Licensing under convex costs

J EconDOI 10.1007/s00712-013-0333-9

Licensing under convex costs

Arijit Mukherjee

Received: 19 May 2011 / Accepted: 9 January 2013© Springer-Verlag Wien 2013

Abstract We show that both the outside and inside innovators license a new product(or drastic process innovation) to all potential licensees in the presence of convexcosts, which occur under decreasing returns to scale technologies. An implicationof our analysis is that a monopolist producer may prefer technology licensing in ahomogeneous goods industry. We also show that an inside innovator’s incentive forinnovation may be higher than that of an outside innovator.

Keywords Licensing · Innovation · Convex cost · Auction · Royalty

JEL Classification D21 · D43 · D45 · L13

1 Introduction

Technology licensing, which helps the innovators to increase their revenues, is verycommon in today’s world. Anand and Khanna (2000) show that licensing occurs inseveral sectors such as chemicals, biotechnology, software, and electrical and non-electrical industries. The evidences of licensing in the chemical and computer indus-tries can also be found in Grindley and Nickerson (1996) and Rivette and Kline (1999)respectively. Moreover, as mentioned in Arora and Fosfuri (2003), often a firm licensestechnology to its potential competitors.

A. Mukherjee (B)School of Business and Economics, Sir Richard Morris Building,Loughborough University, Loughborough, Leicestershire LE11 3TU, UKe-mail: [email protected]

A. MukherjeeCESifo, Munich, Germany

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The seminal papers by Kamien and Tauman (1984) and Kamien and Tauman (1986)consider licensing by an outside innovator, there are latter works (Gallini and Winter1985; Rockett 1990a; Sen and Tauman 2007) showing licensing by an inside innova-tor.1 The general conclusion of this literature is that, if the products are homogeneous,full knowledge diffusion does not occur under drastic innovations2 (where the oppor-tunity costs of the licensees are zero), irrespective of outside or inside innovator.3 Theoutside innovator auctions only one license4 and the inside innovator does not license;however, full knowledge diffusion generally occurs under non-drastic innovation, andthe outside and the inside innovators license to all (or most of the) licensees (Sen andTauman 2007). We show that these results on drastic innovations may be the artefactof constant returns to scale technologies. We show that full knowledge diffusion of adrastic innovation occurs in the case of decreasing returns to scale technologies.

We consider licensing of a new product in the presence of convex costs, whichoccurs under decreasing returns to scale technologies. Licensing of a new product isthe simplest way to capture the situation of zero opportunity costs of the licensees,which occur under licensing of a drastic process innovation. We show that full knowl-edge diffusion occurs—for both outside and inside innovators. An implication of ouranalysis is that a monopolist final goods producer has the incentive for technologylicensing in a homogeneous goods industry with convex costs even if that createsproduct-market competition. Our result on outside innovator suggests that if the inno-vator is not capable of producing the innovated product, it may not prefer to create amonopoly of that product in the presence of convex costs.

We also show that an inside innovator’s incentive for innovation may be higher thanthat of an outside innovator. More firms under inside innovator compared to outsideinnovator tend to reduce the total revenue for the former innovator than the latterinnovator by creating higher product-market competition. However, more firms underinside innovator tend to reduce the total costs compared to an outside innovator byspreading the outputs over more firms. If the diseconomies of scale are sufficientlylarge, so that the cost saving through output diversification is very important, thelatter effect dominates the former and the inside innovator’s incentive for innovationis higher than that of the outside innovator.

Our result on inside innovator complements the literature showing the rationalefor licensing by a monopolist producer. The existing reasons provided so far for thisphenomenon are demand expansion (Shepard 1987; Farrell and Gallini 1988; Wangand Yang 2003), choosing competitor (Rockett 1990b), strategic tariff (Mukherjeeand Pennings 2006), product differentiation (Wang and Yang 1999; Mukherjee andBalasubramanian 2001), network externalities (Economides 1993), entry deterrence

1 Inside (outside) innovator implies that the innovator is (is not) a producer of the product.2 See Arrow (1962) for discussions on drastic and non-drastic innovations.3 See Sen and Tauman (2007) for an extensive review of this literature. See, Kamien (1992) and Mukherjee(2009) also for surveys on technology licensing.4 Sen and Stamatopoulos (2009a) show that an outside innovator earns the same profit from selling onelicense or multiple licenses of a drastic innovation.

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(Gallini 1984; Eswaran 1994)5 and vertical pricing (Mukherjee et al. 2008). Our reasonis based on diseconomies of scale, thus providing a different rationale for licensing bya monopolist producer.

Our reason for multiple licensing is different from that of Arora and Fosfuri (2003).They show that significant knowledge diffusion through licensing occurs if there aremultiple licensers. The trade-off between the revenue effect and the rent dissipationeffect creates the incentive for multiple licensing in their analysis. In contrast, multiplelicensing in our analysis helps cost reduction in the industry by diversifying outputsamong several firms. This benefit from cost reduction creates the incentive for multiplelicensing in our analysis.

The implications of convex costs were discussed extensively in the Industrial Orga-nization literature (Tirole 1988), yet the licensing literature did not pay much atten-tion to this aspect. Sen and Stamatopoulos (2009b) show the implications of scaleeconomies on licensing by an outside innovator with two potential licensees. Theyshow that the innovator may license a drastic innovation to both the licensees. Hence,our analysis of the outside innovator case extends their analysis to n licensees, where ngo beyond two. Besides this, our analysis shows the implication of decreasing returnsto scale technology on the licensing strategy of an inside innovator. Further, we exam-ine the incentive for innovation by the outside and inside innovators.

Considering constant returns to scale technologies, Sen and Stamatopoulos (2009a)show that an outside innovator may license a drastic innovation to all potentiallicensees. However, the innovator earns the same profit from selling one license ormultiple licenses of a drastic innovation. Hence, licensing to one licensee weaklydominates licensing to multiple licensees.6 In contrast, we show that licensing to alllicensees is the strictly dominant strategy of an outside innovator. Moreover, we deter-mine the licensing strategy of an inside innovator and also compare the incentive forinnovation of the outside and inside innovators.

The motivation of our paper is theoretical and it comes from the limitation of Sen andTauman (2007) focusing on the constant returns to scale technologies. Their qualitativeresults under drastic innovation differ significantly in the presence of decreasing returnsto scale technologies. Although our paper is theoretical in nature and considers thecase of a drastic innovation, our results may still provide explanations for the licensingcontracts with downpayment and royalty, which is the dominant licensing contractin many industries. For example, looking at the USA industries, Rostoker (1984)points out that most of the licensing contracts in the transportation industry involvedownpayment and royalties.7 Looking at the British industries, Taylor and Silberston(1973) mention that the licensing contracts in basic and bulk chemicals and in the(non-electrical) plant and machinery field involve combinations of lump-sum androyalties.

5 In contrast to licensing, Gabszewicz and Tarola (2012) consider acquisition as an alternative strategy forpreventing competition. Marjit et al. (2000) compare profitability of licensing and acquisition in a Cournotoligopoly.6 A cost of licensing the technology will make selling one license as the innovator’s optimal policy.7 Decreasing returns to scale can be observed in the transportation sector (Xu et al. 1994).

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We acknowledge that not all the innovations are drastic and scale economiesare one of the several factors affecting licensing strategies. However, like Sen andStamatopoulos (2009b), our paper certainly points out that it is important to lookat the technological characteristics of the industries while determining the licensingstrategy, since the licensing strategies determined so far under the constant returns toscale technologies may not be valid always. Hence, empirical works uncovering theeffects of scale economies on the licensing strategies deserve more attention.

In sum, the contributions of this paper are as follows. In contrast to the existingliterature, it shows that full knowledge diffusion of a new product (or drastic processinnovation) occurs under both outside and inside innovators in the presence of decreas-ing returns to scale technologies. The existence of positive fixed-fee and output royaltyin the equilibrium licensing contract provides an explanation for the most commonlyobserved licensing contracts in several industries even for licensing a new product.Finally, in contrast to the existing literature, it shows that an inside innovator’s incen-tive for innovation can be higher than an outside innovator.

It is arguable that if there are plant level diseconomies of scale, a firm may preferto operate multiple plants if the costs of setting up multiple plants are not very high.Since we consider that each firm uses one plant, it is implicit in our analysis that thecosts of setting up more than one plant are prohibitive.8

The remainder of the paper is as follows. Sections 3 and 4 consider the casesof outside and inside innovators. Section 4 compares the incentive for innovationunder outside and inside innovators. Section 5 discusses welfare implication. Section 6concludes.

2 The outside innovator

Consider an innovator, called I , which invented a new product. However, I cannotproduce the product. There are n ≥ 1 symmetric potential licensees, which can producethe product if obtain license from I . To capture zero opportunity costs of the licensees(which occur under drastic process innovations) in the simplest way, we assume thatthe potential licenses have no existing technologies for the product invented by theinnovator. Like Sen and Tauman (2007), we assume that I licenses the technology tothe potential licensees through “auction plus royalty” where I determines the numberof licenses to auction (possibly with a minimum bid) and also announces the level ofroyalty so that the up-front fixed-fee that a licensee pays is its winning bid.

The outputs of the licensees are perfect substitutes, and the inverse market demandfunction for the product is

P = a − q, (1)

where P is price and q is the total output.

8 Similar assumption can be found in the literature on ex-port-platform foreign direct investment, wherethe high cost of opening up a new plant prevents a firm from operating multiple plants (Fumagalli 2003).

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We assume that, if granted license, the i th licensee can produce the product of the

innovator at the total cost of Ci = cq2i

2 , i = 1, 2, . . . , n. If c = 0, our analysis issimilar to the constant marginal cost case of Sen and Tauman (2007). The licenseescannot produce the product if not granted license.

We consider the following game. At stage 1, I announces k number of licensesto auction through a sealed bid English auction, where 1 ≤ k ≤ n. At stage 2, Idetermines the output royalty, r, that a licensee needs to pay. At stage 3, the licenseessimultaneously and independently decide whether or not to purchase a license, andhow much to bid under auction. The highest bidders get license. The ties are resolvedrandomly. At stage 4, the potential licensees, which get the license, choose their outputssimultaneously. If only one potential licensee obtains the license, he produces like amonopolist. We solve the game through backward induction.

If I auctions k licenses, where 1 ≤ k ≤ n, and charges the per-unit output royalty r ,the i th licensee, i = 1, 2, . . . , k, determines his output to maximise the followingexpression:

Maxqi

(a − q − r)qi − cq2i

2− Fi , (2)

where Fi is the equilibrium bid by the i th licensee. The equilibrium output of the i thlicensee is qi = a−r

k+1+c , i = 1, 2, . . . , k. The equilibrium outputs of the licensees areassumed to be positive, i.e., a>r .

The profit of the i th licensee, i = 1, 2, . . . , k, is πi = (2+c)(a−r)2

2(k+1+c)2 . In the Nashequilibrium of the bidding game, the i th potential licensee, i = 1, 2, . . . , k, will bid

Fi = (2+c)(a−r)2

2(k+1+c)2 . If k = n, I can guarantee this equilibrium bid by specifying aminimum bid (Kamien et al. 1992). I does not need to specify the minimum bid fork < n.

I maximises the following expression to determine the equilibrium royalty rate:

Maxr

k(Fi + rqi ) = Maxr

k(2 + c)(a − r)2

2(k + 1 + c)2 + kr(a − r)

k + 1 + c. (3)

The equilibrium royalty rate is r∗ = a(k−1)c+2k , which is less than a.

If I auctions k licenses, the equilibrium profit of I is π∗I = a2k

2c+4k . If k>1, this profit

is more than the profit of a single-plant monopolist. We get that∂π∗

I∂k = a2c

2(c+2k)2 > 0,implying that I auctions n licenses.

The following proposition is now immediate.

Proposition 1 In the presence of convex costs, Ci = cq2i

2 , n potential licensees andzero opportunity costs of the licensees for getting the license, an outside innovatorauctions n licenses. The equilibrium bid by each licensee, the equilibrium royalty and

the equilibrium profit of the innovator are respectively F∗ = (2+c)a2

2(c+2n)2 , r∗ = a(n−1)c+2n ,

and π∗I = na2

2c+4n .

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Due to the decreasing returns to scale technologies, diversification of outputs tomultiple firms increases the industry profit by reducing the total costs of production.This induces the innovator to auction multiple licenses. However, multiple licensestend to reduce the profit of the innovator by increasing product-market competition.The innovator uses the royalty to soften competition between the licensees. Hence,decreasing returns to scale technologies induce the innovator to auction n licenses, androyalty helps to soften product-market competition as well as to extract profit fromthe licensees along with the equilibrium bids by the licensees.

If c = 0, it implies that there is no cost of production. It is then intuitive that this sit-uation in our analysis gives the result that is similar to the result under constant returnsto scale technology with zero marginal cost of production, since the costs of produc-tion are zero in both the situations. Formally, we get that if c = 0, the equilibriumprofit of the innovator is π∗

I = a2

4 , and it does not depend on the number of licenses.9

This is in line with multiple licensing contracts of Sen and Stamatopoulos (2009a).However, convexity in the cost function breaks this multiplicity in the licensingcontracts.

3 The inside innovator

Now consider a problem like Section 2 with the exception that the innovator is aninsider, i.e., also a producer. The innovator is a monopolist without licensing.

If I auctions k licenses, where 1 ≤ k ≤ n, and charges the per-unit output royaltyr , I and the i th licensee, i = 1, 2, . . . , k, determine their outputs to maximise thefollowing expressions:

MaxqI

(a − q)qI − cq2I

2+

k∑

i=1

(Fi + rqi ) (4)

Maxqi

(a − q − r)qi − cq2i

2− Fi . (5)

The equilibrium output of I and the i th licensee are qI = a(1+c)+kr(1+c)(2+c+k)

and qi =a(1+c)−r(2+c)(1+c)(2+c+k)

, i = 1, 2, . . . , k. The outputs of all firms are assumed to be positive,

i.e., r <a(1+c)(2+c) .

The profit generated in I and the profit of the i th licensee, i = 1, 2, . . . , k, are

respectively π̂∗I = (2+c)[a(1+c)+kr ]2

2(1+c)2(2+c+k)2 and πi = (2+c)[a(1+c)−r(2+c)]2

2(1+c)2(2+c+k)2 . In the Nash equi-librium of the bidding game, the i th potential licensee, i = 1, 2, . . . , k, will bid

Fi = (2+c)[a(1+c)−r(2+c)]2

2(1+c)2(2+c+k)2 . If k = n, I can guarantee this equilibrium bid by specify-ing a minimum bid. I does not need to specify the minimum bid for k < n.

9 Under the constant returns to scale technologies, Erutku and Richelle (2007) show that an outside innovatorearns the profit of a monopolist producer, irrespective of drastic and non-drastic innovations.

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I maximises the following expression to determine the equilibrium royalty rate:

Maxr

πI + k(Fi + rqi ) = Maxr

(2 + c)[a(1 + c) + kr ]2

2(1 + c)2(2 + c + k)2

+k(2 + c)[a(1 + c) − r(2 + c)]2

2(1 + c)2(2 + c + k)2 (6)

+kr [a(1 + c) − r(2 + c)](1 + c)(2 + c + k)

.

The equilibrium royalty rate is r∗ = ak(1+c)2

(2+c)[k+c(2+c+2k)] , which is less than a(1+c)(2+c) .

If I auctions k licenses, the total income of I (which is the sum of the profit generated

in firm I and the income of firm I from licensing) is π∗I = [k+c(2+c)(1+k)]a2

2(2+c)[k+c(2+c+2k)] . We

get that∂π∗

I∂k = a2c3

2[k+c(2+c+2k)]2 > 0, implying that I auction n licenses.The following proposition is now immediate.

Proposition 2 In the presence of convex costs, Ci = cq2i

2 , n potential licensees andzero opportunity costs of the licensees for getting the license, an inside innovator auc-tions n licenses. The equilibrium bid by each licensee, the equilibrium royalty and the

equilibrium total income of the innovator are respectively F∗ = (2+c)a2c2

2[n+c(2+c+2n)]2 , r∗ =an(1+c)2

(2+c)[n+c(2+c+2n)] and π∗I = [n+c(2+c)(1+n)]a2

2(2+c)[n+c(2+c+2n)] .

Licensing creates two effects. On the one hand, it creates product-market com-petition, which tends to reduce the industry profit. On the other hand, it tendsto increase the industry profit by diversifying outputs among multiple firms, thusreducing the total cost of production. The benefit from licensing increases withmore licenses. Since output royalty softens product-market competition, the inno-vator can choose the output royalty in a way to make the cost-reducing effectstronger than the competition effect, thus making licensing to all potential licenseesprofitable.

If c = 0, the inside innovator does not benefit from licensing, as in Sen and Tauman(2007).

Proposition 2 shows that a monopolist final goods producer licenses under convexcosts, even if licensing creates product-market competition. Thus, it complementsthe literature, mentioned in the introduction, showing that a monopolist final goodsproducer may prefer to create competition through licensing.

We show in both Propositions 1 and 2 that the competition softening effect encour-ages the innovators to use output royalties. However, the comparison of the outputroyalties shows that the output royalty charged by the outside innovator is higher than

the output royalty charged by the inside innovator if a(n−1)c+2n >

an(1+c)2

(2+c)[n+c(2+c+2n)]or −4c−4c2−c3−2n−2cn+cn2

(2+c)(c+2n)(2c+c2+n+2cn)> 0, which holds for n → ∞ but does not hold for

n → 1. Hence, if c > 0, the incentive forsoftening competition is higher under

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the outside innovator that under the inside innovator if there is a large number oflicensees. If there are few licensees, the output distortion under outside innova-tor is not significant, and creates lower royalty under outside innovator comparedto an inside innovator. However, if there are many licensees, it creates significantoutput distortion under outside innovator and encourages the outside innovator tocharge such a high royalty that it is higher under outside innovator than under insideinnovator.

While comparing the fixed-fee under outside and inside innovators, we find

that the fixed-fee under outside innovator, i.e., (2+c)a2

2(c+2n)2 , is higher than the fixed-

fee under inside innovator, i.e., (2+c)a2c2

2[n+c(2+c+2n)]2 . Lower competition under outsideinnovator allows the innovator to extract more through fixed-fee than the insideinnovator.

4 The incentive for innovation

Assume that an innovator needs to invest K to invent the technology. An inno-vator innovates if its profit after innovation is higher than the cost of innovation.Hence, it follows from the above discussions that an inside innovator innovates ifK <

[n+c(2+c)(1+n)]a2

2(2+c)[n+c(2+c+2n)] and an outside innovator innovates if K < na2

2c+4n . Since[n+c(2+c)(1+n)]a2

2(2+c)[n+c(2+c+2n)] > (<) na2

2c+4n for c > (<) − 1 + √1 − n + n2 ≡ c∗, an inside

innovator’s incentive for innovation can be higher (lower) than that of an outsideinnovator for c > c∗ (c < c∗).10

The following result is now immediate.

Proposition 3 Consider r <a(1+c)(2+c) , so that the outputs are positive under both out-

side and insider innovators. An inside innovator’s incentive for innovation is higher

(lower) than that of an outside innovator if c > c∗ and [n+c(2+c)(1+n)]a2

2(2+c)[n+c(2+c+2n)] > K >

na2

2c+4n (c < c∗ and [n+c(2+c)(1+n)]a2

2(2+c)[n+c(2+c+2n)] < K < na2

2c+4n ).

On the one hand, more firms under inside innovator tend to reduce the total revenuecompared to an outside innovator by creating higher product-market competition.On the other hand, more firms under inside innovator tend to reduce the total costscompared to an outside innovator by spreading the outputs over more firms. If thediseconomies of scale are sufficiently large (small) such that c > (<)c∗, the lattereffect dominates (dominated by) the former and the inside innovator’s incentive forinnovation may be higher (lower) than that of the outside innovator.

Proposition 3 is in contrast to Sen and Tauman (2007), where the inside innovator’sincentive for innovation cannot be higher than the outside innovator.

10 Given the different requirements for positive outputs under outside innovator (which is a > r) and insideinnovator (which is r <

a(1+c)(2+c) ), we assume r <

a(1+c)(2+c) , since a(1+c)

(2+c) < a.

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5 Welfare implication

It is trivial that if innovation occurs only under outside (inside) innovator, consumersurplus and social welfare, which is sum of consumer surplus and the total profits ofthe licenser and the licensees, are higher under outside (inside) innovator.

Now see what will happen if both outside and inside innovators innovate. In thissituation, consumer surplus and welfare under the case of outside innovator are C So =

a2n2

2(c+2n)2 and W o = na2

2c+4n + a2n2

2(c+2n)2 respectively, while consumer surplus and welfare

under the case of inside innovator are C SI N = a2[n+c2(1+n)+c(2+3n)]2

2(2+c)2[c2+n+2c(1+n)]2 and W I N =[n+c(2+c)(1+n)]a2

2(2+c)[n+c(2+c+2n)] + a2[n+c2(1+n)+c(2+3n)]2

2(2+c)2[c2+n+2c(1+n)]2 respectively. The comparison of thesevalues shows that consumer surplus and welfare are higher under the case of insideinnovator than the case of outside innovator. Higher competition under the case ofinside innovator compared to the case of outside innovator creates higher consumersurplus and higher welfare under the former than the latter.

6 Conclusion

We show that both the outside and the inside innovators license a new product (ordrastic process innovation) to all potential licensees under convex costs. Hence, amonopolist producer may have the incentive to license its technology in a homoge-neous goods industry. We also show that an inside innovator’s incentive for innovationmay be higher than that of an outside innovator. Since the industries may experiencedifferent scale of economies, our analysis shows that the licensing strategies derivedunder constant returns to scale technologies may not be useful always.

Like most of the literature, we have considered per-unit output royalty. Recently,San Martin and Saracho (2010) point out that often the firms charge ad valorem royalty.Hence, future research may be directed to determine optimal licensing strategy underad valorem royalty. However, it must be noted that even under the ad valorem royalty,the incentive for output diversification in the presence of decreasing returns to scaletechnologies remains.

We have considered a linear demand function to show our results.11 However, itis intuitive that the use of royalty to soften competition remains even under a generaldemand function. Rockett (1990a) shows an inside innovator’s rationale for usingoutput royalty in a duopoly market with constant returns to scale technologies andgeneral demand function. Sen and Stamatopoulos (2009a) consider a general demandfunction and show an outside innovator’s rationale for using output royalty underconstant returns to scale technology. The arguments in these papers suggest that theincentive for charging output royalties in our analysis under a general demand functionto soften competition remains, since the output royalties help to contract the totaloutputs of the oligopolists towards the monopoly output, thus increasing the total

11 It is worth noting that if we consider a demand function P = a−bq, it does not affect our qualitative result,since the basic trade-off between the competition effect and the cost-reducing effect remains. However, theequilibrium values may depend on the slope, b.

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profits in the industries. Further, whatever be the demand function, the incentive foroutput diversification through multiple licensing under decreasing returns to scaletechnology also remains.

Acknowledgments I thank three anonymous referees of this journal for helpful comments and sugges-tions. The usual disclaimer applies.

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Licensing under convex costs