Product innovation and vertical integration:private and social incentives∗

Luigi Filippini†and Cecilia Vergari‡

June 5, 2013

Abstract

We study the licensing incentives of an independent input producerowning a patented product innovation which allows the downstream firmsto improve the quality of their final goods. We consider a general two-parttariff contract for both outside and incumbent innovators. We find thattechnology diffusion critically depends on the nature of market competi-tion (Cournot vs. Bertrand). Moreover, the vertical merger with eitherdownstream firm is always privately profitable and it is welfare improvingfor large innovations: this implies that not all profitable mergers shouldbe rejected.

In the case of negative royalties (subsidies) exclusive licensing takesplace and vertical integration is never an equilibrium.

Keywords: Patent licensing, two-part tariff, negative royalties, verticaldifferentiation, vertical integration.

JEL Classification: L15, L13, L24.

1 Introduction

There are several examples of innovation in input required for the production offinal goods. The introduction of a faster computer chip; the wheat bread flourmade in the stone mill as 200 years ago;1 the Loro Piana innovation in newmaterials (super-luxury natural textiles).2 These input innovations improve the

∗We have benefitted from useful comments from Vincenzo Denicolò, Salvatore Piccolo,Cinzia Rovesti and Emanuele Tarantino. We also thank seminar audience at Lisbon, UECELisbon Meetings 2012: Game Theory and Applications, at the University of Bologna, and atIIOC 2013, Boston, May 17-19. Any remaining errors are ours.†Istituto di Teoria Economica e Metodi Quantitativi, Dipartimenti e Istituti di Scienze

Economiche, Università Cattolica del Sacro Cuore. email: luigi.filippini@unicatt.it.‡Department of Economics, University of Bologna, Strada Maggiore, 45 I-40125 Bologna

Italy. email: cecilia.vergari@unibo.it.1 It is a case of reswitching of techniques to traditional Italian whole wheat breads made

200 years ago when the local stone milled flour was the only option.2This thread starts with Tasmanian wool in the 1970s, through vicuna (LP was granted

10-year exclusivity), baby cashmere to, most recently, lotus flower cloth (i.e., the fibers ofnelumbo nucifera, an aquatic perennial more commonly known as the lotus).

1

quality of the final good and/or possibly create vertical differentiation in theproduct market.The soybean seed market makes another case study. As pointed out by

Shi and Chavas (2011), recent advances in biotechnology have led agriculturalbiotech firms (who produce the seeds sold to farmers) to differentiate their seedproducts through patented genetic materials. The vertical organization of thisindustry has changed. While biotech firms producing patented genes have reliedextensively on licensing their technologies to seed companies, they have recentlyincreased their use of vertical coordination through integration.We develop a theoretical model to study the incentives of an input innovator

to diffuse its innovation either through licensing or through vertical integration.As a result of the technology diffusion decision the final product market iseither vertically differentiated or homogeneous. We further wonder whether theprivate incentives are in line with the social incentives. The social profitabilityof vertical integration is indeed an important issue with no clearcut results. Onone hand theoretical contributions, like Rey and Tirole (2007), point out thatvertical merger is welfare detrimental because of market foreclosure.3 Someempirical evidence supports this view. For instance, Shi and Chavas (2011), inthe context of the U.S. soybean seed market, develop an empirical investigationof how differentiated seed products are priced in concentrated markets undertwo alternative vertical structures: vertical integration versus licensing. Theyuse a Cournot model of multi-product firms and find that seeds sold throughvertically-integrated structures are priced higher than those that are licensed.On the other hand, however, the merger guidelines do not exclude that verticalmerger could increase social welfare for effi ciency reasons.More precisely, we consider two downstream firms producing and selling a

final output to heterogeneous consumers and two differentiated inputs in theupstream market, a low quality input provided by competitive firms and a highquality patented input provided by an independent input producer. The qual-ity of the final good depends on the quality of the input. Complete technologydiffusion implies a homogeneous final good of high quality, whereas exclusivelicensing implies a vertically differentiated market. We consider a general two-part tariff contract for both outside and incumbent innovators. In particular,the set of possible contracts, that are observable, is such that the fixed fee hasto be non-negative, whereas the per-unit royalty might also be negative. Themotivation for such assumption comes from the observation that, while negativefixed fees would be clearly held to be illegal by antitrust authorities, a nega-tive per-unit royalty, that is a per-unit subsidy, cannot a priori be consideredwelfare detrimental. Most of the literature on licensing focuses on non-negativeroyalties, however some recent contributions point out the private incentivesto subsidize the downstream production (Liao and Sen (2005) and Milliou andPetrakis (2007)), that in some cases are also welfare improving. From an empir-ical point of view, as pointed out by Liao and Sen (2005), per-unit subsidies arerarely observed in licensing contracts, however they might be present implicitely

3We later make a brief review of the literature on vertical integration.

2

in the form of trasmission of knowhow and technical assistance to the licensee.We endogenize market structure allowing the patent holder to vertically inte-grate with either downstream firm. We also provide the welfare analysis.In case of non-negative royalties, we find that technology diffusion critically

depends on the nature of market competition (Cournot vs. Bertrand). Whenfirms compete in the quantities, the innovation is sold to all firms thus ensuringcomplete technology diffusion as well as a high quality homogeneous good inthe market. In contrast, when firms compete in the prices, the innovator has noincentive for complete technology diffusion, rather he prefers exclusive licensingwhich implies a vertically differentiated market and in turn positive industryprofit to extract. In particular the internal patent holder does not license itsinnovation to the rival firm; the external patent holder sells only one license viaa fixed fee. The intuition behind this result relies on Bonanno (1986) where theauthor investigates the effect of the type of competition (Bertrand vs Cournot)on firms’incentives to (vertically) product differentiate. He shows that (withzero production costs) the two firms decide to produce a homogeneous highquality product under Cournot competition and differentiated products underBertrand competition. The reason is that under vertical differentiation firmshave the incentive to improve the quality of their products, since consumerswillingness to pay increases with quality. However, under Bertrand competitionthere is a counteracting tendency as the more similar the products are thetougher the price competition is. Under Cournot competition in constrast thiscounteracting tendency is not at work so that at equilibrium both firms producethe high quality good.As far as the merger profitability is concerned, we show that under Cournot

competition the vertical integration of the upstream inventor with either down-stream firm is always privately profitable. This result is in line with the newmarket foreclosure theory (see Rey and Tirole 2007) according to which verticalintegration allows the monopolist upstream producer to protect its monopolypower. This result is in contrast with Sandonis and Faulì-Oller (2006) that con-sider non-drastic process innovations in a horizontally differentiated Cournotduopoly and find that the merger is privately profitable only for small innova-tions. They point out the commitment problem faced by the vertical merger(that is the insider innovator) which has only one instrument, the licensing con-tract to the rival firm rather than two (a licensing contract to each downstreamfirm), and cannot credibly restrict its output as the new input is transferred atmarginal cost. However this result breaks down in our model as the incentiveto diffuse the innovation makes homogeneous the downstream market.As for social welfare profitability, we find that under Cournot competition

the merger is also welfare improving for large innovations; this implies thatnot all profitable mergers should be rejected. Indeed on one hand, the mergerpushes prices down as it implies the (partial) internalization of the vertical ex-ternality; on the other hand, the merger has an anticompetitive effect becausethe vertically integrated firm is able to (partially) foreclose the rival firm via apositive per-unit royalty. The first effect prevails as long as the quality improve-ment associated with the innovation is suffi ciently large. Thus, a very simple

3

prescriptions arises: the antitrust authority should approve mergers where largeinnovations are involved. Our result is in line with the European Commis-sion, who mentions for example in its recent Guidelines on the assessment ofnon-horizontal mergers (2008): “The Commission may decide that, as a con-sequence of the effi ciencies that the merger brings about, there are no groundsfor declaring the merger incompatible with the common market pursuant toArticle 2(3) of the Merger Regulation. This will be the case when the Com-mission is in a position to conclude on the basis of suffi cient evidence that theeffi ciencies generated by the merger are likely to enhance the ability and incen-tive of the merged entity to act pro-competitively for the benefit of consumers,thereby counteracting the adverse effects on competition which the merger mightotherwise have”. Under Bertrand competition, we find a result of equivalencebetween an external and an internal patent holder, both in terms of private andsocial welfare profitability. Indeed, an external patent holder optimally sells anexclusive license via a fixed fee, so that there is no distortion due to a positiveper-unit royalty (as if the patent holder were vertically integrated). This waythe patent holder maximizes the licensee’s profit and extract this profit up tothe outside option.In case of negative royalties, the main difference occurs under Cournot com-

petition in the vertical separation scenario, that is when the patent holder re-mains out of the market. Namely, the outside inventor optimally sells an ex-clusive license via a two-part tariff contract that specifies a per-unit subsidy.The intuition is that the innovator via the subsidy makes the unique licenseemore aggressive in the Cournot market and reap higher profit via the fixed fee(Fershtman and Judd, 1987). Thus exclusive licensing takes place under bothCournot and Bertrand competition. As for the merger profitability, the patentholder always prefers to stay out of the market and private and social incentivesalways match: vertical integration never takes place.We extend our analysis modelling the vertical relationship between the U

producer and the two retailers through a bargaining setting where firms withmarket power negotiated over a nonlinear (two-part tariff) input price. In thiswe follow previous work as Milliou and Petrakis (2007). They study incentivesfor horizontal mergers in the upstream sector of vertically related industrieswhen bargaining is present and contract types are endogenous. In particular,under the upstream merger scenario they model the bargaining process in sucha way that the U monopolist bargains simultaneously and separately with thetwo D firms. This key assumption captures the fact the upstream monopolisthas an incentive to behave opportunistically during its negotiations. That is,it cannot commit to each downstream firm that it will not negotiate a morefavourable contract with the rival firm. This implies that the per-unit royaltycannot be higher than the marginal cost of input production. This scenario thencoincides with the case of private vertical contracts (see also, among others Reyand Vergé (2008) and Rey and Tirole (2007)) to be compared with observablecontracts with which we deal in the rest of the paper.

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1.1 Related literature

This paper contributes to the literature on licensing a product innovation aswell as to the debate on the competitive effects of vertical integration.From a theoretical viewpoint, most of the literature on optimal licensing

focuses on cost-reducing, process innovations (see Kamien and Tauman, 1986;Katz and Shapiro, 1986; Kamien, Oren and Tauman, 1992; Sen and Tauman,2007; Erutku and Richelle, 2007). To our knowledge, little has been done toinvestigate the issue of licensing a product innovation.4 A first contribution isby Kamien, Tauman and Zang (1988). They consider the introduction of anew product in a Cournot oligopoly focusing on the fixed fee licensing mode.More recently, Lemarie (2005) compare fixed fee versus royalty for a demand-enhancing innovation. We depart from them in what we consider an innovationthat improves the quality of a product.5 Notice also that this literature focuseson contracts that do not allow for negative per-unit royalties, whereas we extendthe analysis to the case of per-unit subsidies.As far as the non-negative royalty case is concerned, we find that under

Cournot competition with homogeneous goods, the external patent holder opti-mally specifies positive per-unit royalties when the innovation is large. This is inline with the wide prevalence of per-unit royalties over fixed fee in practice (seefor instance, Rostoker (1984)). Moreover, Muto (1993) studies optimal licens-ing of a process innovation and shows that Bertrand competition is a rationaleto explain this empirical evidence. We argue that this theoretical result doesnot hold for a product innovation: in our model under Bertrand competitionthe external patent holder prefers fixed fee over royalty licensing. The reasonfor this contrasting result is as follows. Muto (1993)’s general intuition behindthe comparison with Cournot competition (where fixed fee always prevails overroyalty, see Kamien and Tauman, 1986; Katz and Shapiro, 1986; Kamien, Orenand Tauman, 1992) is that the patentee’s profit in the royalty policy depends onthe equilibrium total output level, and this amount is much greater in Bertrandcompetition than in Cournot. In our model in contrast under Bertrand compe-tition the patent holder only sells an exclusive license so that this equilibriumtotal output advantage is not as strong as in Cournot where complete technol-ogy diffusion takes place. Rather under Bertrand, the incentive to set a fixedfee prevails in order to avoid the distorting negative effect of a positive per-unitroyalty.Arya and Mittendorf (2006) studies the incentives of an internal patent

holder to license a product innovation in the presence of a monopolist input

4As pointed out by Kamien et al. (1988) “a product innovation can be regarded as a costreducing innovation by assuming that the new product could have been produced before butwith a suffi ciently high marginal cost that rendered its production unprofitable." However,we argue that the optimal licensing mode may be affected when we move from a process to aproduct innovation.

5Stamatopoulos and Tauman (2008) analyse optimal licensing for an outside inventor ofan innovation that improves the quality of a product and also affects its marginal cost (theinnovation can be classified as product and process at the same time). They consider the logitdemand framework with price competition.

5

producer. In their model the decision to license the innovation implies givingup the monopoly power on the final good, that is letting a Cournot rival down-stream firm enter the market. They find that the licensing incentive is positiveas it affects the input pricing terms in a way that is beneficial for the internalpatent holder. We also find that the internal innovator has incentive to licensethe new good to the rival, however, the reason is to extract some surplus fromthe rival firm that is active in the market independently of the licensing strategy.As for the negative royalty case, to the best of our knowledge the main

contributions are Liao and Sen (2005) and Milliou and Petrakis (2007). Ourpaper is more closely related to Liao and Sen (2005) that study optimal licensingof a process innovation in a Cournot duopoly with homogeneous goods. Theyshow that subsidy-based contracts are optimal under exclusive licensing butnot under complete technology diffusion. They conclude that negative royaltiesare welfare improving with respect to non-negative royalties if the innovation issuffi ciently small. They point out the importance of the cost of exclusion: theadditional cost that the non-licensee firm has to pay under exclusive licensing.As long as this cost is low, exclusive licensing (that arises with negative royalties)is preferred with respect to complete technology diffusion. We depart fromthis paper in what we endogenize market structure so that under non-negativeroyalties vertical integration always takes place whereas under negative royaltiesvertical integration becomes unprofitable. Therefore, in our model allowingfor negative royalties is always welfare improving because vertical integrationimplies a quasi monopoly outcome to be compared with the exclusive licensingscenario that implements a Stackelberg equilibrium.As for the competitive effects of vertical integration, there are two opposite

views. The Chicago School (e.g., Bork, 1978; Posner, 1976) stresses that, in theabsence of effi ciency gains, vertical integration could not increase the profitabil-ity of merging firms. In contrast the new market foreclosure theory points outthat vertical integration can restrict downstream competition. In particular inthe recent survey by Rey and Tirole (2007) it is shown that vertical integrationis always profitable as the U producer can restore its monopoly power throughinput foreclosure but vertical integration is anticompetitive. In our framework,under non-negative royalties, vertical integration remains privately profitablebut the social welfare profitability depends on the innovation size. In contrast,if we allow for negative royalties, vertical integration becomes privately and so-cially unprofitable. Reisinger and Tarantino (2012) extend the theory of Reyand Tirole (2007) by analysing the profitability of vertical integration in thepresence of complementary input producers and show that vertical integrationcan be privately unprofitable. In particular, VI raises the market profit of themerging entity and it is therefore profitable; however, the presence of a com-plementary input implies that part of this larger profit can be extracted by thesupplier of this input. This expropriation effect can render VI unprofitable.The remainder of the paper is structured as follows. In Section 2, we set

up the ante-innovation model. In Section 3 we introduce the product innovationand we study the licensing incentives of an external innovator. In Section 4we consider the optimal strategy of an internal innovator. In Section 5, we

6

compare the private and social incentives. In Section 6 we extend the analysisto Bertrand competition.

2 Model

We consider two firms producing a homogeneous good and competing à laCournot.6 Final output production requires an essential input provided by acompetitive upstream market.As far as the demand side is concerned, we assume that there is a continuum

of consumers indexed by θ which is uniformly distributed in the interval [0, 1].Thus, θ is a taste parameter. Each consumer has a unit demand and buys eitherone unit of a good of quality s at price p or buys nothing at all. Consumer’sutility takes the following form:

U(θ) =

{θs− p, if consumer type θ buys

0, if does not buy

The demand for the good is then

Q (p) = 1−(ps

)⇐⇒ p (Q) = s (1−Q) , (1)

where Q = q1 + q2 and p/s is the fraction of consumers with a taste parameterless than θ, that is the fraction of consumers not buying the good.7 For futurereference we define the consumer surplus as

CS (s) =

1∫ps

(θs− p) dθ =(p− s)2

2s(2)

As for the supply side, the essential input of quality s is produced at zerofixed cost fL = 0 and at constant marginal cost c = 0 and it is sold at thecompetitive price w = 0. In this framework quality is assimilated to input. TheD firm i profit function is: πi = pqi. D firms compete in the quantities, thenthe Cournot duopoly equilibrium is (superscript C stands for Cournot):

qCi (ci = 0, cj = 0; s, s) =1

3(3)

pC =s

3

πCi =s

9(4)

CSC =2

9s

6Given a homogeneous final good, price competition leads to the Bertrand paradox. Weextend the analysis to Bertrand competition in Section 6.

7At equilibrium the market is not covered, that is Q < 1.

7

The D firms’price and profits depend on the quality of the input s. The D firmssell Q = q1 + q2 = 2

3 which is the demand for the input faced by the upstreammarket (perfect vertical complementarity).For future reference (and as a benchmark) consider the monopoly outcome

for this market:8

pm =s

2, qm =

1

2, πm =

s

4> 2πCi .

3 Innovation

Suppose that an independent input producer obtains a patented product inno-vation which allows the downstream firms to improve the quality of their finalgoods. In the upstream market there is now a monopolist selling an input thatameliorates final product quality by ψ > 1 that measures the innovation size.Assume the unique production cost is the fixed cost fH = f > 0 for the highquality input. Marginal cost of production is then zero.We study the licensing incentives of this patent holder. The U firm can sell

the new input either to one or both D firms via a two-part licensing contract(r, F ).9 We consider the set of possible contracts such that r ∈ R and F ≥ 0.Indeed, whereas negative fixed fees would be clearly held to be illegal by antitrustauthorities as they could be a means to strand the rival firm out of the market,a negative per-unit royalty, that is a per-unit subsidy (given that the marginalcost is equal to zero), cannot a priori be considered welfare detrimental. Mostof the literature of licensing focuses on non-negative royalties, however somerecent contributions point out the private incentives to subsidize the downstreamproduction (Liao and Sen (2005) and Milliou and Petrakis (2007)), that in somecases are also welfare improving.According to the U innovator licensing incentives, different cases derive:

1. Complete technology diffusion: both D firms adopt the new input andwe have a homogeneous final good of quality sψ > s. D firms’profitsπi (ci, cj ; sψ, sψ) depends on the two part-tariff contracts ci = (ri, Fi)with i = 1, 2 and i 6= j.

2. Exclusive licensing : only one of the D firms adopt the new input and wehave two final goods of different qualities. The non-innovating firm, sayfirm 1, produces the low quality good thus incurring zero production costsand gains π1 (0, c2; s, sψ); while the innovating firm 2 produces the highquality good and gets π2 (c2, 0; sψ, s).

Under vertical separation, that is the scenario in which the innovator staysout of the market, we develop a three-stage game: first, the innovator offersa contract to each D firm on a take-it-or-leave-it basis; second, the potential

8The monopolist maximization problem would be maxp [p (1− p/s)].9A two-part licensing contract covers both the case of the royalty and the case of the fixed

fee licensing modes: a royalty can be seen as a two-part tariff with F = 0; the fixed fee canbe seen as a two-part tariff with r = 0.

8

licensees decide whether to accept or reject the contract; finally the D firmscompete. Note that all the contracts are observable. Solving backwards, wefind the subgame perfect Nash equilibrium.

3.1 Exclusive licensing

Suppose that only one D firm adopts the new input. Firm 1 does not buy thenew input and produces a final good of quality s1 = s at price p1; firm 2 adoptsthe new input and produces a final good of quality s2 = ψs > s, at price p2with s2 − s1 = s (ψ − 1). The demands for the goods are:

q1 = θ̂ − p1s

(5)

q2 = 1− θ̂, (6)

whereθ̂ =

p2 − p1s (ψ − 1)

.

The inverse demands are:

p1 = s (1− q2 − q1)p2 = s (ψ − ψq2 − q1)

D firms’profits are:

π1 = p1q1,

π2 = (p2 − r2) q2 − F2.

Given the third stage equilibrium quantities,10 under the assumption of non-negative royalties, the U firm chooses the two-part tariff contract for firm 2(r2, F2) such that:

maxr2,F2

ΠU

s.t.r2 ∈[0,s (2ψ − 1)

2

]0 ≤ F2 ≤ π2 (r2, 0; sψ, s)− s

9

with ΠU = (2sψ−s−2r2)s(4ψ−1) r2 + F2 − f . The first constraint comes from the non-

negativity of q2 and the second constraint (binding at equilibrium) ensures thatfirm 2 has the incentive to get the license rather than the status quo. The

solution is a contract such that rEL2 = 0 and FEL2 =(36ψ2−16ψ+1)(ψ−1)s

9(4ψ−1)2 .11

10More formal details on the solution of this and the other subgames are in Appendix.11This is because ∂/∂r2

((2sψ−s−2r2)s(4ψ−1) r2 + π2 (r2, 0; sψ, s)− s

9− f

)< 0, that is the incen-

tive is to set a royalty as low as possible. Given the non-negative constraint, the solution isr = 0.

9

Remaining equilibrium variables are (superscript EL stands for exclusive li-censing):

qEL1 =ψ

(4ψ − 1), qEL2 =

(2ψ − 1)

(4ψ − 1), QEL =

(3ψ − 1)

(4ψ − 1)

pEL1 =ψs

(4ψ − 1), pEL2 =

(2ψ − 1)ψs

(4ψ − 1),

πEL1 (s, sψ) =ψ2s

(4ψ − 1)2 < πEL2 (sψ, s) =

s

9

ΠELU (s, sψ) =

(36ψ2−16ψ+1)(ψ−1)s9(4ψ−1)2 − f.

ΠELU is the U patent holder equilibrium profit under exclusive licensing, when

selling via a two-part tariff, which reduces to a fixed fee, the new input to onlyone D firm. For completeness we provide equilibrium consumer surplus underexclusive licensing:

CSEL =(ψ+4ψ2−1)sψ2(4ψ−1)2 .

Consider now the case of negative royalties, the optimal contract under ex-clusive licensing is then such that:

rNeg2 = −s4, FNeg2 =

1

36s (9ψ − 4) . (7)

As ∂∂r2

ΠU = (s+4r2)(1−2ψ)(4ψ−1)2s ≥ 0 ⇐⇒ r2 ≤ − s4 , the objective is maximized at a

negative value of r2, namely r2 = − s4 .12 Equilibrium variables are:

π2 (r2, 0; sψ, s) =1

9s, π1 (0, r2; s, sψ) =

1

16s

q1 (0, r2; s, sψ) =1

4, q2 (r2, 0; sψ, s) =

1

2, QEL =

3

4

p1 (0, r2; s, sψ) =1

4s, p2 (r2, 0; sψ, s) =

1

4s (2ψ − 1) ,

ΠELnegU =

1

72s (72ψ − 17)− f

CSELneg =1

32(4ψ + 5) s

PSELneg =1

16s (16ψ − 1)− f

SWELneg =3

32s (12ψ + 1)

The equilibrium we get coincide with Stackelberg where the licensee has a firstmover advantage. The intuition behind this optimal subsidy relies on the in-centive of the U monopolist to make the D licensee more aggressive (strategicdelegation literature, Fershtman and Judd, 1987). This is in line with Liao andSen (2005).12ΠU is concave wrt r2.

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3.2 Complete technology diffusion

Suppose the U firm decides to sell the new input to both D firms via a two-parttariff (r, F ). The U firm maximization problem is, under nonnegative royalties:

maxF1,r1,F2,r2

{r1q1 (r1, r2; sψ, sψ) + r2q2 (r1, r2; sψ, sψ) + F1 + F2 − f}

s.t.π1 (r1, r2; sψ, sψ)− F1 ≥ π1 (0, r2; s, sψ)

π2 (r2, r1; sψ, sψ)− F2 ≥ π2 (0, r1; s, sψ)

r1 ≥ 0, r2 ≥ 0, F1 ≥ 0, F2 ≥ 0

where π1 (0, r2; s, sψ) is defined in (15). Here the outside option for each firm isnot buying the new input given that the rival firm does. Solving the completetechnology diffusion subgame, we find that the optimal contract is: r1 = r2 = rT =

(sψ−26sψ2+16sψ3)64ψ2−14ψ+4 , FT

(rT , rT

)=

(248ψ2−17ψ−272ψ3+256ψ4+1)sψ4(32ψ2−7ψ+2)2 if ψ ≥ ψT ,

r1 = r2 = 0, FT (0, 0) = (ψ−1)(16ψ−1)sψ9(4ψ−1)2 if ψ < ψT .

(8)

where rT =(sψ−26sψ2+16sψ3)

64ψ2−14ψ+4 ≥ 0 ⇐⇒ ψ ≥ ψT = 1.585 6. This means thatwhen the innovation is small the inventor’s incentive is to set a per-unit priceas low as possible, that is the optimal contract is a fixed fee.13 In other words,given the positive outside option of producing the low quality good at zeromarginal cost, for small innovations the inventor cannot set a positive per unitroyalty. In contrast for large innovations we have a positive per-unit royalty.14

If we allow for negative royalties, the optimal contract is the first line of (8)for any innovation size. Note that for a small innovation size (ψ < ψT ), theU monopoist is willling to subsidize the D production, however in this case theexplanation is not the incentive to make the D affi liates more aggressive but thepresence of the outside option: each D firm can always produce the low qualitygood at zero marginal cost and make positive profit. To induce them to buy thesmall innovation the inventor cannot set a positive per unit royalty.Equilibrium magnitudes are for ψ ≥ ψT (superscript T stands for complete

13Under a non-constrained maximization problem, for a small innovation size, the U mo-nopolist would have the incentive to set a negative per-unit royalty, that is to subsidize theproduction of the new product and to extract market profits via the fixed fee. This kindof policy has been analysed among others by Liao and Sen (2005) and Milliou and Petrakis(2007).14This result is in line wih Sen and Tauman (2007).

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technology diffusion), and for any innovation size under negative royalties:

qT (sψ) =(4ψ+16ψ2+1)2(32ψ2−7ψ+2) , Q

T =(4ψ+16ψ2+1)(32ψ2−7ψ+2)

pT =(16ψ2−11ψ+1)ψs(32ψ2−7ψ+2)

πT (sψ) =(4ψ+16ψ2+1)

2sψ

4(32ψ2−7ψ+2)2

πT (sψ)− FT(rT , rT

)= 25(4ψ−1)2sψ2

4(32ψ2−7ψ+2)2

ΠTU (sψ) = 2rT qT (sψ) + 2FT

(rT , rT

)− f =

(16ψ2−16ψ+1)sψ2(32ψ2−7ψ+2) − f

CST =(4ψ+16ψ2+1)

2sψ

2(32ψ2−7ψ+2)2 ,

SWT =3

2(16ψ2−6ψ+1)(4ψ+16ψ2+1)sψ

(32ψ2−7ψ+2)2 − f.

Whereas for ψ < ψT , in case of non-negative royalties, equilibrium magnitudesare:

qT (sψ) =1

3, QT =

2

3

pT (sψ) =sψ

3

πT (sψ) =sψ

9

πT (sψ)− FT (0, 0) =ψ2s

(4ψ − 1)2

ΠTU (sψ) = 2FT (0, 0)− f = 2 (ψ−1)(16ψ−1)sψ

9(4ψ−1)2 − f,

CST =2

9ψs,

SWT =4

9ψs− f

Comparing equilibrium variables under exclusive licensing and completetechnology diffusion, we find the following results.

Proposition 1 Under Cournot competition, (i) in case of non-negative roy-alties, the external patent holder always prefers complete technology diffusion,namely, ΠT

U (sψ)− ΠELU (sψ) > 0. The optimal contract is specified in (8). (ii)

If we allow for negative royalties, exclusive licensing always prevails over com-plete technology diffusion, namely, ΠT

U − ΠELnegU < 0. The optimal contract is

specified in (7).

Under the case of non-negative royalties, two remarks are worthy. First, incontrast with a process innovation, even large product innovations are sold toboth firms as they both survive in the market. A consequence of the innovator’s

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preferences towards technology diffusion is that the downstream market is notvertically differentiated, as only the high quality input is sold to both firms.In other words, the downstream market is characterized by a homogeneoushigh quality good. Secondly, note that as the contracts offered to each D firmare observable, the U monopolist could implement the monopoly outcome.15

However the U monopolist has not incentive to implement this outcome becauseof the presence of the D firms’outside options.Under the case of negative royalties, the private incentives about technol-

ogy diffusion reverse. Subsidizing the unique licensee allows the innovator toconquer most downstream market by selling the high quality good. Negativeroyalties on one hand make stronger competition because downstream produc-tion is subsidized; on the other hand, negative royalties by inducing exclusivelicensing make milder competition because vertical product differentiation arises(whereas homogeneous goods prevail under non-negative royalties).

3.3 Bargaining

We next extend our analysis by considering a bargaining process on the verticalcontracts. More precisely, the U producer decides how many licenses and then itbargains on the contract with each firm separately. Let α ∈ [0, 1] be the patentholder’s bargaining power.Consider first the case of non-negative royalties. Suppose U wants to sell the

new input to both firms. Negotiations occur simultaneously and independently:U and firm 1 negotiate over the (non-negative) contract (r1, F1) while U andfirm 2 negotiate over the contract (r2, F2). Each bargaining process is modelledas a two stage game such that they first choose ri and then Fi.16 This waythe bargaining process is not distorsive. Actually, they first determine the sizeof the pie (setting ri) and then they divide (setting Fi), however by backwardinduction, they first choose F for any given r (that is they first decide how todivide the joint surplus) and then they choose r (that is they determine thejoint surplus). In a simultaneous bargaining process instead how you decide todivide the pie (F ) would affect the size of the pie (and viceversa).17 Consider

15The U monopolist could indeed set a royalty such that each firm produces half monopolyquantity:

q1 =1

3sψ(sψ − 2r1 + r2) =

1

4

q2 =1

3sψ(sψ − 2r2 + r1) =

1

4

⇐⇒ r1 = 14sψ, r2 = 1

4sψ. This implies the following total quantity, price, U profit and indus-

try profit: Q = 12, p(12

)= sψ

(1− 1

2

)= 1

2ψs; ΠmU = 1

4sψ 1

2+ 2

(116ψs− 25

16(4ψ − 1)−2 ψ2s

),

Π = 14ψs = πm.

16We follow Milliou and Petrakis (2007).17This is the same timing implicitly used in the case of no-bargaining. Actually, by backward

induction we first find the optimal fixed fees (F1 and F2) maximising the U objective function(16), and then substitute F1 and F2 in the objective function and maximise wrt r1 and r2.Both contracts are bilaterally effi cient. Note however that the two maximization problems are

13

the bargaining process of U with firm 1, given the outcome of the bargainingprocess of U with firm 2 (r2, F2). The objective function is:

(r1q1 (r1, r2; sψ, sψ) + r2q2 (r1, r2; sψ, sψ) + F1 + F2 − r2q2 (r2, 0; sψ, s)− F2)α

(π1 (r1, r2; sψ, sψ)− F1 − π1 (0, r2; s, sψ))1−α

Solving by backward induction, we first maximize with respect to F1 and wefind F1 (r1, r2) = A(r1,r2)+αB(r1,r2)

9ψs(4ψ−1)2 , where A (r1, r2) and B (r1, r2) are long ex-pressions defined in Appendix. We next substitute this solution in the objectivefunction and maximize with respect to r1. The symmetric solution is r1 = r2 =

rBa = 0 and the optimal fixed fee is then F1 = F2 = FBa = α 19(ψ−1)(16ψ−1)sψ

(4ψ−1)2 .Note that in the bargaining process the optimal contract specifies a zero per-unit royalty versus a positive royalty under full bargaining power. This resultsin a downstream duopoly. The key assumption to this result is the simultane-ity of the bargaining process that introduces an incentive for the U monopolistto behave opportunistically during the separated negotiations with the two Dfirms. In particular, U is not able to commit to each D firm that it will notnegotiate a more favourable contract with the rival D firm. This in turn impliesthat each D firm will never agree on a per-unit royalty that is higher than themarginal cost of production of the new input. This result is in line with Milliouand Petrakis (2007). Equilibrium price, profits, consumer surplus and welfareare then:

pBa =1

3ψs

ΠBaU = 2F = 2α

1

9(ψ−1)(16ψ−1)sψ

(4ψ−1)2 − f,

πBa1 (0, 0; sψ, sψ)− FBa =sψ

9

(4ψ − 1)2 − α (16ψ − 1) (ψ − 1)

(4ψ − 1)2 ,

CSBa =2

9ψs,

SWBa =4

9ψs− f,

with πBa1 (0, 0; sψ, sψ)−FBa1 > 0 ⇐⇒ α < (4ψ−1)2(16ψ−1)(ψ−1) always as

(4ψ−1)2(16ψ−1)(ψ−1) >

1.18

Consider now tha case of negative royalties. Under exclusive licensing theU monopolist and firm 2 negotiate over the contract (r2, F2). The objectivefunction is:

(r2q2 (r2, 0; sψ, s) + F2 − f)α(π2 (r2, 0; sψ, s)− F2 −

s

9

)1−αnot equivalent for α = 1 because in the no-bargaining case U outside option is not considered(formally the objective functions never equalize).18Check whether (sψ)2

9ψs− α 1

9(ψ−1)(16ψ−1)sψ

(4ψ−1)2 is incentive compatible: (sψ)2

9ψs−

α 19(ψ−1)(16ψ−1)sψ

(4ψ−1)2 ≥[(sψ+r2)

2

(4ψ−1)2s

]r2=0

⇐⇒ (sψ)2

9ψs− α 1

9(ψ−1)(16ψ−1)sψ

(4ψ−1)2 − (sψ)2

(4ψ−1)2s =(− 19

)(4ψ − 1)−2 (α− 1) (ψ − 1) (16ψ − 1) sψ > 0.

14

The solution is r2 = − s4 , F2 = 172s (18αψ − 17α+ 9). This solution comes from

the simultaneous maximization of the above objective w.r.t. r2 and F2. Underexclusive licensing the U objective is to maximize firm 2 downstream profit,because this is its only source of surplus. We could thus expect that the per-unit royalty negotiated under this bargaining process coincide with rNeg2 = − s4 ,that is the solution of exclusive licensing with full bargaining power. Note thatfor α = 1 the contract with bargaining power coincide with the contract withfull bargaining power. Given that exclusive licensing with negative royaltiesalways prevails over complete technology diffusion under full bargaining power,we can extend this result to α < 1, given that the equilibrium per-unit royaltyis the same.Extending the private solution to a bargaining process does not change the

technology diffusion incentives: under non-negative royalties, complete tech-nology diffusion prevails, whereas under negative royalties, exclusive licensingdominates.

4 Vertical integration

We have considered so far, the case of an external innovator, i.e. the U firm doesnot sell the final good in the D market. Suppose now that the U producer andone of the two D firms, say firm 2, merge, in this case the vertically integrated(VI) firm is an internal patent holder and its profit consists of two parts: theprofit from selling the new input to the rival D firm 1 (if it decides to license)and the profit from selling the high quality final good 2.We consider the following three-stage game: first, the patent holder offers a

contract to D firm 1, D firm 1 decides whether to accept it and finally marketcompetition takes place.Proceeding backwards, given the third stage equilibrium quantities, the VI

firm offers firm 1 the two-part tariff contract (r1, F1) such that:

maxr1,F1

{πV I (0, r1; sψ, sψ) + r1q1 (r1, 0; sψ, sψ) + F1}

s.t.π1 (r1, 0; sψ, sψ)− F1 ≥ π1 (0, 0; s, sψ)

r1 ∈[0,sψ

2

], F1 ≥ 0

where π1 (0, 0; s, sψ) = ψ2s(4ψ−1)2 , is firm 1 outside option, that is firm 1 profit if it

does not buy the new input thus selling the low quality final good and incurringper-unit cost equal to zero. If the VI firm is constrained to nonnegative fees, itwill optimally let the nonaffi liate firm to produce a positive quantity as low aspossible (up to its outside option) and the optimal contract is then:

rCV I =s(ψ (4ψ − 1)− 3ψ

√ψ)

2 (4ψ − 1)> 0, FCV I = 0. (9)

15

Equilibrium magnitudes are:

qCV I(0, rCV I ; sψ, sψ

)=

(4ψ2 − ψ

√ψ − ψ

)2 (4ψ − 1)ψ

,

qC1(rCV I , 0; sψ, sψ

)=

√ψ

(4ψ − 1),

QCV I = qCV I(0, rCV I ; sψ, sψ

)+ qC1

(rCV I , 0; sψ, sψ

)=

(4ψ2 − ψ + ψ

32

)2 (4ψ − 1)ψ

,

pCV I =ψ(4ψ −

√ψ − 1

)s

2 (4ψ − 1),

πCV I(0, rCV I ; sψ, sψ

)=

(4ψ2 − ψ

√ψ − ψ

)2s

4 (4ψ − 1)2ψ

,

πC1(rCV I , 0; sψ, sψ

)− FCV I =

sψ2

(4ψ − 1)2 ,

ΠCV I = πCV I

(0, rCV I ; sψ, sψ

)− f + rCV Iq

C1

(rCV I , 0; sψ, sψ

)+ FCV I =

(16ψ2−13ψ+1)sψ4(4ψ−1)2 ,

CSCV I =(4ψ+

√ψ−1)

2sψ

8(4ψ−1)2 ,

PSCV I = sψ2

(4ψ−1)2 +(16ψ2−13ψ+1)sψ

4(4ψ−1)2 ,

SWCV I =

(4ψ+√ψ−1)

2sψ

8(4ψ−1)2 + sψ2

(4ψ−1)2 +(16ψ2−13ψ+1)sψ

4(4ψ−1)2 ,

where PSCV I denotes the producer surplus, that is industry profit under verticalintegration.Note that the optimal contract is the same if we allow for negative royalties,

as the VI firm has always incentive to set a positive royalty.We gather the above results as follows.

Proposition 2 Under Cournot competition, the internal patent holder alwayssells the innovation to the rival firm. The optimal contract is the two-part tariffspecified in (9).

Proposition 2 contains two results. The first concerns the firm’s incentiveto license the product innovation thus preferring a homogeneous high qualitygood market rather than a vertically differentiated market in which the internalpatentee would be the high quality firm competing with a low quality producer.The intuition relies on Bonanno (1986) where the author investigates the effectof the type of competition (Bertrand vs Cournot) on firms’incentives to (verti-cally) product differentiate. He shows that (with zero production costs) the twofirms decide to produce a homogeneous high quality product under Cournotcompetition and differentiated products under Bertrand competition.19 Arya19This result also explains the patent holder’s incentives under Bertrand competition, Sec-

tion 6.

16

and Mittendorf (2006) studies the incentives of an internal patent holder tolicense an innovation on the final good in the presence of a monopolist inputproducer. In their model the decision to license the innovation implies givingup the monopoly power on the final good, that is letting a Cournot rival firmenter the market. They find that the licensing incentive is positive as doing sothe internal patent holder creates a "weak" rival (as the rival has the additionalmarginal cost equal to the per-unit royalty) that induces the external supplierto make better pricing conditions. We also find that the VI (input) innovatorhas incentive to license the new good to the rival, however, the reason is not toaffect the input pricing terms but to extract some surplus from the rival firmthat is active in the market independently of the licensing strategy.The second result of Proposition 2 concerns the optimal licensing mode of

the internal patent holder. San Martin and Saracho (2010) consider a non dras-tic process innovation and show that in a Cournot duopoly with homogeneousgoods the optimal licensing mode is an ad valorem royalty, that is a profit shar-ing agreement. This result does not hold in our model: as we prove in theAppendix the ad valorem licensing mode is dominated by the two-part tariffcontract defined in (9). The intuition behind this result is as follows. In SanMartin and Saracho (2010) the ad valorem royalty is optimal as the internalpatentee introduces the process innovation that increases total quantity (it letthe rival firm to produce with a more effi cient technology) and then appropri-ates the rival’s profit up to its outside option. In our model if the innovationis diffused via an ad valorem royalty total output does not increase (as underhomogeneous goods it is independent of the quality, see the status quo equilib-rium quantity, expression 3). This implies that industry profits correspond tothe duopoly profits that are shared according to α. Whereas under two-part tar-iff the internal patentee can at least partially internalize the negative externalitycoming from competition and approach the monopoly outcome.20

5 Private and social profitability of vertical in-tegration

We next consider the merger profitability as well as the social welfare compar-ing the vertical integration scenario with the vertical separation scenario (i.e.,external patent holder) where complete technology diffusion takes place.As far as the private profitability is concerned, the merger is always profitable

under non-negative royalties, namely:

ΠCV I−

(πT (sψ)− FT

(rT , rT

)+ ΠT

U (sψ))

=

5(256ψ4−57ψ2−112ψ3−7ψ+1)sψ2

4(32ψ2−7ψ+2)2(4ψ−1)2 > 0 for ψ ≥ ψT ,(ψ−1)(16ψ−1)sψ

36(4ψ−1)2 > 0 for ψ < ψT ..

20We do not extend formally the VI scenario to bargaining as we expect this would notchange the insider innovator’s incentives. We expect again a zero fixed fee and a positiveroyalty rCV I (α) ≤ rCV I .

17

This result is in line with the new market foreclosure theory according to whichVI allows the U producer monopolist to protect its monopoly power. Moreprecisely Rey and Tirole (2007) point out that vertical integration is a deviceto restore the U monopoly power that cannot be exerted under vertical separa-tion when contracts are unobservable. In our model, contracts are observable,however due to the strategic outside option (πi (0, rj ; s, sψ)) the external patentholder has not incentive to implement the monopoly outcome. Under the caseof negative royalties, however, the private incentives reverse, namely verticalintegration becomes unprofitable:

ΠELnegU −ΠC

V I (sψ) =(190ψ−614ψ2+864ψ3−17)s

72(4ψ−1)2 > 0.

As for the social profitability of VI, we make the following comparisons. For

ψ < ψT , CSCV I − CST =(4ψ+

√ψ−1)

2sψ

8(4ψ−1)2 − 29ψs = ψs

((4ψ+

√ψ−1)

2

8(4ψ−1)2 − 29

)< 0.

For ψ > ψT , CSCV I − CST = sψ

((4ψ+

√ψ−1)

2

8(4ψ−1)2 − (4ψ+16ψ2+1)2

2(32ψ2−7ψ+2)2

)> 0 ⇐⇒

ψ > ψ∗ = 3.407. We can conclude that for ψ > ψ∗ both the industry profitand consumer surplus are higher under VI, that is vertical integration is welfareimproving. In contrast for ψ < ψ∗, the result is ambiguous as consumer surplusis lower but producer surplus is higher under VI rather than the non-mergercase. However direct computations of social welfare (SW = CS + PS) revealthe following.

SWCV I − SWT < 0 ⇐⇒ ψ < ψ∗.

Compare next the two vertical scenarios, considering the case that, undervertical separation, the external patent holder does not have full bargainingpower:

ΠCV I −

(πBa − FBa + ΠBa

U

)= (5−4α)(ψ−1)(16ψ−1)sψ

36(4ψ−1)2 > 0,

SWCV I − SWBa =

(31ψ2−5ψ−80ψ3−18ψ

32+72ψ

52

)s

72(4ψ−1)2 < 0.

We gather our results in the following proposition.

Proposition 3 (i) Under non-negative royalties, vertical integration is alwaysprivately profitable. However, as long as the patent holder has full bargainingpower, VI is welfare improving if and only if the innovation is suffi ciently large,namely ψ > ψ∗; in contrast, in case the external patent holder does not have fullbargaining power, then VI turns out to be welfare detrimental for any innovationsize. (ii) Under negative royalties, vertical integration becomes privately andsocially unprofitable.

Consider first the case of non-negative royalties and full bargaining power.Sandonis and Faulì-Oller (2006) consider non-drastic process innovations in ahorizontally differentiated Cournot duopoly and study the patentee’s incentives

18

to merge with either firm in the market. They show that the merger is privatelyprofitable for small innovations and it is welfare improving for large innovations.They argue that all profitable mergers are welfare detrimental, this also holds forhomogeneous goods. More precisely, Sandonis and Faulì-Oller (2006) individ-uate the following trade-off: an internal patentee (VI case) better internalizesmarket profit with respect to an external one, however the internal patenteecan only use one instrument (a contract for the other firm in the market) ratherthan two (a contract for each firm in the market). An external patentee hastwo instruments but it has to care about firms’outside option which dependson the royalty, in particular it increases with the royalty and decreases with theinnovation size. They find that the balance of these two effects depend on theinnovation size: the merger is privately profitable for small innovations, in factfor large innovations the outside option faced by the external patentee is lowand so it has little relevance with respect to the availability of two instruments.In our model the merger is profitable for any innovation size ψ. In particular, incontrast with Sandonis and Faulì-Oller (2006) the internal patentee approachesthe monopoly outcome as the innovation size ψ increases. In fact, the VI firmhas incentive to reduce as much as possible the quantity produced by the non-affi liate, in order to internalize as much as possible the vertical externality (inthe limit, if allowed, the VI firm would completely foreclose the rival firm com-pensating it via a negative fixed fee). However the VI firm is constrained by thenonaffi liate outside option: the higher is ψ the lower is this outside option andso the lower the quantity that the nonaffi liate produces and in turn the more theVI firm approaches the monopoly outcome. This foreclosure incentive does nottake place in Sandonis and Faulì-Oller (2006), where the VI firm has gains fromthe market served by the rival firm, i.e., the VI firm prefers the duopoly (withvery differentiated goods) rather than a close-monopoly . The outside option forthe external patentee is πi (0, rj ; s, sψ) decreasing in ψ and increasing in rj . The

outside option for the internal patentee is π1 (0, 0; s, sψ) = ψ2s(4ψ−1)2 independent

of r and decreasing in ψ. For low ψ, both outside options are large. There is thesame negative effect of Sandonis and Faulì-Oller (2006) related to the externalpatentee and so vertical integration dominates vertical separation. For high ψ,both outside options are small, however also in this case vertical integrationdominates as the internal patentee approches the monopoly outcome.As for social welfare, in contrast with Sandonis and Faulì-Oller (2006) we

find that, under homogeneous goods, vertical integration is privately and so-cially profitable for high quality improvements. Vertical integration has twoopposite effects on prices: on one hand, VI pushes prices down as it impliesthe (partial) internalization of the vertical externality; on the other hand, VIhas an anticompetitive effect because the VI firm is able to (partially) foreclosethe rival firm via a positive per-unit royalty. The first effect prevails for ψ high(and we have pCV I − pT < 0), the opposite effect prevails for ψ low (and we havepCV I − pT > 0).Whenever the external patent holder does not have full bargaining, as one

could expect the private profitability of VI countinues to hold. However, from

19

the social welfare point of view, VI becomes detrimental. Indeed, in this caseunder vertical separation, complete technology diffusion and the duopoly equi-librium occurs for any value of ψ, which is always preferred to a close-monopolysolution, pCV I − pBa < 0.Under negative royalties the private and social incentives change. From a

private point of view, the U monopolist by vertically integrating (at least par-tially) internalizes the vertical externality, but on the other hand, it reducesthe set of licensing policies given that under VI it trasfers internally the inputat the marginal cost. At equilibrium vertical separation and exclusive licensingalways prevails because the Stackelberg equilibrium arises and the innovatorgets the first mover advantage. As for the social welfare, under vertical sepa-ration, the double marginalization problem disappears because the innovatingfirm is subsidized and the non-innovating firm buys the input from competitivefirms, however goods are vertically differentiated; in contrast under VI the mar-ket is characterized by a homogeneous high quality good, however the VI firmforecloses the rival and approches the monopoly outcome. In other words, theStackelberg equilibrium is always preferred with respect to a quasi-monopoly.Therefore private and social incentives match when negative royalties are al-lowed. We conclude that allowing for negative royalties is always welfare im-proving. Liao and Sen (2005) study licensing of a process innovation and showthat negative royalties are welfare improving with respect to non-negative roy-alties if the innovation is suffi ciently small. They point out the importance ofthe cost of exclusion: the additional cost that the non-licensee firm has to payunder exclusive licensing. As long as this cost is low, exclusive licensing (thatarises with negative royalties) is preferred with respect to complete technologydiffusion.

6 Bertrand competition

We next extend our analysis to Bertrand competition.21 This extension allowsus to analyse a post-innovation scenario with product differentiation. Indeedthe innovator has no incentive to sell its product innovation to both firms, asunder Bertrand competition with homogeneous goods, market profits are equalto zero and therefore he could not extract any surplus from the licensees.Ex-ante, as goods are homogeneous, equilibrium price is equal to marginal

cost, set to zero. Therefore, the market is covered, i.e., the demand is equal toone; firms make zero profit and social welfare coincide with consumer surplusthat is:

SWB (s) = CSB (s) =

1∫0

(θs) dθ =s

2.

Under Bertrand competition, the patent holder will sell the new input to onlyone firm (otherwise the Bertrand paradox applies). We thus analyse the optimal

21We focus on the case of full bargaining power as we expect the same differences to ariseunder partial bargaining power as in Cournot.

20

contract under exclusive licensing, considering in turn the case of an externalpatentee and the case of an internal patentee.Suppose as before that firm 1 is the non-innovating firm that produces a final

good of quality s1 = s at price p1; firm 2 is the innovating firm that produces afinal good of quality s2 = ψs > s, at price p2. Solving the game for the externalpatent holder we find that the optimal contract is:

rB2 = 0, FB2 =(ψ − 1) sψ

(4ψ − 1). (10)

A product innovation is sold via a fixed fee, this is due to the fact that therival firm has zero marginal cost.22 This result is in contrast with a processinnovation (Muto, 1993). Remaining equilibrium variables are:

qB1 =ψ

(4ψ − 1), qB2 =

2ψ

(4ψ − 1), QB =

3ψ

(4ψ − 1)(11)

pB1 =(ψ − 1) s

4ψ − 1, pB2 =

2ψs (ψ − 1)

4ψ − 1, (12)

π1 (s, sψ) = (ψ−1)sψ(4ψ−1)2 = π2 (sψ, s) = (ψ−1)sψ

(4ψ−1)2

ΠBU (s, sψ) = (ψ−1)sψ

(4ψ−1) − f.

PSB = (ψ−1)(4ψ+1)sψ(4ψ−1)2 − f (13)

ΠBU is the U patent holder profit under exclusive licensing, when selling via

a two-part tariff, which reduces to a fixed fee, the new input to only one Dfirm. For completeness we provide equilibrium consumer surplus under Bertrandcompetition:

CSB = (4ψ+5)sψ2

2(4ψ−1)2 . (14)

Consider next the case of an internal patentee, in particular assume that theU producer and firm 2 merge. The VI firm does not sell the innovation to therival so that we have an equilibrium such that the VI firm produces the highquality good at zero costs and compete with the rival firm 1 that produces thelow quality good at zero cost. Equilibrium quantities, prices and CS are as in(11), (12) and (14), equilibrium profits and producer surplus are:

πB1 = (ψ−1)sψ(4ψ−1)2 , π

BV I = 4 (ψ−1)sψ

2

(4ψ−1)2 ,

PSBV I = (ψ−1)(4ψ+1)sψ(4ψ−1)2 − f.

As for the merger private and social profitability under Bertrand competitionwe find that:

πBV I −(π2 (sψ, s)− FB2 + ΠB

U (sψ))

= 0, SWBV I − SWB = 0.

22 If the rival firm had a positive marginal cost of production, say c > 0, (so that theinnovation would include the product as well as the process) then the optimal contract would

be r2 = c2> 0, F2 =

4ψs2(ψ−1)+4csψ+c2(1−4ψ)4(4ψ−1)s .

21

The patent holder is indifferent between staying out of the market and verticallyintegrate. The same holds from the social welfare point of view.Extending the analysis to negative royalties would not change our results be-

cause under Bertrand competition, the U innovator has no incentive to subsidizethe downstream production.The above results are gathered in the following proposition.

Proposition 4 Under Bertrand competition the product innovation is nevercompletely diffused. (i) An external patent holder prefers to sell an exclusivelicensing via a fixed fee rather than a royalty (the optimal contract is specified in(10)). (ii) An internal patent holder does not license its innovation. (iii) Theequilibrium prices, quantities and social welfare are independent of whether thepatent holder stays out of the market or vertically integrate with either firm.

We conclude our analysis by comparing the private and social profitabilityof Cournot vs Bertrand competition. In case of non-negative royalties, underCournot competition, at equilibrium vertical integration takes place and tech-nology diffusion is complete, so that we have a homogeneous high quality goodin the market; in constrast under Bertrand competition, at equilibrium exclu-sive licensing occurs, so that the market is vertically differentiated. We find thatfrom the firms’point of view, the producer surplus is higher under Cournot thanunder Bertrand competition, namely:23

ΠCV I − πBV I =

((16ψ2−13ψ+1)sψ

4(4ψ−1)2 − 4 (ψ−1)sψ2

(4ψ−1)2

)=

sψ

4 (4ψ − 1)2 (3ψ + 1) > 0,

PSCV I − PSBV I =sψ

4 (4ψ − 1)2 (3ψ + 5) > 0.

As for consumer surplus and welfare, the comparison depends on the qualityimprovement, in particular total social welfare is higher under Cournot thanunder Bertrand competition for ψ suffi ciently high:

SWCV I − SWB =

(11ψ−21ψ2−2ψ

32+8ψ

52

)s

8(4ψ−1)2 > 0 ⇐⇒ ψ > 6.28

This result is clearly linked to the fact that on one hand, under Cournot,competition is milder than under Bertrand where both qualities stays on sale; onthe other hand, under Cournot, complete technology diffusion arises so that theaverage quality of the final good is higher than under Bertrand competition. Inthis static setting traditional conclusions about price and quantity competition(Singh and Vives,1984) may be reversed. This is in line with Arya, et al. (2008)that show that the vertically integrated producer may set a higher input priceunder Bertrand competition than under Cournot competition. In our frameworkhowever this result relies on the technology diffusion incentives which in turndetermine the average quality of the final good.

23This is in line with Singh and Vives (1984).

22

Finally, comparing the private and social profitability of Cournot vs Bertrandcompetition in the light of negative royalties, we find that Cournot always pre-vails over Bertrand competition. Under negative royalties, exclusive licensingarises in both competition modes, therefore the quality improving effect whichis at work with non-negative royalties when complete technology diffusion takesplace under Cournot is absent. However, under Cournot competition the equi-librium outcome coincides with Stackelberg and Cournot competition is moreprofitable than Bertrand. In particular:24

PSELneg − PSB =1

16s (16ψ − 1)− (ψ−1)(4ψ+1)sψ

(4ψ−1)2 =(40ψ−96ψ2+192ψ3−1)s

16(4ψ−1)2 > 0

CSELneg − CSB =1

32(4ψ + 5) s− (4ψ+5)sψ2

2(4ψ−1)2 =

(− 1

32

)(4ψ − 1)

−2(8ψ − 1) (4ψ + 5) s < 0

SWELneg − SWB =3

32s (12ψ + 1)−

((4ψ+5)sψ2

2(4ψ−1)2 + (ψ−1)(4ψ+1)sψ(4ψ−1)2

)=

(44ψ−224ψ2+384ψ3+3)s32(4ψ−1)2 > 0

The producer surplus is higher under Cournot than under Bertrand, and thisis in line with traditional results. Consumer surplus is instead higher underBertrand, this is also in line with the idea that Bertrand implies tougher com-petition than Cournot. However, overall, Cournot is preferred with respect toBertrand: the industry profitability overcompensate the loss of consumer sur-plus.

7 Conclusion

We have analysed the optimal licensing strategy of an upstream input innova-tor producing a new input which improves the quality of the final goods. Wehave considered a duopoly downstream market and two-part tariff contractswith non-negative fixed fees and either non-negative per-unit royalties or per-unit subsidies. In case of non-negative royalties, we have shown that, underCournot competition complete technology diffusion takes place and the inno-vator always prefers to be inside the market as the vertical merger with eitherdownstream firm is always privately profitable. It is also welfare improving forlarge innovations. Extending the analysis to the case in which the externalpatent holder does not have full bargaining power, the private profitability ofvertical integration continues to hold, whereas vertical integration turns out tobe welfare detrimental. Under Bertrand competition, in contrast with Cournot,complete technology diffusion never takes place and we find an equivalence re-sult between vertical integration and vertical separation from both the privateand social welfare point of view. Allowing for per-unit subsidies completelyreverse our results. Exclusive licensing takes place under both modes of compe-tition (Bertrand and Cournot), and vertical integration becomes privately and

24QB −QEL = 3ψ(4ψ−1) −

34

= 34

(4ψ − 1)−1 > 0, qB1 −14

= ψ(4ψ−1) −

14

= 14

(4ψ − 1)−1 >

0, qB2 −12

= 2ψ(4ψ−1) −

12

= 12

(4ψ − 1)−1 > 0.

23

socially unprofitable. Our conclusion is that negative royalties are always wel-fare improving. Even though per-unit subsidies are rarely observed in licensingcontracts, there are examples in the personal computer industry that might fitthis scenario. Gawer and Henderson (2007) explore Intel’s strategy with re-spect to complements. They find that Intel, as provider of microprocessors, anessential input of the personal computer, may have the incentive to subsidizecomplements’production by the development and widespread dissemination ofintellectual property.In the licensing game we have assumed that firms’outside option in case of an

external patent holder and complete technology diffusion is such that if the firmdoes not get the innovation the rival firm does. This means that the outsideoption depends on the royalty rate. Things are much different if we assumethat either both firms get the innovation or neither does.25 Indeed, in this casethe outside option is constant with respect to the per-unit royalty so that theupstream innovator via a two-part tariff contract is able and finds it profitableto implement the monopoly outcome. This result has been proved by Inderstand Shaffer (2009) studying optimal two-part tariff contracts in vertical relationswith independent asymmetric firms. Li and Wang (2010) obtains the same resultfor the specific framework of patent licensing. Clearly, the vertical integrationwould not be privately profitable anymore as under vertical separation industryprofit would be maximal. In contrast, vertical integration would result to bewelfare improving as a quasi monopoly would be better than a monopoly.

References

[1] Arya,A. and Mittendorf, B., 2006, Enhancing vertical effi ciency throughhorizontal licensing, Journal of Regulatory Economics, 29: 333-342.

[2] Arya, A., Mittendorf, B. and Sappington, D., 2008, Outsourcing, verticalintegration, and price vs. quantity competition, International Journal ofIndustrial Organization, 26, 1-16.

[3] Bonanno, G. 1986, Vertical differentiation with Cournot competition, Eco-nomic Notes 15 (2), 68-91.

[4] Bork, R.H., 1978, The antitrust paradox: a policy at war with itself. BasicBooks. New York.

[5] Erutku, C. and Richelle, Y., 2007, Optimal licensing contracts and the valueof a patent, Journal of economics and management strategies16, 407-436.

[6] European Commission 2008, Guidelines on the assessment of non-horizontalmergers under the Council Regulation on the control of concentrationsbetween undertakings, Offi cial Journal of the European Union, 18.10.2008.

25Somewhat in line with Gonzales and Ayala (2012) this could be a scenario where thedownstream firms act cooperatively when purchasing the new input from the upstream inno-vator.

24

[7] Fershtman and Judd, 1987, Equilibrium incentives in oligopoly.The Amer-ican Economic Review, 77, 927-940.

[8] Gawer, A. and Henderson, R., 2007. Platform Owner Entry and Innovationin Complementary Markets: Evidence from Intel, Journal of Economics &Management Strategy, 16: 1—34.

[9] Gonzàles, A. and Ayala, L. 2012, Does input purchase cooperation fosterdownstream collusion? SDT 358, Universidad de Chile.

[10] Inderst, R. and Shaffer, G. 2009, "Market power, price discrimination, andallocative effi ciency in intermediate-goods markets," Rand Journal of Eco-nomics, 40: 658-672.

[11] Kamien, M.I. and Tauman, Y. 1986, Fees versus royalties and the privatevalue of a patent. Quarterly Journal of Economics 101, 471-492.

[12] Kamien, M.I., Oren, S.S., and Tauman Y., 1992, Optimal licensing of cost-reducing innovation, Journal of Mathematical Economics 21, 483-508.

[13] Kamien, M. Tauman, Y. and Zang, I., 1988, Optimal License Fees for aNew Product, Mathematical Social Sciences, 16, 77-106.

[14] Katz, M., Shapiro, C., 1986, How to license intangible property, QuarterlyJournal of Economics 101, 567-589.

[15] Lemarie, S., 2005, Vertcal integration and the licensing of innovation witha fixed fee or a royalty. Working paper GAEL 2005-17.

[16] Li, C. and Wang J., 2010, Licensing a vertical product innovation, Eco-nomic Record, 86: 517-527.

[17] Liao C., and Sen D. (2005), Subsidy in licensing: optimality and welfareimplications, The Manchester School, 73, 281—299.

[18] Milliou C. and Petrakis, E. (2007), Upstream horizontal mergers, verticalcontracts, and bargaining, International Journal of Industrial Organization25, 963—987.

[19] Muto, S., 1993, On Licensing Policies in Bertrand Competition, Games andEconomic Behavior 5 , 257-267.

[20] Posner, R.A., 1976, Antitrust law, University of Chicago Press, Chicago.

[21] Reisinger and Tarantino (2012), Vertical integration with complementaryinputs

[22] Rey, P. and J. Tirole, 2007, A Primer on Foreclosure, in M. Armstrong andPorter, R.(eds.),Handbook of Industrial Organization, vol.3, pp. 2145-2220,Elsevier, Amsterdam.

25

[23] Contribution to a book:

[24] Rey, P. and T. Vergé, 2008, Economics of Vertical Restraints, in Handbookof Antitrust Economics, chapter 9, The MIT Press, 353-390.

[25] Rostoker, M.D., 1984. A survey of corporate licensing. IDEA: The Journalof Law and Technology 24, 59-92.

[26] Sandonis, J. and R. Faulì-Oller, 2006, On the competitive effects of verticalintegration by a research laboratory, International Journal of IndustrialOrganization 24, pp.715-731.

[27] San Martìn, M. and A.I. Saracho, 2010, Royalty licensing, Economics Let-ters 107, 284-287.

[28] Sen, D. and Tauman,Y., 2007, General licensing schemes for a cost-reducinginnovation, Games and Economic Behavior 59, 163-186.

[29] Shi, G. and Chavas J., 2011, The effects of vertical organization on thepricing of differentiated products. Journal of Agricultural and ResourceEconomics, 36: 448-464.

[30] Singh, N. and X. Vives, 1984, Price and Quantity Competition in a Differ-entiated Duopoly, RAND Journal of Economics, 15, 546-554.

[31] Stamatopoulos, G. and Tauman, Y. (2008), Licensing of a quality-improving innovation, Mathematical Social Sciences, 56, 410-438.

8 Appendix

8.1 Exclusive licensing subgame

Given demands (5) and (6), Cournot competition leads to the following thirdstage quantity and price equilibrium:

q1 (0, r2; s, sψ) =sψ + r2s (4ψ − 1)

,

q2 (r2, 0; sψ, s) =(2sψ − s− 2r2)

s (4ψ − 1),

p1 (0, r2; s, sψ) =(sψ + r2)

(4ψ − 1),

p2 (r2, 0; sψ, s) =(2ψ − 1) (sψ + r2)

(4ψ − 1),

with q2 (r2, 0; sψ, s) > 0 ⇐⇒ s(2ψ−1)2 > r2. Firm1 profit is then:

π1 (0, r2; s, sψ) =(sψ + r2)

2

(4ψ − 1)2s

(15)

26

As for firm 2:

π2 (r2, 0; sψ, s) =ψ (s− 2sψ + 2r2)

2

(4ψ − 1)2s

The U firm chooses the two-part tariff contract for firm 2 (r2, F2) such that:

maxr2,F2

ΠU

s.t.r2 ∈[0,s (2ψ − 1)

2

]F2 ≤ π2 (r2, 0; sψ, s)− s

9

with ΠU = (2sψ−s−2r2)s(4ψ−1) r2 + F2 − f . The first constraint comes from the non-

negativity of q2 and the second constraint (binding at equilibrium) ensures thatfirm 2 has the incentive to get the license rather than the status quo. The

solution is a contract such that rEL2 = 0 and FEL2 =(36ψ2−16ψ+1)(ψ−1)s

9(4ψ−1)2 .26

8.2 Complete technology diffusion subgame

The U firm maximization problem is:

maxF1,r1,F2,r2

{r1q1 (r1, r2; sψ, sψ) + r2q2 (r1, r2; sψ, sψ) + F1 + F2 − f} (16)

s.t.π1 (r1, r2; sψ, sψ)− F1 ≥ π1 (0, r2; s, sψ)

π2 (r2, r1; sψ, sψ)− F2 ≥ π2 (0, r1; s, sψ)

r1 ≥ 0, r2 ≥ 0, F1 ≥ 0, F2 ≥ 0

where π1 (0, r2; s, sψ) is defined in (15).27 Here the outside option for each firmis not buying the new input given that the rival firm does. πi (ri, rj ; sψ, sψ) andqi (ri, rj ; sψ, sψ) denote the third stage equilibrium D firm i profit and quantitywhen both firms produce the high quality good, namely:

πi (ri, rj ; sψ, sψ) =(sψ − 2ri + rj)

2

9ψs, (17)

qi (ri, rj ; sψ, sψ) =1

3sψ(sψ − 2ri + rj) . (18)

p (ri, rj ; sψ, sψ) = sψ

(1− 1

3sψ(sψ − 2ri + rj)−

1

3sψ(sψ − 2rj + ri)

)26This is because ∂/∂r2

((2sψ−s−2r2)s(4ψ−1) r2 + π2 (r2, 0; sψ, s)− s

9− f

)< 0, that is the incen-

tive is to set a royalty as low as possible. Given the non-negative constraint, the solution isr = 0.27Following Sandonis and Faulì-Oller (2006), we restrict our attention to non-negative roy-

alties and fixed fees.

27

As the two constraints are binding at equilibrium, we have

F1 (r1, r2) = π1 (r1, r2; sψ, sψ)− π1 (0, r2; s, sψ) ,

F2 (r1, r2) = π2 (r2, r1; sψ, sψ)− π2 (0, r1; s, sψ) ,

with πi (0, rj ; s, sψ) =(sψ+rj)

2

(4ψ−1)2s . The maximization problem, thus becomes:

maxr1,r2{r1q1 (r1, r2; sψ, sψ) + r2q2 (r2, r1; sψ, sψ) + F1 (r1, r2) + F2 (r1, r2)− f} .

Comparing total quantities, prices, profits and consumer surpluses, underexclusive licensing vs. complete technology diffusion, we find the following rank-ings.

for ψ > 1.585 6 QEL > Q > QT , π2 (sψ, s) > πT (sψ)− F ∗ (r∗, r∗) > π1 (s, sψ) ,for ψ < 1.585 6 QEL > Q = QT , π2 (sψ, s) > πT (sψ)− F ∗ (r∗, r∗) = π1 (s, sψ)∀ψ pEL1 < p < pT < pEL2 , CST > CSEL > CS, SWT > SWEL

8.3 Bargaining

Consider the bargaining process of U with firm 1, given the outcome of thebargaining process of U with firm 2 (r2, F2). The objective function is:

(r1q1 (r1, r2; sψ, sψ) + r2q2 (r1, r2; sψ, sψ) + F1 + F2 − r2q2 (r2, 0; sψ, s)− F2)α

(π1 (r1, r2; sψ, sψ)− F1 − π1 (0, r2; s, sψ))1−α

Solving by backward induction, we first maximize with respect to F1 and we findF1 (r1, r2) = A(r1,r2)+αB(r1,r2)

9ψs(4ψ−1)2 , whereA (r1, r2) = 3 (4ψ − 1)((−2)

(r21 − r1r2 + r22

)+ ψ22s (r2 − 2r1) + ψ

(sr1 − 2sr2 − 8r1r2 + 8r21 + 2r22

))andB (r1, r2) = 2r1r2−2r21−5r22+16s2ψ4+ψ3s (8r2 − 16r1 − 17s)+ψ

(16r21 − 4sr2 − 16r1r2 − sr1 + 13r22

)+

ψ2(8sr1 − 4sr2 + 32r1r2 + s2 − 32r21 − 8r22

). We next substitute this solution in

the objective function and maximize with respect to r1. The symmetric solutionr1 = r2 is:28

rB =s

(8ψ − 5) (ψ − 1)

(ψ

(−1

2

)(8ψ2 − 4ψ + 5

)− 3

2

√ψ2(4ψ2 − 8ψ + 5

)(4ψ − 1)

2

)< 0

So that we conclude that the equilibrium royalty is rB = 0.

28The sign of the derivative w.r.t. r1 is the sign

of (4ψ−1)2(2r+sψ)ψ2s2(1−16ψ)(ψ−1)+r2(8ψ−5)(ψ−1)+rψs(8ψ2−4ψ+5)

> 0 ⇐⇒ s2 (1− 16ψ) (ψ − 1) +

r2 (8ψ − 5) (ψ − 1) + rψs(8ψ2 − 4ψ + 5

)> 0 ⇐⇒ r < r−, r > r+, with

r− = s(8ψ−5)(ψ−1)

(ψ(− 12

) (8ψ2 − 4ψ + 5

)− 3

2

√ψ2(4ψ2 − 8ψ + 5

)(4ψ − 1)2

)< 0

and r+ = s(8ψ−5)(ψ−1)

(ψ(− 12

) (8ψ2 − 4ψ + 5

)+ 3

2

√ψ2(4ψ2 − 8ψ + 5

)(4ψ − 1)2

)> 0.

28

8.4 Vertical integration

Consider the quantity competition between the VI firm and firm 1 producingthe high quality final good. The VI firm has zero variable production costs asthe new input is trasferred at the marginal cost c2 = 0, whereas firm 1 incursmarginal cost r1. The third stage equilibrium quantities and profits are:

qV I (0, r1; sψ, sψ) =1

3sψ(sψ + r1)

q1 (r1, 0; sψ, sψ) =1

3sψ(sψ − 2r1) ≥ 0 ⇐⇒ r1 ≤

sψ

2

pV I = sψ (1− qV I (0, r1; sψ, sψ)− q1 (r1, 0; sψ, sψ)) =1

3(sψ + r1)

πV I (0, r1; sψ, sψ) =(sψ + r1)

2

9sψ− f

π1 (r1, 0; sψ, sψ) =(sψ − 2r1)

2

9ψs

where qV I (0, r1; sψ, sψ) and q1 (r1, 0; sψ, sψ) are obtained from expression (18)substituting properly ri and rj ; πV I (0, r1; sψ, sψ) and π1 (r1, 0; sψ, sψ) are ob-tained from expression (17) substituting properly ri and rj . The VI firm offersfirm 1 the two-part tariff contract (r1, F1) such that:29

maxr1,F1

{πV I (0, r1; sψ, sψ) + r1q1 (r1, 0; sψ, sψ) + F1}

s.t.π1 (r1, 0; sψ, sψ)− F1 ≥ π1 (0, 0; s, sψ)

r1 ∈[0,sψ

2

],

where π1 (0, 0; s, sψ) = ψ2s(4ψ−1)2 , obtained from (15) is firm 1 outside option,

that is firm 1 profit if it does not buy the new input thus selling the low qualityfinal good and incurring per-unit cost equal to zero. As the first constraint isbinding at equilibrium, we have:

maxr1{πV I (0, r1; sψ, sψ) + r1q1 (r1, 0; sψ, sψ) + π1 (r1, 0; sψ, sψ)− π1 (0, 0; s, sψ)}

The optimal contract is then:

r∗1 =sψ

2, F ∗1 = − ψ2s

(4ψ − 1)2

If we let the VI firm to set negative fees, the vertical merger implements themonopoly outcome by inducing the nonintegrated firm to produce a nil quantity

29 In the following maximization problem we do not restrict the VI firm to set nonnegativefees. This allows us to make clear its incentives. We next solve the maximization problemconstrained to nonnegative fees.

29

(foreclosure) and compensating it for the outside option. Equilibrium magni-tudes are:

qV I (0, r∗1 ; sψ, sψ) =1

2= qm

q1 (r∗1 , 0; sψ, sψ) = 0

pV I =1

2ψs = pm

πV I (0, r∗1 ; sψ, sψ) =1

4ψs

π1 (r∗1 , 0; sψ, sψ)− F ∗1 =ψ2s

(4ψ − 1)2

ΠV I = πV I (0, r∗1 ; sψ, sψ)− f + r∗1q1 (r∗1 , 0; sψ, sψ) + F ∗1 =1

4ψs− ψ2s

(4ψ − 1)2 − f

However negative fees would be clearly held to be illegal by antitrust authorities.It is clear from the analysis above that the VI firm wants to restrict as muchas possible the quantity produced by the non affi liate firm so as to (at least)partially internalize the vertical externality.If the VI is constrained to nonnegative fees, it will optimally let the nonaf-

filiate firm to produce a positive quantity q1 (r1) : π1 (r1) = ψ2s(4ψ−1)2 . Given the

Cournot equilibrium quantities, we have(1

3(sψ + r1)− r1

)1

3sψ(sψ − 2r1) =

ψ2s

(4ψ − 1)2

⇐⇒ r1 =

(ψs (4ψ − 1)− 3sψ

√ψ)

2 (4ψ − 1).

8.5 Ad valorem royalty

Suppose (under Corunot competition) the internal patentee (the VI firm) de-cides to sell the innovation via an ad valorem royalty, that is a profit sharingagreement. Firms’profits are then: πPS1 = (1− β) pq1, πPSV I = pqV I + βpq1where β ∈ [0, 1] is the ad valorem royalty. The equilibrium ad valorem royaltyis the β such that the licensee is indifferent between buying and not buying theinnovation, i.e.:

β : (1− β)sψ

9=

ψ2s

(4ψ − 1)2

⇐⇒ β∗ =(ψ − 1) (16ψ − 1)

(4ψ − 1)2 ∈ (0, 1) .

Industry profit is 2 sψ9 , πPSV I = sψ

9 + (ψ−1)(16ψ−1)(4ψ−1)2

sψ9 andΠ∗V I−πPSV I = (ψ−1)(16ψ−1)sψ

36(4ψ−1)2 >

0.

30

8.6 Social welfare comparisons

For completeness we provide equilibrium social welfare under exclusive licensingand non-negative royalties:

SWEL = PSEL+CSEL =

((36ψ2−16ψ+1)(ψ−1)s

9(4ψ−1)2 + ψ2s(4ψ−1)2 +

s

9

)+

(ψ+4ψ2−1)sψ2(4ψ−1)2

where CSEL =

p2−p1s(ψ−1)∫p1s

(θs− p1) dθ +

1∫p2−p1s(ψ−1)

(θsψ − p2) dθ. The remaining social

welfare comparisons are: for ψ < 1.585 6, SWT−SWEL = s(29ψ +

(2 ψ2

(4ψ−1)2 + 2 (ψ−1)(16ψ−1)ψ9(4ψ−1)2

))−

s

(((36ψ2−16ψ+1)(ψ−1)

9(4ψ−1)2 + ψ2

(4ψ−1)2 + 19

)+

(ψ+4ψ2−1)ψ2(4ψ−1)2

)> 0; for ψ > 1.585 6,

SWT − SWEL = s

(ψ

(4ψ+16ψ2+1)2

2(32ψ2−7ψ+2)2 +

(2 25(4ψ−1)2ψ24(32ψ2−7ψ+2)2 +

(16ψ2−16ψ+1)ψ2(32ψ2−7ψ+2)

))−

s

(((36ψ2−16ψ+1)(ψ−1)

9(4ψ−1)2 + ψ2

(4ψ−1)2 + 19

)+

(ψ+4ψ2−1)ψ2(4ψ−1)2

)> 0. SWC

V I−SWEL =

s

((4ψ+

√ψ−1)

2ψ

8(4ψ−1)2 + ψ2

(4ψ−1)2 +(16ψ2−13ψ+1)ψ

4(4ψ−1)2

)−s((

(36ψ2−16ψ+1)(ψ−1)9(4ψ−1)2 + ψ2

(4ψ−1)2 + 19

)+

(ψ+4ψ2−1)ψ2(4ψ−1)2

).

For high values of ψ (ψ > 3.407 8), social welfare ranking is SWCV I > SWT >

SWEL. For ψ < 3.407 8, we have SWT > SWCV I and SW

T > SWEL.In case of negative royalties,

SWELneg − SWT =3

32(4ψ−127ψ2+1548ψ3−3840ψ4+8192ψ5+4)s

(32ψ2−7ψ+2)2 > 0

CSEL − CST =5

32(129ψ2−28ψ−460ψ3+256ψ4+4)s

(32ψ2−7ψ+2)2 > 0 ⇐⇒ ψ > 1.505 9

PSELneg − PST =(76ψ−513ψ2+3472ψ3−6400ψ4+12 288ψ5−4)s

16(32ψ2−7ψ+2)2 > 0

8.7 Bertrand competition

The demands for the goods are the same as in (5) and (6), in particular:

q1 =p2 − p1s (ψ − 1)

− p1s, q2 = 1− p2 − p1

s (ψ − 1).

D firms profits are:

π1 = p1q1, π2 = (p2 − r2) q2 − F2.

31

Bertrand competition leads to the following third stage prices and quantity equi-librium:

p1 (0, r2; s, sψ) =sψ − s+ r2

4ψ − 1,

p2 (r2, 0; sψ, s) =2ψ (sψ − s+ r2)

4ψ − 1

q1 (0, r2; s, sψ) =(sψ − s+ r2)ψ

(ψ − 1) (4ψ − 1) s

q2 (r2, 0; sψ, s) =2ψs (ψ − 1)− r2 (2ψ − 1)

(ψ − 1) (4ψ − 1) s

with q2 (r2, 0; sψ, s) > 0 ⇐⇒ r2 <2ψs(ψ−1)(2ψ−1) .

30 Firm 1 profit is then:

π1 (0, r2; s, sψ) =(sψ − s+ r2)

2ψ

(4ψ − 1)2

(ψ − 1) s

As for firm 2:

π2 (r2, 0; sψ, s) =(2ψs (ψ − 1)− r2 (2ψ − 1))

2

(4ψ − 1)2

(ψ − 1) s− F2

The U firm chooses the two-part tariff contract for firm 2 (r2, F2) such that:

maxr2,F2

ΠU

s.t.r2 ∈[0,

2ψs (ψ − 1)

(2ψ − 1)

]F2 ≤ (2sψ−r2+2ψr2−2sψ2)

2

(4ψ−1)2(ψ−1)s − (sψ−s+r2)2ψ(4ψ−1)2(ψ−1)s

with ΠU = 2ψs(ψ−1)−r2(2ψ−1)(ψ−1)(4ψ−1)s r2 + F2 − f . The first constraint comes from the

non-negativity of q2 and the second constraint (binding at equilibrium) ensuresthat firm 2 has the incentive to get the license rather than the outside option,that we assume to be equal to π1 (0, r2; s, sψ) > 0.31 ∂

∂r2ΠU = − 2r2ψ

s(ψ−1)(4ψ−1) ≥0 ⇐⇒ r2 ≤ 0.

30This inequality is more stringent than the corresponding one under Cournot competition.Namely, s(2ψ−1)

2>

2ψs(ψ−1)(2ψ−1) .

31 In fact firm 2 can always refuse the offer of the patent holder knowing that he will makethe offer to the rival firm. Therefore the outside option is not zero (the status quo profit),rather it is positive and equal to the low quality firm’s profit.

32