Some implications of pricing bundles

Some Implications of Pricing Bundles

Xiao Huang,1 Mahesh Nagarajan,2 Greys Sošic3

1 Department of Decision Sciences and MIS, John Molson School of Business, Concordia University, Montreal,QC H3G 1M8, Canada

2 Operations and Logistics Division, Sauder School of Business, University of British Columbia, Vancouver, B.C.V6T 1Z2, Canada

3 Department of Information and Operations Management, Marshall School of Business, University of Southern California,Los Angeles, California 90089

Received 3 February 2011; revised 30 October 2012; accepted 1 February 2013DOI 10.1002/nav.21531

Published online 6 March 2013 in Wiley Online Library (wileyonlinelibrary.com).

Abstract: In this article, we consider a problem in which two suppliers can sell their respective products both individually andtogether as a bundle, and study the impact of bundle pricing. Four pricing models (centralized, decentralized, coop–comp, andcomp–coop) are analyzed with regard to the competition formats and sequences. As one would expect, the firms are always betteroff when pricing decisions are centralized. However, rather surprisingly, we find that firms may be worse off if the bundle pricesare set in a cooperative way; we provide analytical characterization of those instances. Numerical studies show that these insightsalso hold for some nonlinear demand. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 60: 237–250, 2013

Keywords: bundle, cooperation, competition, game theory, pricing

1. INTRODUCTION

Consider two products, 1 and 2, that are characterized bythree demand streams—the first two, being direct, are theindependent demands for the two products; the third demandis for the bundled product that involves both products 1 and 2.Examples of this type of setting are numerous; for instance,computer components can be purchased individually, or asa kit. Dell, a virtual assembler, sells such kits of products,many of whom can be directly purchased from its suppliers(for example, Sony monitors). Another classic example is airtickets—consider three cities, A, B, and C. Tickets can be pur-chased for itineraries A-B, B-C, and the bundle A-B-C, withthe first two possibly being operated by different airlines.

Literatures in economics, transportation, operations, andmarketing have studied the pricing and inventorying of suchproducts. In addition, questions on existence of equilibriaand welfare implications have been studied. For example, inthe operations literature, complementary products producedby competing firms have been analyzed in the context ofassembly systems [7, 12, 13]. We, on the other hand, in thisnote do a complete analysis of the effects of cooperation and

Correspondence to: G. Sošic ([email protected])

competition in a model with two products, say 1 and 2, thatcan be sold individually or as a bundle, and when demand islinear and deterministic.

There are several articles that are directly or tangentiallyrelated to this work. Netessine and Zhang [9] look at theeffects of competitively inventorying such products and werepossibly the first to introduce such a model to the litera-ture. Fang and Wang [6] consider a model with n partiallycomplementary products in which they assume that the man-ufacturers’ prices for satisfying the common demand exceedthe prices for filling their individual demands and show thatall the manufacturers first satisfy the common demand, andonly then their individual demand. Fang [5] extends the studyby allowing the price for common demand to be either higheror lower than the individual demand. Perakis et al. [10] lookat a “price of anarchy”-type analysis with complements andpoint out welfare insights. Armstrong and Vickers [2] con-sider a model in which both firms offer both products andcompete for customers in all markets (for individual productsand bundles). Our interest is to study the effects of bundlingalone, so we assume that each firm offers a single product.

There is a large literature on bundling and tying of prod-ucts. The findings from that body of work have implications in

© 2013 Wiley Periodicals, Inc.

238 Naval Research Logistics, Vol. 60 (2013)

various settings. Much of this literature ignore the existenceof individual products in the market, or the ramifications ofpartial cooperation on the pricing of the bundle; our analysisin this article does this. For a good survey of bundling andwelfare ramifications, please see http://www.justice.gov/atr/public/hearings/ip/chapter_5.htm#ii.

In the transportation economics literature, Bilotkach [3]and Czerny [4] are two examples. The latter looks at avery similar model in the context of airline networks. Mostrecently, Armstrong and Road [1] also study scenarios inwhich bundle discount could be set by an integrated firmor two individual firms. While [1] is more relevant to sym-metric firms and customer welfare, we in this article aim tocharacterize a full picture of the equilibria, and identify areasin which cooperation may harm firm-side profit. As far aswe know, the insights we get from our analysis seem to beunreported in the literature.

2. ASSUMPTIONS

Consider a population of customers with size M and twocomplementary products indexed 1 and 2. A customer maydemand product 1, product 2, both products, or no productsat all, with probability γ01, γ10, γ11, and γ00, respectively.The potential markets for product 1 and product 2 alone aredenoted as a1 = Mγ01 and a2 = Mγ10, and the potentialmarket for the bundle is b = Mγ11.

The parameters a1, a2, and b, therefore, partially reflectthe demand correlation between the two individual products;for example, large b with small a’s implies strong positivecorrelation, while large a’s and small b suggest negativecorrelation.

Suppose that product i is sold by firm i with individualdemand Di = ai − λipi , i = 1, 2; and that the bundle hasdemand DB = b−λBpB . To simplify the analysis, we assumethat λ1 = λ2 = λB = 1 and that the cost of each productis zero. These assumptions are made mainly for the ease ofpresentation and without much loss of generality. In fact,a nonzero cost can be readily incorporated into the modelthrough a slight revision of our demand functions. Similarargument applies to the λ’s.1

Without loss of generality, let us assume a1 ≤ a2, and letdenote p = p1 + p2. We will also use notation x ∧ y =min{x, y} and x ∨ y = max{x, y}.

1 The assumptions may affect certain threshold conditions, butwould not change the equilibrium structure or the key insights.Although discussions of the impact of cost or demand elasticityon the results could be further expanded, this somewhat devi-ates from the main focus of this short note, and thus will not beelaborated.

3. THE MODEL

We consider three separate cases: the centralized model, inwhich a single decision maker sets prices that maximize therevenue obtained from sale of all three products; the decen-tralized model, in which each firm, i, selects prices pi andpB

i (so that the bundle price is pB = pB1 + pB

2 ) that maxi-mize its individual profit; and the cooperative cases, in whichfirm i sets price pi that maximizes its profit from sellingproduct i, while the price of the bundle is determined coop-eratively, so that the profit from the sale of the bundle ismaximized.

In each of the above cases, we fully characterize the pricesset by the decision makers. The insights we get from analyz-ing the equilibrium outcomes seem to be interesting. The firstinsight, that may seem somewhat evident, is that the firms arebetter off when they cooperate when pricing both the bundleand the individual products. The second insight is surpris-ing and has important welfare implications. When the twofirms cooperate on the bundle pricing, depending on the val-ues of the parameters, there are a significant set of instanceswhere their overall profit can be lower than in the compet-itive case. We analytically characterize when this happens.The direct implication of this fact is that one has to be carefulwhen deciding policies on allowing firms to cooperate whenpricing bundles.

We now analyze each of the three cases in detail.

3.1. Centralized Model

We first assume that a single decision maker maximizesthe total profit, that is, sets prices p1, p2, and pB that,

maxp1,p2,pB

p1(a1 − p1)+ + p2(a2 − p2)

+ + pB(b − pB)+

s.t .p1 + p2 ≥ pB ≥ pi f or i = 1, 2

p1, p2, pB ≥ 0

where (·)+ = max{·, 0}.As it is usually the case in practice, we assume that the

price of the bundle does not exceed the total of its compo-nents (otherwise bundle customers would buy the two itemsseparately), or falls below any individual price (in which casecustomers demanding only product i could buy the bundleand drop product j ). The main result for this case is thefollowing.

PROPOSITION 1 (Centralized Solution): Suppose that acentralized decision maker maximizes the total profit fromselling individual products and the bundle. The optimal pricesare given as follows:

Naval Research Logistics DOI 10.1002/nav

Huang, Nagarajan, and Sošic: Some Implications of Pricing Bundles 239

Range of the bundle market b p1 p2 pB

[0, (

√2 − 1)a2

] a1

2

a2

2–

((√

2 − 1)a2, a2

]and

a1

2

a2 + b

4

a2 + b

4b ≥ 2a1 − a2

((√

2 − 1)a2, a2

]and

a1 + a2 + b

6

a1 + a2 + b

6

a1 + a2 + b

6b ≤ 2a1 − a2

(a2, a1 + a2]a1

2

a2

2

b

2

(a1 + a2, (

√3 + 1)a1 + a2

] b + 2a1 − a2

6

b + 2a2 − a1

6

2b + a1 + a2

6

((√

3 + 1)a1 + a2, ∞]

–a2

2

b

2.

Specifically, (i) if b ≤ a2, then pB = p2; (ii) if a2 <

b ≤ a1 + a2, then p2 < pB = b/2 < p1 + p2; and (iii) ifa1 + a2 < b, then pB = p1 + p2.

Note that, when a1 + a2 ≥ b, we have p ≥ pB ; that is,higher demand for individual products leads to their higherprice. On the other hand, a1 + a2 a1; when demand for an individual product is very low,it is priced out of the market.

3.2. Decentralized Model

In this section, we assume that firm i determines unilat-erally pi and pB

i that maximize her total profit, which iscomprised of the profit from selling individual product, i,and her share of the profit from selling the bundle,

pi(ai − pi)+ + pB

i (b − pB)+,

under constraints similar to the ones used in the previous case:pB

j ≥ pi − pBi ≥ pB

j − pj and pi , pBi ≥ 0.

Clearly, the profit obtained from the sale of the bundledepends on the price of the other firm as well, hence the pricethat firm i sets for its share of the bundle is obtained by usingthe Nash equilibrium (NE). The main result in this case isgiven by the next proposition.

PROPOSITION 2 (Decentralized Equilibrium): If eachfirm maximizes its own profit, the equilibrium prices are givenas follows:

Range of the bundle market b Equilibria (p1, p2, pB1 , pB

2 )

b ≤ 3

4a2,

• b < a1/2 pa : Bundle not offered

• 4a2 − 3a1

6≤ b <

5a1 − 2a2

4, pb: Full offerings; Deep

discount on bundle4 + 7

√2

20 + 14√

2a1 + 4

20 + 14√

2a2 ≤ b

• 5a1 − 2a2

4≤ b,

4 + 5√

2

2(6 + 5√

2)a2 ≤ b pc: Full offerings; Deep

discount on bundle3

4a2 ≤ b ≤ 3

4(a1 + a2), pd : Full offerings;

Moderate discount onbundle

3

4(a1 + a2) ≤ b ≤ 3

4(√

2 + 1)(a1 + a2),

• 2a1 + 6a2

7≤ b ≤ 6 + 5

√2

8(a1 + a2) pe: Full offerings; No

discount on bundle

• 3

4(a1 + (

√2 + 1)a2) < b pf : Only one product

and bundle offered in themarket

• 3

4((

√2 + 1)a1 + a2) < b pg : Only one product

and bundle offered in themarket

3

4(√

2 + 1)(a1 + a2) < b pf , pg : Only one productand bundle offered in themarketph: Only bundle offered inthe market

where

pa =(a1

2,a2

2, −, −

)

pb =(

a1 + a2 + 2b

7,a1 + a2 + 2b

7,

4a1 − 3a2 + b

7,

4a2 − 3a1 + b

7

)

pc =(

a1

2,a2 + 2b

5,

3b − a2

5,

2a2 − b

5

)

pd =(

a1

2,a2

2,b

3,b

3

)

pe =(

p,a1 + a2 + 2b

5− p,

4a1 − a2 + 3b

5− 2p,

2a2 − 3a1 − b

5+ 2p

),



wherea1

2∨ 4a1 − (6 + 5

√2)a2 + 8b

20≤ p

≤ (2 + √2)a1

4∧ 2a1 − 3a2 + 4b

10

pf =(

p, −,2a1 + b − 4p

3,b − a1 + 2p

3

), where

a1

2≤ p ≤ (2 + √

2)a1

4∧ 2a1 − 3(

√2 + 1)a2 + 4b

10.

pg =(

−,a2 + 2b − 3p

5,

3b − a2 − 2p

5,

2a2 − b + 4p

5

),

where(1 + √

2)a1

2∨ 8b − (6 + 5

√2)a2

12≤ p

≤ 4b − 3a2

6.

ph =(

−, −,b

3,b

3

)

The result suggests that bundle will not be offered in equi-librium when the market for it is small (i.e., b ≤ a1/2).Conditioned on the bundle being offered, the bundle pricepB can be as low as p2 when the market is relatively small(i.e., such that p2 or p3 is the NE), or as high as the sum ofthe individual product prices, p, when the market is moder-ate (i.e., such that p5 or p8 is the NE). However, contrary tothe results obtained in the centralized case, deeper discount(pB < p) will occur as the bundle market size increasesfurther. This, together with the “bundle-not-being-offered”scenario, reflects how competition can distort equilibriumprices from the first-best outcomes.

3.3. Cooperative Models

Finally, we assume that the two firms jointly determinepB which maximizes the profit from selling the bundle,pB(b − pB), and that each firm, i, selects the price of indi-vidual product, pi , that maximizes her profit, pi(ai − pi). Inour analysis, we assume that the firms split the profit fromselling the bundle equally among themselves. Note that thisallocation of profits can be justified by using several differ-ent approaches—it belongs to the core of the correspondingcooperative game, it corresponds to the Shapley value allo-cations, and to divisions obtained by using most reasonablenotions of cooperative bargaining.

We first assume that the firms set the price of the bun-dle first, and then determine their individual prices. Thus, themodel have two stages: the first is cooperative, and the secondis competitive. This coop–comp model leads to the followingresult.

PROPOSITION 3 (Coop-Comp Equilibrium): When thefirst stage of the model is cooperative, and the secondis competitive, the firms make the following decisions inequilibrium:

Range of the bundle market b p1 p2 pB

b ≤ a1b

2

b

2

b

2

a1 < b ≤ a2a1

2

b

2

b

2

a2 < b ≤ a1 + a2a1

2

a2

2

b

2.

a1 + a2 < b p ∈[

a1

2,b − a2

2

]b

2− p

b

2.

Now, suppose that the players first determine their individ-ual prices, and then jointly decide the bundle price. Thus, themodel again have two stages: the first is competitive, and thesecond is cooperative. This comp–coop model leads to thefollowing result.

PROPOSITION 4 (Comp-Coop Equilibrium): When thefirst stage of the model is competitive, and the second iscooperative, the firms make the following decisions.2

Range of the bundle market b (p1, p2, pB)

(√3

2+ 1

)b ≤ a2

(a1

2,a2

2, −

)

b ≤ a1 ≤ a2 ≤(√

3

2+ 1

)b

(2a2 + b

6,

2a2 + b

6,

2a2 + b

6

)

a1 ≤ b ≤ a2 <

(√3

2+ 1

)b

(a1

2,

2a2 + b

6,

2a2 + b

6

)

a1 ≤ a2 ≤ b < a1 + a2

(a1

2,a2

2,b

2

)

2 For areas with multiple equilibria, only the one with maximumtotal profit is shown.



Range of the bundle market b (p1, p2, pB)

a1 + a2 ≤ b

(3a1 − a2 + b

8,

3a2 − a1 + b

8,

≤ a1 + a2 + 4√3a1

a1 + a2 + b

4

)

a1 + (√

3 + 1)a2 ≤ b

(a1

2, −,

b

2

)

(√

3 + 1)a1 + a2 ≤ b

(−,

a2

2,b

2

)

For both the coop–comp and comp–coop models, there ispB = p2 ≥ p1 when b ≤ a2, and p2 < pB a2. We now further compare the two models in deter-mining the timing sequence that is most beneficial for thefirms. It leads to the following result.

COROLLARY 1: Suppose that the firms cooperativelydetermine the price of the bundle.

1. For small or large bundle market (b ≤ a2 or b ≥a1 + a2), the firms are jointly better off when theydetermine their individual prices before pricing thebundle.

2. For bundle market of moderate size (a2 ≤ b ≤a1 + a2), the firms generate the same profits undereither sequence of pricing.

The corollary suggests that, overall, comp–coop can be abetter mechanism than coop–comp. Further more, the tim-ing sequence matters only when there is an obvious differ-ence between the individual markets and the bundle market.Specifically, a firm should always attend to pricing in its mainmarket first.

4. COMPARISONS OF THE MODELS

In this section, we look at the models over the space ofparameter values, and compare their effects on the differentparties in the market. Should multiple equilibria exist in someregion/model, we pick the one that generates maximum totalprofit for comparison.

We first compare the decentralized model describedin Section 3.2 with the centralized model described inSection 3.1.

PROPOSITION 5 (Centralized vs. Decentralized):

• Both centralized and decentralized models charge thesame individual prices (pi and pj ) when the bundle

market is extremely small (i.e., b ≤ (√

2 − 1)a2), orlarge (i.e., b ≥ (

√3 + 1)a1 + a2).

• Otherwise, decentralized firms charge higher pricesthan the centralized model for both the bundle (pB)and for individual products (pi and pj ).

Thus, when demand for the bundle is in the extremes, thecompeting firms charge the same prices as the centralizeddecision maker. However, once the demand for the bundle ismore moderate, competition leads to higher prices.

Next, we compare the cooperative models describedin Section 3.3 with the centralized model described inSection 3.1.

PROPOSITION 6 (Centralized vs. Coop–Comp):

• Both centralized and coop–comp models charge thesame prices (individual and bundle) when the bundlemarket size is moderate (i.e., a2 ≤ b ≤ a1 + a2) orextremely large (e.g., b ≥ (

√3 + 1)a1 + a2).

• The coop–comp model charges weakly lower individ-ual prices than the centralized model when the bundlemarket is not too large (i.e., b ≤ a1 + a2).

• The coop–comp model charges lower bundle pricesthan the centralized model unless the bundle marketis moderately large (i.e., b ≥ a2).

PROPOSITION 7 (Centralized vs. Comp–Coop):

• Both centralized and comp–coop models charge thesame prices (individual and bundle) when the bundlemarket size is moderate (i.e., a2 ≤ b ≤ a1 + a2) orextremely large (e.g., b ≥ (

√3 + 1)a1 + a2).

• The comp–coop model charges weakly higher indi-vidual and bundle prices than the centralized modelwhen the bundle market is relatively small (i.e., b ≤a2), and weakly lower individual and bundle pricewhen the bundle market is large (i.e., b > a1 + a2).

The comparison yields quite contrary results than the onebetween the decentralized and centralized model. To beginwith, both cooperative models arrive at the same equilibriumas the centralized solution under moderate bundle marketsize. If this does not happen, the cooperative models veryoften achieve lower individual or even bundle prices (unlessthe bundle market is large) than the centralized solution.Thus, the distortion brought by the “partial” cooperation isthe opposite of that induced by the full competition.

Given the traditional wisdom that cooperation protectsits members from internal friction and improves the overallwelfare, a natural question to ask is, then, does this “partial”cooperation also leads the firms towards a better positionthan full competition? We investigate this next, and obtainour main result.



THEOREM 1 (Decentralized vs. Cooperative Models):The effects of the cooperation can be described as follows:

• The firms are better off only when the bundle marketis sufficiently large (e.g., b ≥ 3

4a2).• Cooperation, in any sequence, may hurt the profit

of the firms when the bundle market is small (e.g.,b < a1

2 ).

The theorem above shows that the cooperation on bundlepricing can be a double-edged sword. The effects of coop-eration depend greatly on the relative sizes of the demandsfor individual products and their relationship to the demandfor the bundle. Proper cooperation will help firms retain theright share in the bundle market, which justifies the impacton the pricing of their individual product. However, when thebundle market is less lucrative, either form of cooperation onbundle pricing will create redundant constraint on individualcomponent pricing, that drives down the overall profit. There-fore, decision about cooperation on bundle product pricing isnot straightforward.

4.1. Robustness Test

We acknowledge that Propositions 5–7, and Theorem 1were derived on the basis of linear demand functions. Nev-ertheless, throughout extensive numerical analysis, we findthat these insights sustain under certain nonlinear demandsas well. To highlight this fact, consider linear-power demandwhere Di = (ai −pi)

α and DB = (b−pB)α with α > 1. Thelinear-power demand has been used in a number of marketingand operations articles (e.g., [8, 11, 14]) concerning comple-mentary products, therefore quite relevant to the bundlingcontext. In addition, this demand form does not affect theassumption in Section 2 or the analysis with the four mod-els. The equilibria can then be derived in a similar fashion.Numerical studies show that the comparative results amongthe four models generally hold under linear-power demand.We illustrate this through a simple example below:

EXAMPLE 1: Consider two products with individual mar-kets a1 = 1 and a2 = 2, respectively. The power isα = 2. Then, the demand for product i in its individualmarket is Di = [max{0, 1 − pi}]2, and for the bundle isDB = [max{0, b − pB}]2. We allow the bundle market sizeb to be taken within [0, 16], with minimum step 0.1. Equi-librium solutions of the four models generate the followingobservations.

In comparing the centralized model with decentralizedmodel, (p1

c, p2c) = (p1

d , p2d) when b ≤ 0.8 or when

b > 8.5; otherwise, (p1c, p2

c, pBc) ≤ (p1

d , p2d , pBd

).Hence, Proposition 5 also holds.

In comparing the centralized model with coop–compmodel, we find that (p1

c, p2c, pBc

) = (p1coop−comp,

p2coop−comp, pB coop−comp

) when b ∈ [2, 3.2] and b > 5.

pBc> pB coop−comp

when b < 2 and pBc ≤ pB coop−comp

when b ≥ 3. When b ≤ 3.2, there is also p1c ≥

p1coop−comp and p2

c ≥ p2coop−comp. These are consistent with

Proposition 6.In comparing the centralized model and the comp–coop

model, when b ∈ [2, 3] and b ≥ 4.9, (p1c, p2

c, pBc) =

(p1comp−coop, p2

comp−coop, pB comp−coop). Overall, the individ-

ual and bundle prices in comp–coop are weakly higher whenb ≤ 2 and weakly lower when b ≥ 3. These observationsconform with Proposition 7.

Finally, compare the profits under centralized and the twocooperative models. We find that πd ≥ π comp−coop whenb ≤ 1 (in particular, “<” holds for b ∈ [0.8, 1)), andπd > π coop−comp when b < 1.5. When b ≥ 1.5, bothcooperative models yield higher total profit than the decen-tralized model. This reinforces our findings in Theorem 1 thatcooperation might hurt the overall profit of the firms. �

5. CONCLUSIONS

Our analysis demonstrates that one cannot reach obvi-ous conclusions on whether cooperation is beneficial whenbundling products as compared to competition. The analysisdemonstrates that this entirely depends on the relative sizesof the market and their interaction in terms of the parametervalues. The firms may be worse off as a result of cooperation;the situations in which this happens are characterized by thebundle market size being below a certain threshold. More-over, the timing sequence of making a cooperative decisionis also important. We believe these findings can be usefulfor researchers in operations, information economics, andtransportation sciences. From our analysis, we believe thatthe nature of our results will continue to hold in more com-plicated settings with a network of products with possiblebundles of nodes having their own demands.

APPENDIX

The following Lemma will be used in the proofs of propositions.

LEMMA 1: For given x, y, λ > 0 and y ≥ q ≥ 0, the solution to

maxp

π = maxp

p(x − p)+ + λ(p + q)(y − q − p)+

s.t.p ≥ 0

is

• when 0 < x ≤ y − q,1. if x − q ≤ y − 2q < (

√(1 + λ)/λ+ 1)x, then p∗ = x+λ(y−2q)

2(1+λ)

and π∗ = [x+λ(y−2q)]24(1+λ)

+ λq(y − q);

2. if (√

(1 + λ)/λ + 1)x ≤ y − 2q, then p∗ = y−2q2 and

π∗ = λ(y−2q)2

4 + λq(y − q).• when 0 ≤ y − q ≤ x,

1. if y − q ≤ x < (√

1 + λ + 1)y − 2q, then p∗ = x+λ(y−2q)2(1+λ)

and

π∗ = [x+λ(y−2q)]24(1+λ)

+ λq(y − q);

2. if (√

1 + λ + 1)y − 2q ≤ x, then p∗ = x2 and π∗ = x2

4 .



PROOF: For 0 < x ≤ y − q, consider the following two options:

1. Suppose p ≤ x ≤ y − q, then π = −(1 + λ)p2 + [x +λ(y−2q)]p+λq(y−q). The first-order condition (FOC) solutionp∗ = x+λ(y−2q)

2(1+λ)sustains (p∗ ≤ x) iff x ≤ y − 2q ≤ ( 1

λ+ 2)x.

Otherwise, p∗ = x and π∗ = x(y − x).2. Suppose x 2x. Otherwise, p∗ = x and π∗ = x(y − x).

We now compare the above two options. Obviously, when x ≤ y − q < 2x

there should be p∗ = x+λ(y−2q)2(1+λ)

, and when y − q > ( 1λ

+ 2)x there should

be p∗ = y/2. When 2x ≤ y − q ≤ ( 1λ

+ 2)x, comparing π(x+λ(y−2q)

2(1+λ)) =

[x+λ(y−2q)]24(1+λ)

+ λq(y − q) vs. π(y/2) = λ(y − 2q)2/4 + λq(y − q) one can

verify that the former is greater when (

√1+λλ

+ 1)x > y − 2q.The case with 0 ≤ y − q ≤ x can be proved in a similar fashion. �

PROOF OF PROPOSITION 1: The centralized problem is formulated as

(CP) : maxp1,p2,pB

�c(p1, p2, pB)

= maxp1,p2,pB

p1(a1 − p1)+ + p2(a2 − p2)

+ + pB(b − pB)+

s.t. pB ≤ p1 + p2

pB ≥ p1

pB ≥ p2

Given (p1, p2), the decision for pB should follow

pB =

⎧⎪⎪⎨⎪⎪⎩

1.b/2, if p1 + p2 ≥ b/2 , p1 < b/2 and p2 < b/2;2.p1, if p1 ≥ b/2 and p1 ≥ p2;3.p2, if p2 ≥ b/2 and p1 < p2;4.p1 + p2, if p1 + p2 < b/2.

We assume that a1 ≤ a2. Then, we have the following analysis.

1. Suppose pB = b/2. Then (CP) is equivalent to the globalmaximum of the following three sub-problems:a. maxp1,p2 p1(a1 −p1) s.t. p1 +p2 ≥ b/2, p1 ≤ b/2, p2 ≤ b/2,

and p2 ≥ a2. The stationary point is p∗1 = a1/2 if b ≥ 2a2;

otherwise, the feasible solution set is empty.b. maxp1,p2 p2(a2 − p2) s.t. p1 + p2 ≥ b/2, p1 ≤ b/2, p2 ≤

b/2, and p1 ≥ a1. The stationary point is p∗2 = a2/2 if

b ≥ max{a2, 2a1}; p∗2 = b/2 if 2a1 ≤ b ≤ a2; otherwise,

the feasible solution set is empty.c. maxp1,p2 p1(a1 − p1) + p2(a2 − p2) s.t. p1 + p2 ≥

b/2, p1 ≤ b/2, and p2 ≤ b/2. The stationary points are(p∗

1 , p∗2) = (a1/2, a2/2) if a2 ≤ b ≤ a1 + a2; (p∗

1 , p∗2) =

(b+a1−a2

4 , b+a2−a14 ) if b ≤ 3a1 + a2; (p∗

1 , p∗2) = (

a12 , b

2 ) ifa1 ≤ b ≤ a2; and (p∗

1 , p∗2) = ( b

2 , b2 ) if b ≤ a1 .

Comparing the objective values of each problem above, we candetermine that

Scenario: p1 p2 pB �c

b ≤ a1 b/2 b/2 b/2 a1b/2 + a2b/2 − b2/4

a1 ≤ b ≤ a2 a1/2 b/2 b/2 a21/4 + a2b/2

a2 ≤ b ≤ a1 + a2 a1/2 a2/2 b/2 a21/4 + a2

2/4 + b2/4

a1 + a2 ≤ b b/2 a2/2 b/2 a22/4 + b2/4.

2. Suppose now that pB = p1, and solve maxp1,p2 p1(a1 − p1)+ +

p1(b −p1)+ +p2(a2 −p2) s.t. p1 ≥ b/2, p1 ≥ p2. First consider

a1 ≤ b ≤ (√

2 + 1)a1. Note that (a1 + b)/4 ≤ b/2. By Lemma 1,the optimal solution is (p∗

1 , p∗2) = (b/2, a2/2) when b ≥ a2 ≥ a1

and (p∗1 , p∗

2) = (b/2, b/2) when a1 ≤ b ≤ a2. If b > (√

2 + 1)a1,then the optimal solution is (p∗

1 , p∗2) = (b/2, a2/2) when b ≥ a2

and (p∗1 , p∗

2) = (b/2, b/2) when (√

2 + 1)a1 < b ≤ a2. Nowconsider b < a1 ≤ a2. If b < a1 ≤ (

√2 + 1)b, then the optimal

solution is (p∗1 , p∗

2) = ((b+a1)/4, a2/2) when b+a1 ≥ 2a2 (doesnot hold) and (p∗

1 , p∗2) = ((a1 +a2 +b)/6, (a1 +a2 +b)/6) when

b + a1 < 2a2. If (√

2 + 1)b < a1, then the optimal solution is(p∗

1 , p∗2) = ((a1 + a2 + b)/6, (a1 + a2 + b)/6).

Scenario p1 p2 pB �c

a2 ≤ b b/2 a2/2 b/2 a1b/2 + a22/4

a1 ≤ b < a2 b/2 b/2 b/2 a1b/2 + a2b/2 − b2/4

b ≤ a1 ≤ a2 (a1 + a2 + b)/6 (a1 + a2 + b)/6 (a1 + a2 + b)/6 (a1 + a2 + b)2/12

3. Similarly, for pB = p2, solve maxp1,p2 p2(a2 − p2)+ + p2(b −

p2)+ + p1(a1 − p1) s.t. p2 ≥ b/2, p2 ≥ p1. First consider

a2 ≤ b ≤ (√

2 + 1)a2. Note that (a2 + b)/4 ≤ b/2. By Lemma 1,the optimal solution is (p∗

1 , p∗2) = (a1/2, b/2). If b > (

√2+1)a2,

then the optimum is still (p∗1 , p∗

2) = (a1/2, b/2). Now consider

b < a2. If b < a2 ≤ (√

2 + 1)b, then the optimal solu-tion is (p∗

1 , p∗2) = (a1/2, (b + a2)/4) when b ≥ 2a1 − a2 and

(p∗1 , p∗

2) = ((a1 +a2 +b)/6, (a1 +a2 +b)/6) when b < 2a1 −a2.If (

√2 + 1)b < a2, then the stationary point is (p∗

1 , p∗2) =

(a2/2, a1/2).


a2 ≤ b a1/2 b/2 b/2 a2b/2 + a21/4

b < a2 < (√

2 + 1)b a1/2 (a2 + b)/4 (a2 + b)/4 (a2 + b)2/8 + a21/4

and b ≥ 2a1 − a2

b < a2 < (√

2 + 1)b (a1 + a2 + b)/6 (a1 + a2 + b)/6 (a1 + a2 + b)/6 (a1 + a2 + b)2/12and b < 2a1 − a2

(√

2 + 1)b ≤ a2 a1/2 a2/2 a2/2 a21/4 + a2

2/4



4. Finally, for pB = p1 + p2, we solve:a. maxp1,p2 (p1 +p2)(b−p1 −p2) s.t. p1 +p2 ≤ b/2, p1 ≥

a1, p2 ≥ a2. The stationary point is p∗1 ∈ [a1, b/2 − a2]

and p∗2 = b/2 − p∗

1 if a1 + a2 ≤ b/2. Otherwise, thefeasible solution set is empty.

b. maxp1,p2 p2(a2 − p2) + (p1 + p2)(b − p1 − p2) s.t.p1 + p2 ≤ b/2, p1 ≥ a1. The stationary point is(p∗

1 , p∗2) = (

b−a22 , a2

2 ) if b ≥ 2a1 + a2; (p∗1 , p∗

2) =

(a1, b/2 − a1) if 2a1 ≤ b ≤ 2a1 + a2; otherwise, thefeasible set is empty.

c. maxp1,p2 p1(a1 − p1) + p2(a2 − p2) + (p1 + p2)(b −p1 − p2) s.t. p1 + p2 ≤ b/2, p1 ≥ 0, p2 ≥ 0. Thestationary points are (p∗

1 , p∗2) = (

2a1−a2+b6 , 2a2−a1+b

6 )

if b ≥ a1 + a2; (p∗1 , p∗

2) = (a1−a2+b

4 , a2−a1+b4 ) if

3a1 +a2 ≥ b ≥ a2 −a1; (p∗1 , p∗

2) = (0, a2+b4 ) if b ≥ a2;

and (p∗1 , p∗

2) = (0, b2 ) if b ≤ a2.

Consequently, we have


b ≤ a2 − a1 0 b/2 b/2 a2b/2

a2 − a1 ≤ b ≤ a1 + a2a1 − a2 + b

4

a2 − a1 + b

4b/2

a21 + a2

2 + b2 − 2a1a2 + 2a1b + 2a2b

8

a1 + a2 ≤ b ≤ 2a1 − a2 + b

6

2a2 − a1 + b

6

a1 + a2 + 2b

6

a21 + a2

2 + b2 + a1b + a2b − a1a2

6≤ (

√3 + 1)a1 + a2

(√

3 + 1)a1 + a2 ≤ bb − a2

2

a2

2

b

2

a22 + b2

4.

Now, comparing the four tables that we have so far yields:

Scenario Optimal bundle pricing decision

b ≤ a2 pB = p2

a2 < b ≤ a1 + a2 pB = b/2

a1 + a2 0 and w ≥ u, the solution to

maxp,q

p(x − p)+ + q(y − q)+

s.t. u ≤ p − q ≤ w

p ≥ 0

q ≥ 0

is:

(i) if 2u ≤ x − y ≤ 2w, then (p∗, q∗) = ( x2 , y

2 );

(ii) if x − (√

2 + 1)y ≤ 2w < x − y, then (p∗, q∗) =( x+y4 + w

2 , x+y4 − w

2

);

(iii) if 2w < x − (√

2 + 1)y < x − y, then (p∗, q∗) =(x2 , x

2 − w ∼ x2 − u

);

(iv) if (√

2 + 1)x − y ≥ 2u > x − y, then (p∗, q∗) =( x+y4 + u

2 , x+y4 − u

2

);

(v) if x − y ≤ (√

2 + 1)x − y < 2u, then (p∗, q∗) =( y2 + u ∼ y

2 + w, y2

).

PROOF: On R2+, define A = {(p, q) : u ≤ p − q ≤ w}, B = {(p, q) :

p − q > w}, and C = {(p, q) : p − q < u}. Clearly, So = (x/2, y/2) is theoptimal solution if it is in A. Otherwise,• if So ∈ B and (x/2−y/2

√2, y/2+y/2

√2) /∈ B, then the optimal solution

is( x+y

4 + w2 , x+y

4 − w2

);

• if So ∈ B and (x/2−y/2√

2, y/2+y/2√

2) ∈ B, then the optimal solutionis

(x2 , x

2 − w ∼ x2 − u

);

• if So ∈ C and (x/2 + x/2√

2, y/2 − x/2√

2) /∈ C, then the optimal solu-tion is

( x+y4 + u

2 , x+y4 − u

2

);

• if So ∈ C and (x/2+x/2√

2, y/2−x/2√

2) ∈ C, then the optimal solutionis

( y2 + u ∼ y

2 + w, y2

); �

PROOF OF PROPOSITION 2: Firm i needs to determine pi and pBi

given pj and pBj :

maxpi ,pB

i

pi(ai − pi)+ + pB

i (b − pBj − pB

i )+

s.t. pi + pj ≥ pBi + pB

j

pBi + pB

j ≥ pi

pi ≥ 0

pBi ≥ 0

If some firm announces a pBj ≥ b, it indicates that this firm does not

intend to supply the bundle market, thus the only equilibrium shall be(p∗

1 , p∗2) = (a1/2, a2/2) and pB∗ ≥ b. the optimal response is p∗

i =ai/2∧pB

j ∨ (pBj −pj )

+ and pBi ∗ = 0. Other than this trivial case, consider

pBj < b. Then, by Lemma 2, the optimal reaction function is



1. Distinct full line—both individual item i and the bundle B willbe offered. The bundle price will be distinct from pi the price ofproduct i, or the total of product i and j , pi + pj :

(p∗i , pB∗

i ) =(

ai2 ,

b−pBj

2

)if 2pB

j − 2pj ≤ ai − b + pBj ≤ 2pB

j

and pBj < b;

2. Marginal full line—both individual item i and the bundle B willbe offered. The bundle price is the same as the firm’s individualitem, i.e., pi = pB :

(p∗i , pB∗

i ) =(

ai+b+pBj

4 ,ai+b−3pB

j

4

)if ai − (

√2 + 1)(b − pB

j ) ≤2pB

j < ai − b + pBj and pB

j < b;3. Bundle free—bundle product will be priced out of the market,

pB ≥ b:

(p∗i , pB∗

i ) =(

ai2 , ai

2 − pBj ∼ ai

2 − pBj + pj

)if 2pB

j < ai −(√

2 + 1)(b − pBj ) ≤ ai − b + pB

j and pBj < b;

4. Discount-free full line—both individual item i and the bundle B

will be offered. Bundle discount does not exist , that is, pB =pi + pj :

(p∗i , pB∗

i ) =(

ai+b+pBj

−2pj

4 ,ai+b−3pB

j+2pj

4

), if (

√2+1)ai −b+

pBj ≥ 2pB

j − 2pj ≥ ai − b + pBj and pB

j ≤ b − ai ;5. Product-i free—product i will be priced out of the market, that is,

pi ≥ ai :

(p∗i , pB∗

i ) =(

b+pBj

−2pj

2 ∼ b+pBj

2 ,b−pB

j

2

)if (

√2 + 1)ai − b +

pBj < 2pB

j − 2pj and pBj ≤ b − ai .

We next analyze possible equilibria. Denote pmn as equilibrium pricing whenfirm 1 uses strategy m (indexed as above) and firm 2 is with strategy n, respec-tively, where m, n ∈ {1, 2, 3, 4, 5}. Note that not all equilibria is feasible. Takep13, for example, in equilibrium bundle cannot be offered and priced out ofmarket at the same time. The feasible equilibria are listed below (Italic termin a price vector indicates dummy pricing, i.e., the particular item will bepriced out of market, and the price itself is just for technical purpose.).

p11 =(

a1

2,a2

2,b

3,b

3

).

Feasible when3

4a2 ≤ b ≤ 3

4(a1 + a2).

p22 =(

a1 + a2 + 2b

7,a1 + a2 + 2b

7,

4a1 − 3a2 + b

7,

4a2 − 3a1 + b

7

),

Feasible when4a2 − 3a1

6

≤ b <5a1 − 2a2

4and

4a1 + (4 + 7√

2)a2

20 + 14√

2≤ b.

p33 =( a1

2,a2

2, pB

1 ,a2

2− pB

1

), where (

√2 + 1)2b

− (√

2 + 1)a1 < pB1 < a1 − b.

Feasible when b <a1

2.

p44 =(p1,

a1 + a2 + 2b

5− p1,

4a1 − a2 + 3b

5

− 2p1,2a2 − 3a1 − b

5+ 2p1

), where

a1

2≤ p1 ≤ (2 + √

2)a1

4

and4a1 − (6 + 5

√2)a2 + 8b

20≤ p1 ≤ 2a1 − 3a2 + 4b

10.

Feasible when3

4(a1 + a2) ≤ 2a1 + 7a2

6≤ b ≤ 6 + 5

√2

8(a1 + a2).

p55 =(

p1,2

3b − p1,

b

3,b

3

), where

(√

2 + 1)

2a1

< p1 <4b − 3(

√2 + 1)a2

6.

Feasible when b >3

4(√

2 + 1)(a1 + a2).

p12 =(

a1

2,a2 + 2b

5,

3b − a2

5,

2a2 − b

5

).

Feasible when 5a1 − 2a2 ≤ 4b and4 + 5

√2

2(6 + 5√

2)a2 ≤ b ≤ 3

4a2.

p15 =(

a1

2, − a1

2+ 2

3b,

b

3,b

3

).Feasible when

3

4(a1 + (1 + √

2)a2) < b.

p51 =(

− a2

2+ 2

3b,

a2

2,b

3,b

3

).Feasible when

3

4((1 + √

2)a1 + a2) < b.

p45 =(

p1,a1 + 2b − 5p1

3,

2a1 + b − 4p1

3,b − a1 + 2p1

3

),

wherea1

2≤ p1 ≤ (2 + √

2)a1

4

and p1 <2a1 − 3(

√2 + 1)a2 + 4b

10.

Feasible when3

4(a1 + (1 + √

2)a2) < b.

p54 =(

p1,a2 + 2b − 3p1

5,

3b − a2 − 2p1

5,

2a2 − b + 4p1

5

),

where p1 ≥ (1 + √2)a1

2, p1 ≥ 8b − (6 + 5

√2)a2

12,

and p1 ≤ 4b − 3a2

6.

Feasible when3

4((1 + √

2)a1 + a2) < b.

Note that p15 and p51 are degenerated cases for p45 and p54, respectively.The equilibria are summarized in the following table:

Range of the bundle market b (p1, p2, pB1 , pB

2 )

b ≤ 3

4a2,

b < a1/2 pa

4a2 − 3a1

6≤ b <

5a1 − 2a2

4,

4a1 + (4 + 7√

2)a2

20 + 14√

2≤ b pb

5a1 − 2a2

4≤ b,

4 + 5√

2

2(6 + 5√

2)a2 ≤ b pc

3

4a2 ≤ b ≤ 3

4(a1 + a2), pd

3

4(a1 + a2) ≤ b ≤ 3

4(√

2 + 1)(a1 + a2),

2a1 + 7a2

6≤ b ≤ 6 + 5

√2

8(a1 + a2) pe

3

4(a1 + (

√2 + 1)a2) < b pf



Range of the bundle market b (p1, p2, pB1 , pB

2 )

3

4((

√2 + 1)a1 + a2) < b pg

3

4(√

2 + 1)(a1 + a2) < b pf

pg

ph

where

pa =( a1

2,a2

2, −, −

)pb =

(a1 + a2 + 2b

7,a1 + a2 + 2b

7,

4a1 − 3a2 + b

7,

4a2 − 3a1 + b

7

)

pc =(

a1

2,a2 + 2b

5,

3b − a2

5,

2a2 − b

5

)

pd =(

a1

2,a2

2,b

3,b

3

)

pe =(

p,a1 + a2 + 2b

5− p,

4a1 − a2 + 3b

5− 2p,

2a2 − 3a1 − b

5+ 2p

),

wherea1

2∨ 4a1 − (6 + 5

√2)a2 + 8b

20≤

p ≤ (2 + √2)a1

4∧ 2a1 − 3a2 + 4b

10

pf =(

p, −,2a1 + b − 4p

3,b − a1 + 2p

3

),

wherea1

2≤ p ≤ (2 + √

2)a1

4∧ 2a1 − 3(

√2 + 1)a2 + 4b

10.

pg =(

−,a2 + 2b − 3p

5,

3b − a2 − 2p

5,

2a2 − b + 4p

5

),

where(1 + √

2)a1

2∨ 8b − (6 + 5

√2)a2

12≤ p ≤ 4b − 3a2

6.

ph =(

−, −,b

3,b

3

).

�

PROOF OF PROPOSITION 3: Clearly, the bundle price that maximizesthe profit is pB = b/2. The decision pi for firm i, given the price of theother firm pj , is therefore:

maxpi

pi(ai − pi)+

s.t. pi ≥ b/2 − pj

pi ≤ b/2

pi ≥ 0.

The optimal response function is

Scenario p∗i

pj ≤ b/2,b ≤ ai b/2b − 2pj < ai b/2b ≤ ai b/2ai < b ai/2

and the equilibrium is, therefore,

(p1, p2) b < a1 a1 ≤ b

b < a2 (b/2, b/2) (a1/2, b/2)

a2 ≤ b < a1 + a2 – (a1/2, a2/2)

a1 + a2 ≤ b – (p, b/2 − p) where p ∈ [ a1

2,b − a2

2]

�

PROOF OF PROPOSITION 4: We start with the second stage: given thecompetitive prices p1 and p2, the cooperative decision pB in the secondstage is determined by:

maxpB

pB(b − pB)

s.t. pB ≤ p1 + p2

pB ≥ p1 ∨ p2

pB ≥ 0.

The optimal decision should follow:

pB∗ =

⎧⎪⎪⎨⎪⎪⎩

b/2, if p1 + p2 ≥ b/2 , p1 < b/2 and p2 < b/2;p1, if p1 ≥ b/2 and p1 ≥ p2;p2, if p2 ≥ b/2 and p1 < p2;p1 + p2, if p1 + p2 < b/2.

Now, we address competitive decision made in the first stage. If firm i

receives a half of the bundle profit, her price pi is determined by:

(Pi) : maxpi

�i = maxpi

pi(ai − pi)+ + 0.5pB(b − pB)+

s.t. pB = pB∗

pi ≥ 0.

• If b ≤ pj , obviously p∗i = ai/2.

• If pj < b/2, first consider b/2 − pj ≤ pi < b/2 such thatpB = b/2. Then, p∗

i = (ai/2 ∧ b/2) ∨ (b/2 − pj ). Alterna-tively, for pi ≥ b/2 such that pB = pi , by Lemma 1 the local

optimum is p∗i = ai+b/2

3 if b < ai ≤ (

√32 + 1)b, b/2 if ai ≤ b,

and ai2 if (

√32 + 1)b ≤ ai . Finally, for pi < b/2 − pj there

will be pB = pi + pj . Again by Lemma 1, the local optimum

is p∗i = ai+(b−2pj )/2

3 ifb−2pj√

3+1< ai ≤ b − 2pj , b/2 − pj if

ai > b − 2pj orb−2pj√

3+1≤ ai . Comparing total profits across the

three regions derives the the optimal solution.• If b/2 ≤ pj < b, consider pi ≥ pj such that pB = pi . Simi-

larly by Lemma 1 for ai ≤ b, the optimal solution is pi = 2ai+b6

if ai ≤ b ≤ (√

3 + 1)ai and ai/2 if (√

3 + 1)ai ≤ b. In either

case pi ≤ b/2 ≤ pj so p∗i = pj . For ai ≥ (

√32 + 1)b,

p∗i = ai/2 ∧ pj = ai/2. For b ≤ ai ≤ (

√32 + 1), p∗

i = 2ai+b6

when 3pj − b/2 ≤ ai , p∗i = pj when 2b ≤ ai ≤ (3pj − b/2) ∧

[2pj +√2pj (b − pj )] or when ai ≤ (3pj −b/2)∧2b, p∗

i = ai/2when 2b ∨ [2pj + √

2pj (b − pj )] ≤ ai ≤ 3pj − b/2. Comparethe total profits gives the following table:



Scenario p∗i pB �i

pj ≤ b/2,(√3

2+ 1

)b < ai ai/2 ai/2 a2

i /4

b ≤ ai ≤(√

3

2+ 1

)b (2ai + b)/6 (2ai + b)/6 (2ai + b)2/24

b − 2pj ≤ ai ≤ b ai/2 b/2 (2a2i + b2)/8

b − 2pj√3 + 1

< ai ≤ b − 2pj (ai + b/2 − pj )/3 (ai + b/2 + 2pj )/3 (ai + b/2 − pj )2/6 + pj (b − pj )/2

0 ≤ ai ≤ b − 2pj√3 + 1

b/2 − pj b/2 b2/8

b/2 ≤ pj ≤ b(√3

2+ 1

)b < ai ai/2 ai/2 a2

i /4

2pj ≤ ai ≤(√

3

2+ 1

)b,

3pj − b/2 ≤ ai (2ai + b)/6 (2ai + b)/6 (2ai + b)2/24pj ∨ 2b ≤ ai ≤ 3pj − b/2 ai/2 ai/2 a2

i /42b ≤ ai ≤ (3pj − b/2) ∧ pj pj pj pj (ai − pj ) + pj (b − pj )/2ai ≤ (3pj − b/2) ∧ 2b pj pj pj (ai − pj ) + pj (b − pj )/2

ai < 2pj ai/2 pj a2i /4 + pj (b − pj )/2

b ≤ pj ai/2 pj a2i /4

where pj = 2pj + √2pj (b − pj ).

It can be verified that the equilibrium is

(p1, p2, pB)

(√3

2+ 1

)b < a1 b ≤ a1 ≤ (

√3

2+ 1)b a1 ≤ b

(√3

2+ 1

)b < a2

(a1

2,a2

2, −

) ( a1

2,a2

2, −

) (a1

2,a2

2, −

)

b < a2 ≤(√

3

2+ 1

)b – (p, p, p)

(a1

2,

2a2 + b

6,

2a2 + b

6

)

where p ∈[

2a2 + b

6,

2a2 + b

6

+√

b2+4a2b−2a22

6

]

a2 ≤ b – –

(a1

2,a2

2,b

2

)when b ≤ a1 + a2;(

a1

2,

2a2 − a1 + b

6,

2a1 + 2a2 + b

6

)when a1 + a2 ≤ b ≤ a1 + (

√3 + 1)a2;(

2a1 − a2 + b

6,a2

2,

2a1 + 2a2 + b

6

)when a1 + a2 ≤ b ≤ (

√3 + 1)a1 + a2;(

3a1 − a2 + b

8,

3a2 − a1 + b

8,a1 + a2 + b

4

)

when a1 + a2 ≤ b ≤ a1 + a2 + 4√3a1;

(a1

2, −,

b

2

)

when a1 + (√

3 + 1)a2 ≤ b;

(−,

a2

2,b

2

)

when (√

3 + 1)a1 + a2 ≤ b;

(−, −,

b

2

).

when (1 + √3)(a1 + a2) ≤ b;



It can be verified that for areas with multiple equilibria, ( 3a1−a2+b8 , 3a2−a1+b

8 ,a1+a2+b

4 ) yields higher total profit than (a12 , 2a2−a1+b

6 , 2a1+2a2+b6 ) or

(2a1−a2+b

6 , a22 , 2a1+2a2+b

6 ), a1 + (√

3 + 1)a2 ≤ b (b−a2

2 , a22 , b

2 ) �

PROOF OF COROLLARY 1: When a2 ≥ a1 > b, the total profit in thecooperative–competitive model equals �coop−comp = b(2a1 + 2a2 − b)/4and �comp−coop = (a2

1 + a22)/4 when (

√3/2 + 1)b ≤ a2 and �comp−coop =

(2a1 + b)(2a2 + b)/12 when b ≤ a1 ≤ a2 ≤ (√

3/2 + 1)b. In either case,�comp−coop ≥ �coop−comp.

When a1 ≤ b ≤ a2, �comp−coop = (a21 + a2b)/4 and �comp−coop =

(a21 + a2

2)/4 when (√

3/2 + 1)b ≤ a2 and �comp−coop = (9a21 + 4a2

2 +10a2b + 4b2)/36 when a1 ≤ b ≤ a2 ≤ (

√3/2 + 1)b. In either case,

�comp−coop ≥ �coop−comp.When a2 ≤ b ≤ a1 +a2, �comp−coop = �coop−comp = (a2

1 +a22 +b2)/4.

When a1 + a2 < b, maxp �comp−coop = (a21 + a2

2 − 2a1b − 2a1a2)/4when b < a2+3a1 and maxp�comp−coop = (a2

2 +b2)/4 when b ≥ a2+3a1.�coop−comp = (5a2

1 + 5a22 + 5b2 − 6a1a2 + 6a1b + 6a2b)/32 when

b ≤ a2 + (1 + 4/√

3)a1 and max �coop−comp = (a22 + b2)/4. Overall

�comp−coop ≥ �coop−comp. �

PROOF OF PROPOSITION 5: The conclusion can be drawn by simplycomparing the pricing decisions in two tables from Propositions 1 and 2. �

PROOF OF PROPOSITION 6:

• If a1 ≤ (√

2 − 1)a2, then the relationship between (pc1, pc

2, pBc)

and(p

comp−coop1 , pcomp−coop

2 , pBcomp−coop) as b increases from (0, a1],(a1, (

√2 − 1)a2],

((√

2 − 1)a2, a2], (a2, a1 + a2] is (>, >, >), (=, >, >), (=, >, >),and (=, =, =), respectively.

• If (√

2 − 1)a2 < a1 ≤ a2, the relationship between (pc1, pc

2, pBc)

and(p


2 , pBcomp−coop) asb increases from (0, (√

2−1)a2],((

√2 − 1)a2, a1], (a1, a2] (a2, a1 + a2] is (>, >, >), (>, >, >),

(≥, >, >), and (=, =, =), respectively.

When b > a1 + a2, comparative result can go two way due to multipleequilibria in the coop–comp model. If only the equilibrium with maximumtotal profit is considered, i.e., (p

coop−comp1 , pcoop−comp

2 ) = (b−a2

2 , a22 ), there

is still pc1 ≥≤ p

coop−comp1 and pc

2 ≥ pcoop−comp2 . �

PROOF OF PROPOSITION 7:

• If a1 ≤ (√

6 − 2)a2, the relationship between (pc1, pc

2, pBc) and

(pcomp−coop1 , pcomp−coop


2−1)a2], ((

√2 − 1)a2, (

√6 − 2)a2], ((

√6 − 2)a2, a2], (a2, a1 + a2],

(a1+a2, (√

3+1)a1+a2], ((√

3+1)a1+a2, ∞] is (=, =, =), (=, <, <), (=, <, <), (=, =, =), (>, >, >) and (=, =, =), respectively.

• If (√

6 − 2)a2 ≤ a1 ≤ a2, the relationship between (pc1, pc

2, pBc)

and(p



2−1)a2], ((

√2−1)a2, (

√6−2)a2], ((

√6−2)a2, a1] (a1, a2], (a2, a1+

a2], (a1 +a2, (√

3+1)a1 +a2], ((√

3+1)a1 +a2, ∞] is (=, =, =),(=, <, <), (<, <, <), (≥, <, <), (=, =, =), (>, >, >), and (=, =, =), respectively.

�

PROOF OF THEOREM 1: For the decentralized model, firm’s profit withrespect to each possible equilibrium price is as follows:

Equilibria πd

pa

a21 + a2

2

4

pb

4a21 + 4a2

2 + 2b2 + 9a1b + 9a2b + 8a1a2

49

pc

a21

4+ 3a2

2 + 2b2 + 7a2b

25

pd

a21 + a2

2

4+ 2b2

9

pe

6a21 + 6a2

2 + 4b2 + 14a1b + 14a2b − 13a1a2

50

pf

a21

4+ 2b2

9

pg

a22

4+ 2b2

9

ph

2

9b2

For the coop–comp model,

π coop−comp

b ≤ a1a1b + a2b

2− b2

4

a1 < b ≤ a2a2

1

4+ a2b

2

a2 < b ≤ a1 + a2a2

1 + a22 + b2

4

a1 + a2 < b p(a1 − p)+ + (b/2 − p)(a2 − b/2 + p)+

+ b2

4 ≤ a22+b2

4

For the comp–coop model,

π comp−coop

b ≤ (√

6 − 2)a2a2

1 + a22

4

(√

6 − 2)a2 ≤ b ≤ a1 ≤ a2(2a1 + b)(2a2 + b)

12



a1 ≤ b ≤ a2, (√

6 − 2)a2 ≤ b9a2

i + 4a2j + 4b2 + 10aj b

36

a1 ≤ a2 ≤ b < a1 + a2a2

1 + a22 + b2

4

a1 + a2 ≤ b ≤ a1

+a2 + 4√3a1

5a2i − 6aiaj + 5a2

j + 6aib + 6aj b + 5b2

32

a1 + (√

3 + 1)a2 ≤ ba2

1 + b2

4

(√

3 + 1)a1 + a2 ≤ ba2

2 + b2

4

We note that multiple equilibria can exist when b < a1; if this occurs, wepick the one that yields maximum joint profit for the firms (i.e., p = a1/2).

Now, compare the profits in decentralized model with those in thecoop–comp model:

Range of the bundle market b πd vs. π coop−comp

b ≤ 3

4a2,

b < a1/2 >4a2 − 3a1

6≤ b <

5a1 − 2a2

4,

4a1 + (4 + 7√

2)a2

20 + 14√

2≤ b >

5a1 − 2a2

4≤ b,

4 + 5√

2

2(6 + 5√

2)a2 ≤ b >

3

4a2 ≤ b ≤ 3

4(a1 + a2), <

3

4(a1 + a2) ≤ b ≤ 3

4(√

2 + 1)(a1 + a2),

2a1 + 7a2

6b ≤ 6 + 5

√2

8(a1 + a2) <

b3

4(a1 + (

√2 + 1)a2) < b

3

4((

√2 + 1)a1 + a2) <4a2 − 3a1

6≤ b <

5a1 − 2a2

4,

4a1 + (4 + 7√

2)a2

20 + 14√

2≤ b ><

5a1 − 2a2

4≤ b,

4 + 5√

2

2(6 + 5√

2)a2 ≤ b <

3

4a2 ≤ b ≤ 3

4(a1 + a2), <

3

4(a1 + a2) ≤ b ≤ 3

4(√

2 + 1)(a1 + a2),

2a1 + 7a2

6b ≤ 6 + 5

√2

8(a1 + a2) <

b3

4(a1 + (

√2 + 1)a2) π comp−coop if (√

6 − 2)a2 < b <a12 . �

ACKNOWLEDGMENTS

This research is partially supported by the Natural Sciencesand Engineering Research Council of Canada (NSERC).

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Some implications of pricing bundles