]. Inthe automotive industry, Japanese firms made dramatic gainsin market share in the 1980s as compared to the US firms byinvesting in quality-improvement efforts [12].

Similarly, the retailer has the opportunity to influ-ence the final demand by choosing the appropriate sell-ing/promotional efforts. Examples of such activities includebreaking bulk, providing shelf spaces, promotional displays,advertising, and other demand enhancing activities [10].In the automotive industry, final demand depends not only onthe quality/brand reputation of the product, but also on thedealer’s selling efforts including after-sales service support.The cost of these promotional efforts is private informa-tion and is directly incurred by the retailer only. In manysituations, the manufacturer may not know the retailer’s effec-tiveness (or willingness) to exert effort to influence demand.

© 2008 Wiley Periodicals, Inc.

Gurnani and Erkoc: Supply Contracts in Manufacturer-Retailer Interactions 201

For instance, in electronic commerce settings, a manufacturerselling to a first-time buyer (retailer) is unlikely to know thebuyer’s ability to use effort to develop the market. In a retailsetting, the manufacturer may not know the incentives forthe retailer to set the appropriate shelf space for his productin the presence of different profit margins from competingproducts. Moreover, since the manufacturer may not be ableto reliably verify the retailer’s effort, we assume that retailereffort is noncontractable in the model. If the retailer decidesto accept the contract terms offered by the manufacturer, shehas the flexibility to jointly choose the effort level and theretail price in order to maximize her own profits.

1.1. Types of Contracts

We compare three different types of supply contracts inthis paper.

1. At one extreme is the wholesale price contract with anarm’s length relation between the manufacturer andthe retailer; the manufacturer invests in product qual-ity and the retailer buys the product at the wholesaleprice and sells it to the consumers with no oblig-ation to the manufacturer. The retailer may acceptthe contract terms in which case she would chooseher selling effort along with the retail price. Whole-sale price contracts are easy to implement and havebeen extensively used in many industries includingfor electricity, electronics, food items, pharmaceuti-cals, etc. Lariviere and Porteus [22] note that in manysupply chains, transactions are “governed by simplecontracts defined only by a per unit wholesale price.”

2. In the second contract type, referred to as the fixed-fee contract, the manufacturer invests in productquality, sells the product at cost to the retailer butcharges a fixed-fee (transfer fee) to the retailer. Notethat in the fixed-fee contract, since the wholesaleprice is set equal to the marginal cost, the manufac-turer derives all of his profits by careful selection ofthe transfer fee. Essentially, the transfer fee servestwo main purposes: It is used as a mechanism todivide the total channel profits; further, the retailerseeking a high-quality product may be willing to paya higher transfer fee in order to reimburse the manu-facturer for the cost of quality-improving activities.Again, on acceptance, the retailer chooses the sellingeffort and the retail price.

3. Finally, we consider the general franchise contractwhere the manufacturer can select the wholesaleprice along with the transfer fee. Franchise contractshave had remarkable success and it is estimated thatmore than 40% of retail sales in the United Statespass through some franchised operations [7, 19, 20].

While the wholesale price contract is a single-tariff price-only contract, the fixed-fee and franchise contracts are two-part tariff contracts with the fixed-fee contract as a specialcase of the general franchise contract.

1.2. Forms of Contracts

While the retailer’s cost of effort is not known to the man-ufacturer, the manufacturer has some priors on the retailerbeing efficient with low costs (l-type) or with high costs (h-type). For instance, a retailer with a strong market presencemay be more cost-effective in influencing demand as com-pared to a retailer with a weaker presence. We consider thefollowing three forms of contracts:

1. If the manufacturer has perfect information about theretailer type, he can offer the optimal contract derivedin the perfect information case. In the imperfect infor-mation case, the manufacturer may still use the samecontract form derived for the perfect information caseby assuming that the retailer is of h-type or of l-type.Essentially, here the manufacturer does not attemptto identify the retailer type but offers a contract formwhich would be optimal if the retailer type is the oneas originally assumed. We refer to such contract formas an individual contract.1

2. We also consider a second form referred to as menucontract with self-selection. The menu contract is aseparating equilibrium contract where each retailerselects the contract expressions designed strictly forher type. Here, the manufacturer offers a menu ofcontracts (with self selection) in order to distin-guish between the retailer type. However, the con-tract structure has some loss in efficiency becauseof the prospect of information rent collected by thel-type retailer.

3. Finally, we consider a pooling contract where, sim-ilar to the individual contract case, the manufac-turer offers a single contract, but now the contractmust satisfy the participation constraint for bothtypes of retailers. The individual contract considered

1 The motivation for using the Individual Contract lies in the factthat the structure of the contract is efficient in the perfect infor-mation case and we check whether it may be the preferred oneeven in the imperfect information case. Consider the case when themanufacturer offers the low-cost Individual contract assuming thatthe retailer is of the low-cost type. If the retailer is indeed of thelow-cost type, the contract form is optimal for the manufacturer.On the other hand, if the retailer is of the high-cost type, she maystill choose to accept the terms offered in the contract. The expectedprofit for the manufacturer would then depend on the probabilityof the retailer types. Similarly, if the manufacturer offers the high-cost Individual contract, the low-cost retailer may also be willing toaccept the contract terms.

Naval Research Logistics DOI 10.1002/nav

202 Naval Research Logistics, Vol. 55 (2008)

above is different from the pooling contract as it isdesigned for one retailer type and does not considerthe individual rationality constraint of the other.

In the analysis of the various types/forms of contracts, wefirst compare within each contract type for the different con-tract forms. Subsequently, we compare across the differentcontract types.

1.3. Research Objectives and Managerial Implications

Since the manufacturer is assumed to be the leader inthis article (the one who designs the terms of the contract),the problem is to choose the contract that maximizes hisexpected profits and ensures that the retailer participates bydoing at least as well as her reservation utility. The limitationof wholesale price contracts is the double marginalizationeffect leading to lower profits for both firms. For the caseof perfect information, we know that the fixed-fee contractis channel optimal and would always dominate the whole-sale price contract [24]. However, when the retailer’s cost ofeffort information is private knowledge, and both the man-ufacturer and the retailer make decisions not only on pricesbut on other demand-affecting variables as well, would thewholesale price contract ever be used? In addition, wouldthe fixed-fee contract (which is channel optimal under per-fect information) be preferred to the general franchise menucontract (with both wholesale price and transfer fees) in theimperfect information case?

In the comparison of the different contracts, we are ableto generate managerial insights related to the type and formfor the contract selection problem for the manufacturer. Thefollowing results are established in the paper:

• We show that the pooling contract is always domi-nated by either the individual or the menu contract forall types of contracts;

• If the manufacturer is constrained to offer contractswith only wholesale prices, then he always prefers tooffer a menu of contracts over the individual contract;

• If the manufacturer is constrained to offer the fixed-fee contract, then it may prefer to offer the individualcontract over the menu of contracts under certainconditions;

• It is never optimal for the manufacturer to excludetransfer fees in the franchise menu contract, that is,the wholesale price menu contract is dominated bythe franchise menu contract. It is important to notethat even though the wholesale price menu contractis dominated, we need the results derived above (thatwholesale price individual contracts will never be pre-ferred to the wholesale menu contract) to eliminate thewholesale price type contract.

• Given a choice between the general franchise menucontract and the fixed-fee contract, under certain con-ditions, the manufacturer may prefer the fixed-feeindividual contract. Here, the manufacturer wouldexclude the h-type retailer and offer fixed-fee indi-vidual contract that is accepted by the l-type retaileronly.

We show that when the manufacturer’s prior on the retailerbeing of the l-type exceeds a certain threshold, it may bepreferable to offer the fixed-fee individual contract as com-pared to the franchise menu contract. Intuitively, the individ-ual contract is efficient for the case of perfect information andas such, the manufacturer may prefer it to the franchise menucontract (which has some loss in efficiency in order to makeit self-selecting for both retailer types) when the likelihoodof the retailer being of the l-type is high. Further, we alsonote that the preference for the fixed-fee individual contractincreases when the reservation utility of the retailer and thedegree of asymmetry in costs is high. As such, the manufac-turer becomes more willing to exclude the high-cost retailerby offering the fixed-fee contract which is only accepted bythe l-type retailer.

To the best of our knowledge, there is no similar study of thecontract selection problem for a manufacturer using differenttypes (wholesale, fixed-fee, and franchise) and forms (indi-vidual, pooling, and menu) of contracts when both playerscan use price and nonprice factors to influence demand.

1.4. Related Literature

The problem of contract design in decentralized distrib-ution channels (supply chains) is of interest to researchersin the operations management, marketing, and economicscommunities. We briefly review some of the key papers indifferent streams of research in this area.

Cachon [3] provides an excellent review of differentaspects of the operations management literature as it pertainsto supply chain contracts. In a article with information asym-metry, Corbett et al. [6] investigate various two-part tariffcontracts under a one-supplier, one-buyer setting where thebuyer’s cost is private information. Their study focuses onsupplier-initiated contracts and evaluates the value of infor-mation for the supplier. In their article, the supplier does notknow the buyer’s internal variable cost; however, they donot consider supplier or buyer investment decisions to influ-ence the demand through higher quality or selling efforts,respectively. In line with their article, we observe that as theinformation gap about retailer type increases, the flexibilityprovided by using transfer fees is reduced. However, we alsonote that the manufacturer may prefer the fixed-fee contractover the general franchise menu contract (two part tariff con-tract) under certain conditions. In a paper with retailer selling



effort, Krishnan et al. [18] consider the effect of retailerpromotions on demand in a stochastic model. They discussthat a buy-back contract alone cannot coordinate the chan-nel as buy-backs reduce the incentive for retailer’s promo-tional efforts. As such, they suggest using buy-backs coupledwith promotional cost-sharing agreements when effort cost isobserved in order to coordinate the channel. Other coordinat-ing mechanisms for the case of observable (but not verifiable)demand, and for the case of verifiable demand are also dis-cussed. Other research in this stream examines the role ofa variety of mechanisms to improve supply chain efficiencyincluding buy-back agreements [26], quantity commitmentcontracts [1], quantity discount contracts [5], informationsharing [13], and revenue sharing [2, 4, 14].

Researchers in marketing and economics have modeledthe role of wholesale prices on the supply contract problem.We refer to the seminal work by McGuire and Staelin [23]and Jeuland and Shugan [17] for additional information.Lal [21] examines franchising arrangements in improvingchannel coordination by focusing on two elements in thecontract – royalty structure and monitoring technology. Theresults indicate that monitoring is necessary if there is fran-chise competition to prevent the free riding problem. Whenthe franchisor also invests in the brand name that influencesthe final demand, the optimal arrangement requires the use ofroyalties. Gal-Or [11] studies the role of monitoring in main-taining quality standards in franchise chains. The findings ofthe work indicate that the franchisor has greater incentive tomonitor the service efforts of franchisees that serve relativelysmall markets, and those that are subject to high demand vari-ability. In contrast to these articles, we assume that retailerefforts cannot be verified and we do not include monitoringin our paper.

Similar to our article, other researchers have also studiedmodels concerning demand that depend on variables otherthan price. For instance, Desiraju and Moorthy [10] considerthe role of pricing and service commitments in achievingchannel coordination when the retailer is better informedabout market conditions. Iyer [16] studies channel coordi-nating mechanisms for the manufacturer when retailers com-pete in price and other nonprice factors such as provisionof product information, free repair, etc. However, neither ofthese articles include the effect of manufacturer investmentin quality improving efforts.

In other work on franchise contracts, Desai and Srini-vasan [9] consider the case when the franchisor has privateinformation about demand and incurs signaling costs to cred-ibly inform the franchisee about the demand potential. Theystudy both linear and nonlinear price contracts and show thata two-part scheme is unable to achieve the first-best solutionand therefore, propose a three-part scheme which is able toachieve the first-best profits. Although Desai and Srinivasanlook at a different problem context as compared to our arti-

cle, we note that the individual fixed-fee contract achieveschannel coordination if the retailer is of the l-type.

In another problem comparing wholesale price contractswith two-part tariff contracts, Ingene and Parry [15] showthe dominance of a quantity discount wholesale price contractover two part-tariff contracts. The problem setting consideredin their article is different from our article as they considerthe case of two competing retailers with perfect information.In addition, their model does not include manufacturer orretailer investment in demand-influencing decisions.

The rest of the paper is organized as follows. In Section 2,we model the wholesale price contract under asymmetric costinformation and determine the optimal individual and menucontracts. In Section 3, we consider two-part tariff contractsand again analyze both the individual and menu contracts. InSection 4, we compare the different contracts and determinethe optimal pricing strategy for the manufacturer. Finally, weconclude and discuss future research in Section 5.

2. WHOLESALE PRICE CONTRACT

In this section, we consider a wholesale-price contractbetween a manufacturer, M selling a product through an inde-pendent retailer, R. The sequence of events in the model is asfollows. The manufacturer offers the product with quality θ

and wholesale price ω to the retailer. There is no negotiationover the contract terms; the retailer may accept the contractor reject it, in which case both sides could walk away.2 If theretailer decides to accept the contract, she can influence thedemand through her promotional/selling efforts, e. Therefore,in addition to setting the retail price, p, she would determinethe optimal selling effort in response to the terms offeredin the contract. The quantity ordered by the retailer is equalto the normalized demand, D (defined later).

While the retailer’s efforts may not be verifiable (and there-fore cannot be contracted upon), the manufacturer anticipatesthe retailer’s actions and acts as a Stackelberg leader indesigning the terms of the contract. Later, we consider thecase of information asymmetry about the retailer’s cost ofeffort and model the adverse selection problem using a menuof contracts. For now, we assume that there is no informationasymmetry in the model.

The normalized retail demand D is assumed to be:

D = α − p + γ e + λθ ,

where α is the market size, γ measures the influence ofretailer’s effort on demand, and λ measures the impact of

2 In the rest of the paper, we implicitly assume that firms commitnot to renegotiate in that if the retailer does not accept the contractterms offered by the manufacturer, there is no transaction betweenthem.



product quality on demand. The retailer’s investment in efforthas a diminishing impact on demand and the cost of effortis assumed to be ηe2/2. Similar downward sloping demandmodels have been used by Desai and Srinivasan [9], Desai [8],and Corbett et al. [6]. For a given contract (θ , ω), the retailer’sobjective is:

�R = Max(p,e)(p − ω)[α − p + γ e + λθ ] − ηe2/2,

which yields p∗ = (α + λθ)η + (η − γ 2)ω

(2η − γ 2), (1)

and e∗ = γ

(2η − γ 2)(α + λθ − ω). (2)

Define K = η

(2η−γ 2). On substituting from above and

simplifying terms, we get:

�∗R(ω, θ) = K

2(α + λθ − ω)2, (3)

and D(ω, θ) = α − p + γ e + λθ = K(α + λθ − ω). (4)

The manufacturer invests in quality improvement efforts thatmay include new high-precision equipment with high relia-bility, fast or flexible equipment, organizational training andrestructuring, etc., which improve the demand potential ofthe product. Another example of manufacturer-initiated effortto improve the demand potential is the investment in brandname, (see Lal [21]). By inferring the retailer’s effort reactionfunction in response to the terms of the contract, the manu-facturer can suitably choose the contract expressions in orderto maximize his profits. Let c be the manufacturer’s unit costof production. Also, similar to the case of retailer investmentin effort, the manufacturer’s cost of quality investment hasdiminishing impact on demand and is assumed to be ζθ2/2.Then, we get:

�M = Max(ω,θ)(ω − c)D − ζθ2

2

= K(ω − c)(α + λθ − ω) − ζθ2

2.

We use the superscript (w) to denote the optimal expressionsfor the wholesale price contract. Solving the manufacturer’sproblem above, we get:

ωw∗ = ζ(α − c)

(2ζ − λ2K)+ c = ζα + c(ζ − λ2K)

(2ζ − λ2K), (5)

and θw∗ = λK(α − c)

(2ζ − λ2K). (6)

The manufacturer’s optimal profit is:

�w∗M = ζK(α − c)2

2(2ζ − λ2K).

From Eqs. (1–4), we get:

pw∗ = ωw∗ + ζK(α − c)

(2ζ − λ2K), (7)

ew∗ = γ ζK(α − c)

η(2ζ − λ2K), (8)

Dw∗ = ζK(α − c)

(2ζ − λ2K), (9)

and �w∗R = K

2

[ζ(α − c)

(2ζ − λ2K)

]2

. (10)

We note from above that θw∗, pw∗, and ew∗, and the prof-its �w∗

M and �w∗R are decreasing in c, as expected. Further,

observe that the optimal terms of the contract offered by themanufacturer depend on the retailer’s cost of effort, η (whichis assumed to be known to the manufacturer in this section).We can show that (θw∗, ωw∗) and (pw∗, ew∗) are all decreas-ing in η, as expected. That is, if the retailer has a lower costof effort, the manufacturer provides a higher quality prod-uct and charges a higher wholesale price. The retailer thenexerts more effort and is able to charge a higher retail priceas well. Also, note that ∂�w∗

M

∂η, ∂�w∗

R

∂η< 0, that is, the manufac-

turers’ and retailer’s profits are decreasing in increasing costof effort. The issue of information asymmetry in the cost ofeffort is addressed next.

2.1. Asymmetric Information on Cost of Effort

As discussed in the previous subsection, the manufacturerand the retailer profits are inversely proportional to the cost ofeffort. Also, if the manufacturer uses the perfect informationcontract, the retailer may have an incentive to misrepresenther true cost of effort. We now consider the case when theretailer’s cost of effort is not known to the manufacturer and,as such, the manufacturer may not know the retailer’s opti-mal choice of effort. For instance, a manufacturer sellingto a first-time buyer (say, retailer) is unlikely to know thebuyer’s ability to use effort to influence demand.3 The retailercould be either of two types: low-cost retailer (l-type) orhigh-cost retailer (h-type), between which the manufacturercannot distinguish. However, the manufacturer has some pri-ors regarding the distribution of retailer-type, and believesthat the retailer is of l-type with probability r , and of h-typewith probability (1 − r). Let ηl and ηh be the cost of effortfor the l-type and h-type retailer, respectively, with ηh > ηl .

3 In our paper, we consider the case when the retailer can observe thequality of the product and infer the manufacturer’s cost of quality.For instance, the retailer may obtain samples of the product beforethe contract is executed in order to judge the quality of the prod-uct. On the other hand, since retailer effort is not verifiable (anddepends on the retailer-type), the cost of effort may not be knownto the manufacturer.



We now consider two forms of wholesale price contractsoffered by the manufacturer. In the first case, the manufac-turer offers an Individual Contract (single contract) using thecontract form derived for the perfect information case. Then,we consider the case where a Menu of Contracts (separatecontracts) is offered with self selection, that is, the retailercorrectly chooses the contract offered to her true type. Acomparison of both contracts is done and we show that themenu of contracts provides higher expected profits for themanufacturer.4

2.1.1. Individual Contract

Consider the contract expressions derived for the case ofperfect information [Eqs. (5) and (6)]. The manufacturer cannow offer a contract (ωw

l , θwl ) using η = ηl (aimed at the

l-type retailer), or a contract (ωwh , θw

h ) using η = ηh (aimedat the h-type retailer). In both cases, the retailer who is notof the true type (as assumed by the manufacturer) will alsoaccept the contract offer if profits exceed the reservation prof-its.5 As such, the manufacturer’s decision would be to selectthe contract that maximizes his expected profits.

Define Ki = ηi

(2ηi−γ 2)for i = h, l. Note that Kh, Kl ≥ 0

if (2ηi − γ 2) ≥ 0 for i = h, l. We need this conditionto ensure that optimal demand, effort, etc. values are non-negative. Also, Kl > Kh if ηh > ηl (as assumed). Further,the difference (Kl −Kh) increases as γ increases, that is, thedegree of asymmetry in costs increases when retailer efforthas a higher impact on demand. Suppose now, the manufac-turer offers a contract designed for retailer of type i(i = h, l).If the retailer’s true type is indeed i, the manufacturer’s profitwill be:

�wM

(ωw

i , θwi

)|R = i) = Ki

(ωw

i − c)(

α + λθwi − ww

i

)−1

2ζ(θwi

)2.

On the other hand if the retailer type is not i but i−, themanufacturer profit is

�wM

(ωw

i , θwi

)|R = i−) = Ki−(ωw

i − c)(

α + λθwi − ωw

i

)−1

2ζ(θwi

)2,

4 We also provide the comparison with a pooling contract which isshown to be always dominated. Details are provided in Appendix-B.5 We assume that the reservation utility for the retailer is zero in thissection. If there is postive reservation utility, the expected profits forthe manufacturer could be lower since the manufacturer has to offerbetter contract terms to induce participation from the retailer. Assuch, the profits derived assuming zero reservation utility representan upper bound on the manufacturer’s expected profits using theindividual contract.

where θwi = λKi(α−c)

2ζ−λ2Kiand ωw

i = ζ(α−c)

2ζ−λ2Ki+ c, respectively.

Finally, the expected profit for the manufacturer is:

�wM

(ωw

i , θwi

) = r�wM

(ωw

i , θwi

∣∣R = l)

+ (1 − r)�wM

(ωw

i , θwi

∣∣R = h),

= E[K].(

(α − c)ζ

2ζ − λ2Ki

)2

− 1

2ζ

((α − c)

λKi

2ζ − λ2Ki

)2

,

= ζ

2

(α − c

2ζ − λ2Ki

)2 (2ζE[K] − λ2K2

i

), (11)

where E[K] = rKl +(1−r)Kh. Let 1(r) denote the differ-ence in expected manufacturer profits for contracts (ωw

l , θwl )

and (ωwh , θw

h ) for a given probability, r . Then,

1(r) = [�w

M

(ωw

l , θwl

)] − [�w

M

(ωw

h , θwh

)],

= r[�w

M

(ωw

l , θwl

)∣∣R = l) − �w

M

(ωw

h , θwh

)∣∣R = l)]

+ (1 − r)[�w

M

(ωw

l , θwl

)∣∣R = h)

− �wM

(ωw

h , θwh

)∣∣R = h)]. (12)

LEMMA 1: For the case of wholesale price individualcontracts, the following hold true:

1. the manufacturer’s expected profits linearly increasein r for i = h, l;

2. 1(r) strictly increases in r;3. There exists a unique r in (0,1) for which 1(r) = 0.

PROOF: All Proofs are in Companion Appendix-A.The result of Lemma 1 follows from the fact that for any

given wholesale price contract, since the l-type retailer ismore efficient and can exert more effort and generate higherdemand, the profit for the manufacturer is higher as comparedto the case when the retailer is of the h-type. The expectedprofits for the manufacturer are the weighted sum depend-ing on the probability, r . Therefore, as r increases, even ifthe h-type individual contract is offered, since retailers ofboth types will participate, the manufacturer is better off withhigher values of r . As shown above, Lemma 1 implies that upto a threshold probability, r , the manufacturer is better off byoffering a contract designed for theh-type retailer (comparingindividual contracts only). Beyond r , the contract designedfor the l-type retailer becomes more appealing. �

2.1.2. Menu of Contracts

We now design the menu of contracts {(ωwl , θw

l ), (ωwh , θw

h )}where (ω

wl , θw

l ) is directed to the l-type retailer and (ωwh , θw

h )

is directed to the h-type retailer (we use ˆ to denote expres-sions for the menu contract). For the menu of contracts to



be correctly designed, they must be self-selecting, [25]. Themanufacturer’s objective is to maximize expected profits:

�wM = Max{(

ωw

l ,θwl

)(ω

w

h ,θwh

)}r[�w

M

(ω

wl , θw

l

∣∣R = l)]

+ (1 − r)[�w

M

(ω

wh , θw

h

∣∣R = h)]

(13)

s.t. �R

((ω

wl , θw

l

)∣∣R = l − type) ≥ U , (14)

�R

((ω

wh , θw

h

)∣∣R = h − type) ≥ U , (15)

�R

((ω

wl , θw

l

)∣∣R = l − type)

≥ �R

((ω

wh , θw

h

)∣∣R = l − type), (16)

�R

((ω

wh , θw

h

)∣∣R = h − type)

≥ �R

((ω

wl , θw

l

)∣∣R = h − type), (17)

where U is the reservation utility of the retailer. The expectedprofit function of the manufacturer comprises of profit termsfor the l-type and h-type retailers. Constraints (14) and (15)are the participation constraints and (16) and (17) ensureself-selection. Let

A = ζ [rKl + (1 − r)Kh](α − c)[2rζKl + 2(1 − r)ζKh − λ2

{rK2

l + (1 − r)K2h

}] .

(18)

Then, the following proposition outlines the optimal solutionto the manufacturer’s problem:

PROPOSITION 1: The solution to the Manufacturer’sProblem (13) depends on the value of the reserva-tion utility, U . For U ≤ Umax, where Umax =Kh

2

[ζ [rKl+(1−r)Kh](α−c)

[2rζKl+2(1−r)ζKh−λ2{rK2l +(1−r)K2

h }]]2

, the optimal whole-

sale price menu contract is as follows:

1. The optimal expressions for the l-type contract are:

ωw∗l = α + λ2KlA

ζ− A, (19)

θw∗l = λKlA

ζ. (20)

2. The optimal expressions for the h-type contract are:

ωw∗h = α + λ2KhA

ζ− A, (21)

θw∗h = λKhA

ζ. (22)

It is easy to show by substituting A from (18) into (19) and(21) that ω

wl and ω

wh are greater than or equal to c, for all r .

As we discuss in the proof, participation constraint for the

l-type retailer (14) is ensured by the other three constraints(15)–(17) and therefore can be omitted.

There exists a threshold value of the reservation utility,Umax, such that, if U is below this value, the contract offeredby the manufacturer automatically satisfies the participationconstraint of the h-type retailer (participation of the l-typeretailer is ensured by the other constraints). However, if thereservation utility exceeds this threshold value, the manu-facturer is not able to optimize his unconstrained objectivefunction but offers the contract ensuring that the retaileraccepts the contract. At some stage, if the reservation util-ity is very high, the manufacturer will not make any profitsand will therefore not offer any contract to the retailer. In therest of the paper, we focus on the case when the reservationutility does not exceed the threshold level and analyze theeffect of r on the contract selection problem for the manu-facturer. The case when the reservation utility exceeds thethreshold value is discussed in section 5.

On comparing the optimal expressions for the two con-tracts, we can see that ω

w∗l > ω

w∗h , and θw∗

l > θw∗h , that is,

the l-type retailer gets a higher quality product but has to paya higher wholesale price to the manufacturer.

The menu of contracts yields an optimal expected profitfor the manufacturer defined in (13). Therefore, as longas the optimal profit (13) exceeds the profit with the indi-vidual contract (using η = ηl or η = ηh), the manufac-turer would offer the menu of contracts. We now discusssome properties of the solution. Note that A in (18) can berewritten as:

A = ζ(α − c)E[K]

2ζE[K] − λ2E[K2] .

Then, for i = l, h, we have

θwi =(α−c)

λKiE[K]2ζE[K] − λ2E[K2] = E[K](2ζ − λ2Ki)

2ζE[K] − λ2E[K2]θwi ,

where ˆθwi is the quality offered in the contract menu in Propo-

sition 1, and θwi is the quality offered in the individual contract

(from equation (6) using K = Ki , i = l, h).

LEMMA 2: For the case of wholesale price menu contract,the following hold true:

1. Product quality in both low-cost and high-cost menucontracts strictly increases in r;

2. Manufacturer offers higher quality to high-costretailer in the menu contract as compared to thequality offered in the individual contract, that is,θwh ≥ θw

h , for any r in [0, 1);



3. Manufacturer offers lower quality to low-cost retailerin the menu contract as compared to the qualityoffered in the individual contract, that is, θw

l ≤ θwl ,

for any r in [0, 1).

Essentially, Lemma 2 indicates that the quality choice ofthe manufacturer increases in the menu contract as his beliefof l-type retailer rises. However, the selection of quality levelfor different retailer types is subject to making the contractself-selecting.

2.1.3. Comparison of Wholesale Price Individual andMenu Contracts

We now compare the manufacturer’s profit for the whole-sale price individual and menu contracts. In the individ-ual contract case, the manufacturer’s expected profit can bewritten (from (11)) for a given r and contract i as:

�wM

(ωw

i , θwi

) = ζ

2

(α − c

2ζ − λ2Ki

)2 (2ζE[K] − λ2K2

i

),

for i = h, l. On the other hand, the expected manufacturerprofit from the menu of contracts (from (13) and Proposition1) is:

Er

[�w

M

] = A

(E[K](α − c − A) + λ2AE[K2]

2ζ

)= (α − c)E[K]

2A = ζ(α − c)2E2[K]

2�,

where � is the denominator in (18). We know that the man-ufacturer profits increase in r in the individual contract case,and derive a similar result for the menu contracts.

LEMMA 3: For the case of wholesale price menu contract,the manufacturer’s expected profit is convex increasing in r .

This leads us to the following key result:

PROPOSITION 2: The manufacturer’s expected profitunder wholesale price menu contract will be no less than theexpected profit using wholesale price individual contracts forany r in [0, 1].

We observe from the results of Proposition 1 that bothconstraints (16) and (17) are binding at optimality, that is,the retailer would be indifferent to picking either of thecontracts offered in the menu contract (designed for l-typeor h-type retailer). This implies that the individual con-tracts can be interpreted as special instances of the menucontract satisfying both constraints (16) and (17), and con-sequently, individual contracts cannot outperform the menucontract.

Figure 1. Comparison of wholesale price individual and menucontracts. [Color figure can be viewed in the online issue, which isavailable at www.interscience.wiley.com.]

The result is illustrated in Fig. 1. Essentially, we note thata risk neutral manufacturer, using a wholesale price con-tract, will always prefer to offer a menu of contracts insteadof choosing an individual contract. While this is true forwholesale price contracts, as we will see in the next section,the menu contract may not always dominate the individualcontract for other types of contracts.

Note that the analytical results derived in this section arefrom the manufacturer’s perspective. If we were to considerchannel profits, our numerical analysis suggests that for thetotal channel, individual contracts may generate higher prof-its for relatively large values of r . But since the manufacturercannot divide/share the higher channel profit due to the lack ofa transfer fee in the contract, the manufacturer would be worseoff using the individual contract even though the retailer isstrictly better off. The question then is, if the manufacturercould use transfer (fixed) fees, would he be always better off?In the next section, we analyze the role of transfer fees in afixed-fee contract.

3. TWO-PART TARIFF CONTRACT

The wholesale price contract derived in the previoussection does not achieve channel coordination (even for theperfect information case) due to the double-marginalizationeffect. We now consider a two-part tariff contract6 where themanufacturer chooses a wholesale price and charges a trans-fer fee which offers the manufacturer additional flexibility

6 Franchise contracts, which include both fixed and variable feesand are referred to as two-part tariff contracts, have been exten-sively studied in the marketing and economics literature (see Refs.8, 9, 19, 20).



to achieve channel coordination. To show this, we model thecentralized problem where the decisions are to choose theoptimal values of (θ , p, e). Define:

�c = Max(θ ,p,e)[α − p + γ e + λθ ]×(p − c)−ζθ2/2−ηe2/2.

which yields

θ∗c = Kλ(α − c)

(ζ − λ2K), (23)

p∗c = c + Kζ(α − c)

(ζ − λ2K), (24)

e∗c = Kγζ(α − c)

η(ζ − λ2K). (25)

Substituting in(�c)above, we get:

�∗c = ζK(α − c)2

2(ζ − λ2K)= B(say).

(26)

Let us now focus our attention on the decentralized prob-lem (we use f to denote the expressions for this contract).Now, the retailer’s problem is defined as follows:

�R = Max(pf ,ef )[α − pf + γ ef + λθf ](pf − ωf )

− η(ef )2/2 − Ff , (27)

which yields �∗R = K

2(α + λθf − ωf )2 − Ff , (28)

when (ωf , θf , Ff ) is offered by the manufacturer.On comparing �∗c with �∗

R above, we note that if themanufacturer offers the contract (ωf , θf , Ff ), channel coor-dination can be achieved by setting ωf ∗ = c, θf ∗ = θ∗c, andFf ∗ = B +ζ(θ∗c)2/2−U , where U is the reservation utilityof the retailer. We call this as the fixed-fee contract as themanufacturer sets wholesale price equal to the marginal costand derives all of his profits by setting the fixed-fee Ff . Notethat the total payment made by the retailer is cD + F , whereD = α − p + γ e + λθ , depends on the various parametersof the model including cost of quality, effort, etc. Thus, as D

increases, the unit cost, (cD + F)/D = c + F/D, decreaseswhich is similar to a quantity discount contract.7

Since the fixed-fee contract achieves channel coordina-tion with perfect information, we determine conditions underwhich fixed-fee contracts may be used by the manufacturerunder asymmetric information. Later, in this section, we alsoconsider the generalized franchise contract where wholesaleprice exceeds the marginal cost and the manufacturer chargesa transfer fee as well.

7 We thank the Associate Editor for suggesting a quantity discountinterpretation to the fixed-fee contract.

3.1. Fixed-Fee Contract

Note that the fixed-fee Ff ∗ above consists of three terms;the middle term is the manufacturer’s cost of providing qual-ity θf ∗. As such, only B represents the actual profit for themanufacturer. The delineation of F into separate terms allowsus to use only the “B” term in the objective function for themanufacturer.

As for the case of the wholesale price contract, we observethat the contract terms depend on η, the retailer’s cost ofeffort. In fact, we can show that the transfer fee Ff ∗ (definedabove) is decreasing in η, that is, an efficient retailer (one witha lower cost of effort) would pay a higher transfer fee. SinceKl > Kh, from equation (30), we see that if the manufac-turer uses the perfect information contract, the l-type retailerhas an incentive to misrepresent her cost of effort and makeabove-reservation profit. We address this issue of asymmetryin cost information next. Similar to section 2.1, the manu-facturer’s priors are r and (1 − r) for the l-type and h-typeretailer, respectively.

3.1.1. Individual Contract

Consider the contract expressions derived for the case ofperfect information [Eqs. (23)–(25)]. The manufacturer cannow offer contracts (perfect information contracts) using ηl

in one contract (l-type aimed at the low-cost retailer) or ηh

in the other (h-type aimed at the high-cost retailer). Fromthe perfect information case, we know that the manufacturercan set the fixed fee such that the retailer makes exactly herreservation profit.

Lets start with the case when the manufacturer offers thecontract using η = ηl , that is, (c, θf

l , Ff

l ). In this case, if theretailer is of the l-type, she would accept the contract andmake the reservation profit. On the other hand, if the retaileris of the h-type, she would make less than her reservationprofit and would therefore not accept the contract aimed atthe l-type retailer. Then, the manufacturer’s expected profit is:

�f ,lM = r

(ζKl(α − c)2

2(ζ − λ2Kl)− U

).

Note that the fixed-fee l-type individual contract is channeloptimal if the retailer is of the l-type.

Now, if the manufacturer offers the contract using η = ηh,that is, (c, θf

h , Ff

h ), if the retailer is of the h-type, she wouldaccept the contract and make the reservation profit. On theother hand, if the retailer is of the l-type, she would alsoaccept the contract and make greater than her reservationprofit. Then, from equation (26), the manufacturer’s expectedprofit is:

�f ,hM = ζKh(α − c)2

2(ζ − λ2Kh)− U .



PROPOSITION 3: For the case of fixed-fee individualcontracts, there exists a unique r in (0, Kh

Kl) such that the

manufacturer would have higher expected profit with thel-type contract (c, θf

l , Ff

l ) if r > r . If r < r , the manu-facturer’s expected profits would be higher with the h-typecontract (c, θf

h , Ff

h ), where

r =[

ζ − λ2Kl

ζ − λ2Kh

(ζKh(α − c)2 − 2U(ζ − λ2Kh)

ζKl(α − c)2 − 2U(ζ − λ2Kl)

)]

The intuition for this result is that if the manufacturer has astrong belief on the retailer being of the l-type (that is, r > r),it is preferred to offer the contract aimed at the l-type retaileronly. In this case, even though only the l-type retailer acceptsthe contract offer, since the fixed fee gained by the manu-facturer (F

f

l ) is higher than the fixed fee (Ff

h ) derived fromoffering the h-type contract, the manufacturer is better offwith the low-cost individual contract when r is sufficientlyhigh.

3.1.2. Menu of Contracts

We now design the menu of contracts {(c, θ f

l , F f

l ),(c, θ f

h , F f

h )} where (c, θ f

l , F f

l ) is directed to the l-typeretailer, and (c, θ f

h , F f

h )} is directed to the h-type retailer(again, we useˆto denote expressions for the contract menu).From the definition of F

f

i , for i = l, h, once the manufacturerselects θ

f

i and Bf

i , the transfer fee Ff

i = Bf

i + ζ θf 2i /2 is

determined.Using (28) and the definitions of Kl and Kh, similar to the

wholesale price menu contract derived in section 2.1.2, themanufacturer’s problem is as follows:

�f

M = Max{(θ

f

l ≥0,Bf

l

)(θ

f

h ≥0,Bf

h

)}[rB

f

l + (1 − r)Bf

h

]s.t.

Kl

2

[α + λθ

f

l − c]2 − B

f

l − ζ θf 2

l /2 ≥ U ,

(29)

Kh

2

[α + λθ

f

h − c]2 − B

f

h − ζ θf 2

h /2 ≥ U ,

(30)

Kl

2

[α + λθ

f

l − c]2 − B

f

l − ζ θf 2

l /2 ≥ Kl

2

× [α + λθ

f

h − c]2 − B

f

h − ζ θf 2

h /2, (31)

Kh

2

[α + λθ

f

h − c]2 − B

f

h − ζ θf 2

h /2 ≥ Kh

2

× [α + λθ

f

l − c]2 − B

f

l − ζ θf 2

l /2. (32)

The optimal solution to the Manufacturer’s Problem isgiven in the following Proposition.

PROPOSITION 4: In the solution to the manufacturer’sproblem above, the optimal fixed-fee menu contract is asfollows:

1. The optimal expressions for the high-cost type con-tract are:

θf ∗h = Max

[λ(Kh − rKl)(α − c)

(ζ(1 − r) − λ2(Kh − rKl)), 0

],

(33)

Bf ∗h = Kh

2

[α + λθ

f

h − c]2 − ζ θ

f 2

h /2 − U . (34)

2. The optimal expressions for the low-cost type con-tract are:

θf ∗l = λKl(α − c)

(ζ − Klλ2), (35)

Bf ∗l = Kl

2

[α + λθ

f

l − c]2 − ζ θ

f 2

l /2 − 1

2

× [Kl − Kh][α + λθ

f

h − c]2 − U . (36)

Note from (33) that θf

h > 0 for r < Kh/Kl , and zerootherwise. The quality offered for the l-type retailer in themenu contract is independent of r and is the same as thequality offered to the l-type retailer in the perfect informationcase. Further, from (36), we note that an increase (decrease)in θ

f

h would decrease (increase) the transfer fee Bf

l for thel-type retailer. Interestingly, the manufacturer can use thequality level for one type of retailer to adjust the transfer feecharged to the other type. Similar to the results in Lemma 2,other observations are presented next:

LEMMA 4: For the case of fixed-fee menu contract, thefollowing hold true:

1. Product quality for the low-cost retailer is indepen-dent of r and is the same as the quality offered to thelow-cost retailer in the fixed-fee individual contract,that is, θ

f

l = θf

l , for any r in (0, 1];2. Product quality for the high-cost retailer is decreas-

ing in r; the manufacturer offers lower quality tothe high-cost retailer in the fixed-fee contract menuas compared to the quality offered in the fixed-feeindividual contract, that is, θ

f

h ≤ θf

h , for any r in[0, 1).

In contrast to the wholesale price menu contract, here,quality for the h-type retailer decreases in r . Intuitively, asr increases, by reducing θ

f

h , the manufacturer increases thetransfer fee charged to the l-type retailer (see (36)). Simi-lar conclusion can be derived if the spread between Kh andKl is large. Finally, we note that for small values of r (i.e.,



r → 0), quality offered to the h-type retailer also approachesthe centralized system solution.

Next, we analyze the expected profit for the manufacturerunder this type of contract. Because of the non-negativityconstraint on θ

f

h , the manufacturer profit function is definedseparately for the two cases. For r < Kh/Kl ;

�f −M � E

[�

f

M(r < Kh/Kl)] = 1

2ζ(α − c)2

×(

rKl

ζ − λ2Kl

+ (1 − r)(Kh − rKl)

(1 − r)ζ − λ2(Kh − rKl)

)− U ;

and, for r ≥ Kh/Kl ,

�f +M � E

[�

f

M(r ≥ Kh/Kl)] = 1

2(α − c)2

×(

Kh + rλ2K2l

ζ − λ2Kl

)− U .

3.1.3. Comparison of Fixed-Fee Individualand Menu Contracts

In Proposition 3, we compared the manufacturer’sexpected profits for the fixed-fee individual contracts. Now,we compare the fixed-fee individual contracts with thefixed-fee menu contract derived in the previous subsection.

First, observe that �f −M is convex increasing in r for a given

reservation utility of the retailer. To prove this, note that thefirst derivative of �

f −M wrt to r is:

∂�f −M

∂r= 1

2(ζ − λ2Kl)

(ζ(α − c)(Kl − Kh)

(1 − r)ζ − λ2(Kh − rKl)

)2

> 0;

The second derivative is:

∂2�f −M

∂r2=

(ζ(α − c)(Kl − Kh)

(1 − r)ζ − λ2(Kh − rKl)

)2

×(

1

(1 − r)ζ − λ2(Kh − rKl)

)>0 for r <Kh/Kl .

The results are given in the following two propositions.

PROPOSITION 5: The fixed-fee h-type individual con-tract is always dominated by either the fixed-fee l-type indi-vidual contract or the fixed-fee menu contract, that is, themanufacturer would never find it optimal to offer the fixed-feeindividual contract for the high-cost retailer.

We know from Proposition 3 that if the probability of l-type retailer is sufficiently high, the l-type individual contractwill bring more profits to the manufacturer compared to theh-type contract. If the manufacturer offers the h-type con-tract, retailer of both high and low cost would be willing to

participate, and hence, the h-type individual contract is a spe-cial case of the menu contract and can do no better than thefixed-fee menu contract.

Since the fixed-fee h-type individual contract is alwaysdominated, we compare the fixed-fee l-type individual con-tract with the fixed-fee menu contract. First, we define U suchthat, U = 1

2ζ(α − c)2U , where U is the reservation utility ofthe retailer.

PROPOSITION 6: There exists a unique rt in (0, Kh

Kl) such

that the manufacturer would have higher expected profit withthe fixed-fee l-type individual contract if r > rt . If r < rt ,the manufacturer’s expected profits would be higher with thefixed-fee menu contract, where

rt = Kh − U (ζ − λ2Kh)

Kl − U (ζ − λ2Kl). (37)

From Proposition 6, we conclude that the individual con-tract designed for l-type retailer yields better expected profitsfor the manufacturer if and only if r > rt . Otherwise, themenu contract outperforms the individual low-cost contract.Essentially, when the manufacturer has a high prior on theretailer being of the low-type, it may be optimal not to dis-tinguish between the retailer types and lose some efficiencyby offering the menu contract. From Proposition 4 we knowthat the efficiency loss is due to the fact that under fixed-feemenu contract, the l-type contract must be designed to pre-vent the l-type retailer from choosing the h-type contract.Consequently, at optimality, we show that constraint (31) isbinding which limits the manufacturer’s pricing and qualitychoices. As such, the manufacturer incurs a cost for screen-ing, which is needed to separate the retailer types. Clearly, theoptimal manufacturer profits from selling to a certain type ofretailer are always higher under perfect information. The gapbetween profits can be interpreted as the fee that the manufac-turer needs to pay to learn the true type of the retailer. Aboveresult indicates that when the probability of l-type retailer issufficiently high, it is not worthwhile for the manufacturer toscreen retailers and to pay this fee. Hence, for large valuesof r , l-type individual contract is preferable to the manufac-turer. The results are illustrated in Fig. 2. On the flip side, wenote that the h-type individual contract is never preferable tothe manufacturer even when r is small. The h-type individualcontract always generates lower profit for the manufacturercompared to the l-type individual contract. As such the con-tract design must include the l-type retailer for any positiver value.

The decision to offer the low-cost individual or menu con-tract depends on the probability rt . We now derive some



Figure 2. Comparison of fixed-fee individual and menu contracts.[Color figure can be viewed in the online issue, which is availableat www.interscience.wiley.com.]

bounds on the values of rt which will be useful in derivingthe subsequent results in the next section.

COROLLARY 1: The critical threshold probability rt isa decreasing function of ζ for a given U , and a decreasingfunction of U (or U ) for a given ζ . Also, the bounds are:Kh

Kl≥ rt ≥ 3Kh

4Kl−Kh.

In Appendix B, we consider the fixed-fee pooling contractand show that is dominated by the menu contract.

3.2. General Franchise Contract

In this section, we consider the general franchise contractwhere the manufacturer sets the wholesale price (at a levelnot necessarily equal to the marginal cost) in addition to thefixed fee charged to the retailer. Specifically, we considertwo forms of franchise contracts: menu contract and poolingcontract. We do not have to consider the franchise individualcontract as it is the same as the fixed-fee individual contract(if there is no information asymmetry, the optimal franchisecontrast is essentially a fixed-fee contract only as wholesaleprice is set equal to production cost). In the analysis below,the franchise menu contract is derived whereas the franchisepooling contract is given in Appendix B and is shown to bedominated by the menu contract.

Similar to the fixed-fee menu contract in section 3.1.2,we design the franchise menu contract {(ωF

l , θ Fl , F F

l ),(ω

Fh , θ F

h , F Fh )} directed to the l-type and h-type retailer,

respectively. We use the superscript F for the expressionsin this contract.

The manufacturer’s problem is as follows:

�FM = Max{(

ωF

l ,θFl ,F F

l

)(ω

F

h ,θFh ,F F

h

)}×

[r

{(ω

Fl − c

)Kl

(α + λθF

l − ωFl

) − ζ

2θ F 2

l + F Fl

}+ (1 − r)

{(ω

Fh − c

)Kl

(α + λθF

h − ωFh

)−ζ

2θ F 2

h + F Fh

}](38)

s.t.Kl

2

[α + λθF

l − ωFl

]2 − F Fl ≥ U , (39)

Kh

2

[α + λθF

h − ωFh

]2 − F Fh ≥ U , (40)

Kl

2

[α + λθF

l − ωFl

]2 − F Fl ≥ Kl

2

× [α + λθF

h − ωFh

]2 − F Fh , (41)

Kh

2

[α + λθF

h − ωFh

]2 − F Fh ≥ Kh

2

× [α + λθF

l − ωFl

]2 − F Fl . (42)

The optimal solution to the manufacturer’s problem is givenin the following proposition.

PROPOSITION 7: In the solution to the manufacturer’sproblem (38), the optimal franchise menu contract is asfollows:

• The optimal expressions for the high-cost type con-tract are:

ωF∗h = c + (α − c)rζ(Kl − Kh)

rζ(Kl − Kh) + (1 − r)Kh(ζ − λ2Kh),

(43)

θ F∗h = (1 − r)λK2

h(α − c)

rζ(Kl − Kh) + (1 − r)Kh(ζ − λ2Kh),

(44)

F F∗h = Kh

2

[α + λθF

h − ωFh

]2 − U . (45)

• The optimal expressions for the low-cost type contractare:

ωF∗l = c, (46)

θ F∗l = λKl(α − c)

(ζ − Klλ2), (47)

F F∗l = Kl

2

[α + λθ

f

l − c]2 − 1

2[Kl − Kh]

×[α + λθF

h − ωFh

]2 − U . (48)

We note that the franchise menu contract is similar to thefixed-fee menu contract derived in section 3.1.2. While the



l-type retailer gets wholesale price equal to the marginal cost,the contract for the h-type retailer includes a wholesale pricethat strictly exceeds marginal cost, that is, the manufacturerfully utilizes the two-part tariff for the h-type retailer. Finally,as shown in Appendix B, the franchise pooling contract isdominated by the menu contract.

4. COMPARISON OF CONTRACT TYPES

In sections 2 and 3, we analyzed wholesale price and two-part tariff contracts, respectively, and showed that while thewholesale menu contract dominated other forms of wholesalecontracts, for the fixed-fee contract, both the l-type individ-ual or menu contract may be offered by the manufacturer. Wenow compare the different contract types in this section.

4.1. Comparison Between Wholesale Price andFranchise Contracts

Note that the wholesale price menu contract (in section2.1.2) is a special case of the general franchise menu contractwith F F

h = F Fl = 0. First, we show that the franchise menu

contract dominates the wholesale price menu contract, thatis, the optimal solution to the franchise menu contract willnot have both F F

h = F Fl = 0.

PROPOSITION 8: It is never optimal for the manufacturerto set transfer fees, F F

h = F Fl = 0, that is, the wholesale

price menu contract is always dominated by the franchisemenu contract.

The above result establishes that the wholesale price menucontract, which is a special case of the general franchise menucontract (when transfer fees are set to zero), will not be offeredas the franchise contract yields higher profits for the manu-facturer. Essentially, as the proof indicates, it is never optimalto set zero transfer fees in the contract. It is however impor-tant to note that even though the wholesale menu contractis dominated, we need the results derived in Section 2 (thatindividual wholesale price contracts are dominated by thewholesale menu contract) in order to eliminate the wholesaleprice only contract.

4.2. Comparison Between Fixed-Fee and GeneralFranchise Contracts

While the wholesale price contract is dominated by thefranchise menu contract, would the fixed-fee contract, whichis channel optimal under perfect information, be preferred tothe general franchise menu contract in the imperfect informa-tion case? To analyze this, we compare the expected profitsfor the manufacturer for the two contracts.

From (38), the expected profit for the manufacturer in thefranchise menu contract is:

�FM = r

(α − c)2ζKl

2(ζ − λ2Kl)+ (1 − r)2

× (α − c)2ζK2h

2(rζ(Kl − Kh) + (1 − r)Kh(ζ − λ2Kh))− U

(49)

The expected profit for the manufacturer by using the fixed-fee low-cost individual contract is given in section 3.1.1.As shown in the following proposition, there exist condi-tions such that the manufacturer would prefer the Individualfixed-fee contract over the more general franchise menu con-tract. As noted earlier in section 3.1.3, we define U such that,U = 1

2ζ(α − c)2U .

PROPOSITION 9:

1. The manufacturer would offer the general franchisemenu contract if r ≤ rf ;

2. The manufacturer would offer the fixed-fee l-typeindividual contract if r > rf ,

where rf = K2h − UKh(ζ − λ2Kh)

K2h − U [Kh(ζ − λ2Kh) − ζ(Kl − Kh)]

.

(50)

It is easy to note that rf < 1 since Kl > Kh. Also,non-negativity of rf is ensured since the maximum reser-vation utility of the retailer is less than the centralized system

Figure 3. Comparison of fixed-fee l-type individual and franchisemenu contracts. [Color figure can be viewed in the online issue,which is available at www.interscience.wiley.com.]



profit for the h-type retailer case. The result is illustratedin Fig. 3. A simple comparison between (37) and (50) willreveal that rf ≥ rt implying that the threshold probabilitybetween the fixed-fee l-type individual contract and the gen-eral franchise contract is no less than the threshold probabilitybetween the former contract and the fixed-fee menu contract.This is intuitive as the general franchise contract providesmore flexibility to the manufacturer compared to the fixed-feemenu contract.

COROLLARY 2: The manufacturer is more likely to offerthe fixed-fee l-type individual contract (as compared to thefranchise menu contract) to a retailer with high reservationutility since rf decreases in U . Also, preference for thefixed-fee contract is stronger when the difference in retailercosts (measured in terms of the difference, Kl − Kh) ishigher.

Essentially, we show that when the manufacturer’s prioron the retailer being of the l-type exceeds a certain threshold,it may be preferable to offer the fixed-fee l-type individualcontract as compared to the franchise menu contract. Then,the manufacturer would exclude the h-type retailer and offeran individual contract that is accepted by the l-type retaileronly. Intuitively, the individual contract is efficient for thecase of perfect information and as such, the manufacturermay prefer it to the franchise menu contract (which has someloss in efficiency in order to make it self-selecting for bothretailer types) when the likelihood of the retailer being of thel-type is high. Further, we also note that the preference for theIndividual fixed-fee contract increases when the reservationutility of the retailer and the degree of information asymme-try in costs is high. As such, the manufacturer becomes morewilling to exclude the h-type retailer by offering the fixed-fee individual contract which is only accepted by the l-typeretailer.

5. CONCLUSIONS AND FUTURE RESEARCH

In this paper, we examine and compare different contract-ing approaches that a manufacturer may adopt selling to aretailer when end-customer demand depends on the man-ufacturer’s chosen quality and the retailer’s selling effort.It has been shown in the literature that franchising offersmore flexibility to the manufacturer as compared to price-only contracts. Nonetheless, both approaches are common inpractice. In our analysis, we examine whether wholesale con-tracts might indeed be preferable to the manufacturer when acontract design requires additional non-monetary terms suchas product quality. Our results indicate that while the gen-eral franchise menu contract dominates the wholesale menucontract, the manufacturer may, at times, prefer to use the

fixed-fee contract as opposed to the general franchise menucontract.

The analytical results derived in the paper are for the casewhen the retailer’s reservation utility is less than a thresholdvalue (Umax). While we do not derive analytical results for thecase when U ≥ Umax, in our numerical analysis, we observedthat as the reservation utility increases up to point beyondUmax, the difference between manufacturer profits for fixed-fee and wholesale contracts continues to decrease and theresults are the same as derived in the paper. Beyond a certainlevel, if the reservation profit continues to increase, fixed-feecontracts will outperform the wholesale contracts since theexpected profits for the manufacturer will also decrease underwholesale contracts. For sufficiently high reservation profits,the manufacturer will ignore the h-type retailer and designfixed-fee contracts for the l-type retailer only. Finally, if thereservation profit is too large, the manufacturer may refuseto offer any contract at all.

In this article, we considered the case when there is nodemand uncertainty and we were able to derive interestingmanagerial insights into the contract selection problem forthe manufacturer. In future research, we plan to generalizethe model in order to study the effect of demand variabil-ity. In addition, if the manufacturer’s quality is not observedat the time of signing the contract, we encounter a doublemoral hazard problem. Other possible extensions include amenu of products offered by the manufacturer with differentquality, and the issue of horizontal competition with multiplemanufacturers and/or retailers in the channel.

APPENDIX

Appendix A: Companion

PROOF OF LEMMA 1: 1. For a given contract type (ωwi , θw

i ), i = h, l,the marginal change in manufacturer’s expected profits with respect to r is

∂�wM

(ωw

i , θwi

)∂r

= (Kl − Kh)

((α − c)ζ

2ζ − λ2Ki

)2

> 0,

since Kl > Kh. Also, the first derivative is independent of r . Therefore,the manufacturer profits linearly increase in r for either type of IndividualContract offered.

2. The derivative of 1(r) with respect to r can be written as:

∂1(r)

∂r= (

�wM

(ωw

l , θwl

)∣∣R = l) − �w

M

(ωw

h , θwh

)∣∣R = l))

+(�w

M(wh, θh)∣∣R = h

) − �wM(wl , θl)|R = h)) > 0,

since for a given retailer type i, contract (wi , θi ) maximizes the manufacturerprofits. Hence, 1(r) increases in r .

3. From 1) and 2) above,

1(0) = �wM(wl , θl)|R = h) − �w

M(wh, θh)|R = h) < 0,

and 1(1) = �wM(wl , θl)|R = l) − �w

M(wh, θh)|R = l) > 0.



Since 1 strictly increases in r , there must exist a r , such that 0 < r < 1,and 1(r) = 0. �

PROOF OF PROPOSITION 1: Using (3) and the definitions of Kl , Kh,we can write constraints (14)–(17) as:

Kl

2

[α + λθw

l − ωwl

]2 ≥ U (A1)

Kh

2

[α + λθw

h − ωwh

]2 ≥ U (A2)

Kl

2

[α + λθw

l − ωwl

]2 ≥ Kl

2

[α + λθw

h − ωwh

]2(A3)

Kh

2

[α + λθw

h − ωwh

]2 ≥ Kh

2

[α + λθw

l − ωwl

]2(A4)

We note that constraints in (A3) and (A4) imply that (α + λθwl − ω

wl ) =

(α + λθwh − ω

wh ). Hence, we can replace the last two constraints with this

equality. It appears that both retailers may be indifferent to the contractoffered to them and as such the menu contract does not distinguish betweendifferent types of retailers. However, suitable use of ε (with ε → 0) in theRHS of (A3) and in the LHS of (A4) can ensure that a separating contractcan be achieved. We first ignore the first two constraints in (A1) and (A2),and solve the optimization problem. Later we note that the optimal solutionto the relaxed problem does satisfy the constraints in (A1) and (A2) whenU ≤ Umax, as assumed in the statement of the Proposition.

From (13), we can write the manufacturer’s profits as:

�wM

(ω

wl , θw

l |R = l) = (

ωwl − c

)Kl

(α + λθw

l − ωwl

) − ζ θw2

l /2, (A5)

�wM

(ω

wh , θw

h

∣∣R = h) = (

ωwh − c

)Kh

(α + λθw

h − ωwh

) − ζ θw2

h /2. (A6)

Using Lagrangean relaxation and substituting from (A5) and (A6) in (13),we get:

�wM = Max{

ωwl ≥0,ωw

h ≥0,θwl

≥0,θwh

≥0,µ)}r

[Kl

(ω

wl − c

)× (

α + λθwl − ω

wl

) − ζ θw2

l /2]

+ (1 − r)[Kh

(ω

wh − c

)(α + λθw

h − ωwh

) − ζ θw2

h /2]

+ µ[(

ωwl − ω

wh

) − λ(θwl − θw

h

)]where µ is the Lagrange multiplier. First order optimality conditions yieldthe following five equations:

∂�wM

∂ωwl

= rKl

(α + λθw

l + c − 2ωwl

) + µ = 0 (A7)

∂�wM

∂ωwh

= (1 − r)Kh

(α + λθw

h + c − 2ωwh

) − µ = 0 (A8)

∂�wM

∂θwl

= r(λKl

(ω

wl − c

) − ζ) − λµ = 0 (A9)

∂�wM

∂θwh

= (1 − r)(λKh

(ω

wh − c

) − ζ) + λµ = 0 (A10)

∂�wM

∂µ= (

ωwl − ω

wh

) − λ(θwl − θw

h

) = 0 (A11)

Solving equations (A7–A11) for ωwl , ωw

h , θwl , θw

h , and µ, we get (19), (20),(21), and (22). Finally, note that the left hand side in (A2) is less than orequal to Umax. Since by assumption U ≤ Umax, the constraint must hold.Moreover, since ηh > ηl , constraint (A2) also implies (A1). In fact, con-straint (A1) is a strict inequality, that is, the low-cost retailer makes profitsin excess of the reservation profit. �

PROOF OF LEMMA 2: i. It is sufficient to show that the derivative ofquality function with respect to r is positive so as to complete the proof ofthis part. Let � represent the denominator in (18). We can write the derivativeof A with respect to r as follows:

A′(r) = ζ(α − c)λ2(Kl − Kh)KlKh

�2> 0

Hence, since A increases in r , quality for any i must be increasing with r aswell.

ii. To see that menu contracts offer higher quality to high-cost retailer,note from (18) that for r = 0, θw

h = θwh . From first part of the proof we

know that θwh increases in r and that θw

h is independent of r . Thus for r ≥ 0,

θwh ≥ θw

h .

iii. Observe from (18) that for r = 1, θwl = θw

l . Using a similar approach

as in part ii), we can show that for r ≤ 1, θwl ≤ θw

l . �

PROOF OF LEMMA 3: The derivative of Er [�wM ] is as follows:

∂Er

[�w

M

]∂r

= 1

2(α − c)(Kl − Kh)A + (α − c)E[K]

2A′ > 0,

since Kl > Kh and A′ > 0. Hence, we show that manufacturer profitsincrease in r . However, the increase is not linear as compared to the indi-vidual contract case. To observe convexity, we compute the second orderderivative:

∂2Er

[�w

M

]∂r2

= ζ(α − c)2(Kl − Kh)2(

(� − E(K)�′)2

�3

)> 0,

where �′ is derivative of � with respect to r . Thus, the above functionreturns a non-negative value implying that the manufacturer’s profit functionis convex in r . �

PROOF OF PROPOSITION 2: First, we consider the difference in theexpected profits using the menu contract and high-cost individual contract(ωw

h , θwh ). Define,

2(r) = Er

[�w

M

] − [�w

M

(ωw

h , θwh

)]From Lemmas 1 and 3, notice that 2(r) is convex in r . Let ′

2(r) denotethe first order derivative of 2(r) with respect to r . Observe that at r = 0,2(0) = 0 and,

′2(0) = 1

2ζ(α − c)2(Kl − Kh)2 λ2Kh

(2ζ − λ2Kh)2> 0.

Since 2(r) is convex and increasing at r = 0, 2(r) must be positive forany r > 0.

Now, consider the low-cost individual contract (ωwl , θw

l ). Let 3(r) =Er [�M ] − [�M(ωw

l , θwl )], where 3(r) is also convex in r . Observe that

3(1) = 0, and

′2(1) = − 1

2ζ(α − c)2(Kl − Kh)2 λ2Kl

(2ζ − λ2Kh)2< 0,

implying that the difference is decreasing at r = 1 and because of convexityit must be decreasing for any r < 0 as well. Thus, 3(r) ≥ 0 for r ≤ 1. �

PROOF OF PROPOSITION 3: The proof follows from comparing themanufacturer’s expected profits �

f ,lM and �

f ,hM . It is easy to verify that r is



less than KhKl

. Note that if U = 0, we get r = KhKl

(ζ−λ2Kl

ζ−λ2Kh

)which is also

less than KhKl

. �

PROOF OF PROPOSITION 4: Let µ1, µ2, µ3, and µ4 be the Lagran-gian multipliers for the four constraints (29–32). Then, we can write theLagrangian as L = �M +µ1[.]+µ2[.]+µ3[.]+µ4[.]. Taking derivatives,we get:

∂L

∂Bf

l

= 0 ⇒ r − µ1 − µ3 + µ4 = 0, (A12)

∂L

∂Bf

h

= 0 ⇒ (1 − r) − µ2 + µ3 − µ4 = 0. (A13)

Adding the foc’s, we get µ1 +µ2 = 1, that is, at least one of constraints (29)or (30) is activated. There exist three possible scenarios here: (i) µ1 > 0only; (ii) µ2 > 0 only; (iii) both µ1 and µ2 > 0. We can see that if both µ1

and µ2 > 0, that is, both constraints (29) and (30) are activated, we get acontradiction from the other two constraints (31) and (32). The same is trueif only µ1 > 0. Therefore, we have µ1 = 0 and µ2 = 1. Substituting in(A12) and (A13), we get µ3 − µ4 = r , which implies that µ3 > 0. Again,we can see that if µ4 > 0 as well, we have a contradiction from (31) and(32). In summary, we have µ2 and µ3 > 0 and µ1 = µ4 = 0. Therefore,constraints (30) and (31) are activated. On rearranging terms, we get

Bf

h = Kh

2

(α + λθ

f

h − c)2 − ζ

θf 2

h

2− U , (A14)

Bf

l = Kl

2

(α + λθ

f

l − c)2 − ζ

θf 2

l

2− 1

2(Kl − Kh)

(α + λθ

f

h − c)2 − U .

(A15)

On substituting Bf

l and Bf

h from above and taking partial derivatives and

equating to zero, we get (33) (non-negativity of θf

h is ensured by using themaximum of zero and the solution using the foc), (34) and (35), (36). �

PROOF OF LEMMA 4: The first part of the proof is from straightfor-ward comparison of (35) with (23) (using K = Kl ). For the second part, thederivative of θ

f

h wrt to r is:

∂θf

h

∂r= −λ(α − c)ζ(Kl − Kh)

(�f )2< 0,

where �f is the denominator in (33). From (33), we also observe that forr ≥ Kh/Kl , θ

f

h = 0.

Finally, note that θf

h |r=0 = θf

h , but since θf

h is decreasing in r , we have

θf

h ≤ θf

h . �

PROOF OF PROPOSITION 5: We note from Proposition 3 that thehigh-cost individual franchise contract is dominated by the low-cost individ-ual contract for r > r . Therefore, we only need to consider the case whenr ≤ r . Since r ≤ Kh/Kl , we need to compare �

f −M with �

f ,hM .

Note that �f −M |r=0 = ζKh(α−c)2

2(ζ−λ2Kh)− U = �

f ,hM (which is independent of

r). But since �f −M is increasing in r , we have �

f −M ≥ �

f ,hM for r ≤ r . �

PROOF OF PROPOSITION 6: We can show that rt (defined above) isless than Kh/Kl . Therefore, we consider the case when r ≤ Kh/Kl . Now,

using U , we can rewrite the profit functions for the manufacturer as:

�f −M = 1

2ζ(α − c)2

(rKl

ζ − λ2Kl

+ (1 − r)(Kh − rKl)

(1 − r)ζ − λ2(Kh − rKl)− U

),

and �f ,lM = 1

2ζ(α − c)2r

(Kl

ζ − λ2Kl

− U

).

Define 4(r) = �f −M − �

f ,lM = 1

2ζ(α − c)2(1 − r)

×(

(Kh − rKl)

(1 − r)ζ − λ2(Kh − rKl)− U

).

Note that the term within the parenthesis of 4(r) is decreasing in r and isequal to zero for r = rt . Therefore, for r > rt , 4 < 0.

For r > Kh/Kl , define,

5 = �f +M − �

f ,lM = 1

2(α − c)2

(Kh + rλ2K2

l

ζ − λ2Kl

− rKlζ

ζ − λ2Kl

)− (1 − r)U ,

= 1

2(α − c)2(Kh − rKl) − (1 − r)U < 0.

�

PROOF OF COROLLARY 1: For a given U , the first derivative of rtwrt ζ is:

∂rt

∂ζ= − (Kl − Kh)(1 + Uλ2)U

(Kl − U (ζ − λ2Kl))2< 0.

Similarly, we can easily show that rt decreases in U for a given ζ . Therefore,the maximum value for rt is achieved when U = 0, that is, rt |U=0 = Kh

Kl.

Now, we note that the reservation utility for the retailer is less than equalto Umax as defined in Proposition 1. Further, corresponding to the maximumreservation utility Umax, we get Umax. Since rt is a decreasing function in U

and ζ , we substitute the highest value Umax and take the limit as ζ → ∞,and get:

limζ→∞ rt = 3Kh

4Kl − Kh

,

which gives the minimum possible value that rt can take. �

PROOF OF PROPOSITION 7: The proof is similar to that ofProposition 4. �

PROOF OF PROPOSITION 8: Let µ1, µ2, µ3, and µ4 be theLagrangian multipliers for the four constraints (39–42). Also, let µ5 andµ6 be the Lagrangian multipliers on the negativity constraints for F F

l

and F Fh , respectively. Then, using (38), we can write the Lagrangian as

L = �M +µ1[.]+µ2[.]+µ3[.]+µ4[.]+µ5[.]+µ6[.]. Taking derivatives,we get:

∂L

∂F Fl

= 0 ⇒ r − µ1 − µ3 + µ4 + µ5 = 0, (A16)

∂L

∂F Fh

= 0 ⇒ (1 − r) − µ2 + µ3 − µ4 + µ6 = 0. (A17)

Adding the foc’s, we get µ1 + µ2 = 1 + µ5 + µ6, that is, at least one ofconstraints (39) or (40) is activated. Also, from (A16), at least one of µ1 andµ3 > 0. We can easily show that µ1 > 0 results in a contradiction (sinceKl > Kh), and therefore, µ2 > 0 and µ3 > 0.



Now suppose that F Fl = F F

h = 0. Then, we note that constraint (42) must

also be binding, and α + λθFl − ω

Fl = α + λθF

h − ωFh . Now

∂L

∂ωFl

= −r(ω

Fl − c

)Kl + (rKl − µ3Kl + µ4Kh)

(α + λθF

l − ωFl

) = 0.

Since µ3 = r + µ4 + µ5, we get ωFl ≤ c since Kl > Kh.

Also, solving for θFl , we get θF

l = λKl (α−ωFl )

(ζ−Klλ2)

. Then, we get α + λθFl −

ωFl = (α − ω

Fl )

ζ

ζ−λ2Kl= α + λθF

h − ωFh .

From, (40), we get

Kh

2

(α + λθF

h − ωFh

)2 = Kh

2

(α − ω

Fl

)2 ζ 2

(ζ − λ2Kl)2= U .

Since ωFl ≤ c, on simplification, we get

U ≥ Kh

2(α − c)2 ζ 2

(ζ − λ2Kl)2>

Khζ(α − c)2

2(ζ − λ2Kl),

which is a contradiction since the right hand side above is the centralizedsystem profit when the retailer is of the high-type. Clearly, we need theretailer’s reservation utility to be less than the centralized system profit orless the manufacturer would choose not to do business with the retailer. �

PROOF OF PROPOSITION 9: We have

�FM − �

f ,lM = (1 − r)

(α − c)2ζ

2

[(1 − r)

K2h

�F− U

]

where, �F = 2(rζ(Kl − Kh) + (1 − r)Kh(ζ − λ2Kh)).

Define, 7 = (1 − r)K2

h

�F − U . Then, we have ∂7∂r

< 0, 7|r=0 > 0, and7|r=1 < 0. Therefore, 7 = 0 for r = rf as defined in the proposition,and strictly negative for all r > rf . �

PROOF OF COROLLARY 2: The result is proved by noting that thepartial derivative of rf (defined in (50)) with U is less than zero. The secondpart of the Corollary follows from the fact that rf decreases as Kl − Kh

difference is higher. �

Appendix B: Pooling Contracts

We prove that pooling contracts will always be dominated for bothwholesale price only and fixed-fee contracts.

Wholesale Price Pooling Contract

In contrast to wholesale price menu contract, a pooling contract speci-fies a single wholesale price ωw

p and quality θwp pair offered to both retailer

types. In a pooling contract, the manufacturer chooses the optimal (ωwp , θw

p )

pair that maximize her expected profits while ensuring that both retailertypes will have incentives to accept the contract. We note that individualcontracts derived in our paper are not pooling contracts as they are designed

for one retailer type and as such they do not consider the individual ratio-nality constraint of the other. Under the wholesale price only contracts, themanufacturer’s problem for pooling is:

�p,wM = Max(

ωwp ,θw

p

)[(rKl + (1 − r)Kh)(ωw

p − c)(

α + λθwp − ωw

p

)]− 1

2ζ(θwp

)2

s.t .1

2Kl

(α + λθw

p − ωwp

)2 ≥ U ,

1

2Kh

(α + λθw

p − ωwp

)2 ≥ U .

The pooling contract is a single contract offered for both types of retail-ers and hence the incentive compatibility constraints are not needed. Weknow from the proof of Proposition 1 that under optimal wholesale pricecontracts, the incentive compatibility constraints (A3 and A4) are binding.Clearly, these constraints are also valid for the above model if they are bind-ing. Hence, the optimal solution to the pooling problem is a feasible solutionto the wholesale menu contract problem (13), that is, ωw

l = ωwh = ωw∗

p and

θwl = θw

h = θw∗p is feasible solution to (13). Consequently, the optimal

manufacturer profit under wholesale pooling contract cannot exceed herprofits with wholesale menu contracts and therefore, the pooling contract isdominated. Therefore, from Proposition 2, we conclude that for the case ofwholesale price contracts, employing menu of contracts is the best strategyfor the manufacturer.

Fixed-fee Pooling Contract

In the case of fixed-fee contracts, pooling is done by offering the fixedfee, F

fp and quality θ

fp . Thus, we define B

fp = F

fp − ζθ

fp /2. The resulting

expected profit function for the manufacturer can be written as follows:

�p,fM = Max(

θfp ≥0,Bf

p

)Bfp

s.t .1

2Kl

(α + λθf

p − c)2 − Bf

p − ζθfp /2 ≥ U ,

1

2Kh

(α + λθf

p − c)2 − Bf

p − ζθfp /2 ≥ U .

It is easy to show that the second constraint must be binding at optimalitysince Kh < Kl . Hence, solving for θ

fp , we get θ

f ∗p = λKh(α−c)

ζ−λ2Kh, and thus,

�p,f ∗M = 1

2 Khζ(α−c)2

ζ−λ2Kh− U .

Next, we compare �p,f ∗M with the manufacturer’s profit for the cases of

fixed-fee individual and menu contracts. First suppose that r < Kh/Kl inwhich case the manufacturer’s expected profit under menu of contracts, �f −

M

is given in section 3.1.2. We note in section 3.1.3 that �f −M is increasing in

r and at r = 0, �f −M (r = 0) = 1

2 Khζ(α−c)2

ζ−λ2Kh− U = �

p,f ∗M , implying that

�f −M ≥ �

p,f ∗M for r in [0, Kh/Kl).

We know from Proposition 6 that the low-cost individual fixed-fee con-tract dominates the menu of contracts for r ≥ rl where 0 < rl < Kh/Kl .The manufacturer’s expected profit under this contract is given by �

f ,lM . At

r = rl , �f ,lM (r = rl) = �

f −M (r = rl) > �

p,f ∗M , and since �

f ,lM increases in

r , it implies that �f ,lM (r > rl) > �

p,f ∗M .

Consequently, we conclude that the pooling fixed-fee contract will alwaysbe dominated by either the fixed-fee menu of contracts or the low-costindividual contract.



Franchise Pooling Contract

The proof is analogous to the wholesale price contract case. In the caseof franchise-fee contracts, pooling is done by offering the fixed fee F F

p , the

wholesale price ωFp , and quality θF

p . The manufacturer’s optimization modelfor the pooling contract is

�p,FM = Max(

ωFp ,θF

p ≥0,F Fp

)E[K]((ωFp −c

)(α + λθF

p − ωFp

)) − ζ

2θFp

2 + F Fp

s.t .1

2Kl

(α + λθF

p2 − ωF

p

)2 − F Fp ≥ U ,

1

2Kh

(α + λθF

p2 − ωF

p

)2 − F Fp ≥ U .

It is easy to show that the second constraint must be binding at optimalitysince Kh < Kl . We know from Proposition 7 that under optimal fran-chise menu contracts, the incentive compatibility constraints (A3 and A4)are binding. Clearly, these constraints are also valid for the above modelif they are binding. Hence, the optimal solution to the pooling problemis a feasible solution to the wholesale menu contract problem, that is,(ωF

l , θFl , F F

l ) = (ωFh , θF

h , F Fh ) = (ωF

p , θFp , F F

p ) is feasible solution to (50).Consequently, the optimal manufacturer profit under franchise pooling con-tract cannot exceed her profits with franchise menu contracts and therefore,the pooling contract is dominated. Therefore, we conclude that for the caseof franchise contracts, employing menu of contracts is the best strategy forthe manufacturer.

ACKNOWLEDGMENTS

The authors would like to thank the Associate Editor andtwo anonymous referees for their constructive suggestionswhich have helped improve the quality and exposition of thepaper.

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Documents

Supply contracts in manufacturer-retailer interactions with manufacturer-quality and retailer effort-induced demand