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Technology licensing in a vertically differentiated duopoly
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Japan and the World Economy 21 (2009) 183–190 www.elsevier.com/locate/jwe
Technology licensing in a vertically differentiated duopoly Changying Li a,b,*, Juan Song c a
Center for Transnational Studies, Nankai University, Tianjin 300071, PR China b Institute of Economics, Nankai University, Tianjin 300071, PR China c School of Law, Linyi Normal University, Linyi, Shandong Province 276000, PR China Received 2 October 2007; received in revised form 10 April 2008; accepted 14 April 2008
Abstract In this paper, we develop a vertically differentiated duopoly model where a high-quality producer competes against a low-quality producer, a la Cournot competition. The high-quality firm has both a new technology and an obsolescent technology. After first deciding whether to license, the firm then chooses which of the two technologies to license. We show that, irrespective of the licensing contract, licensing the new technology is always superior to licensing the obsolescent technology. This finding poses a sharp contrast to the conventional wisdom. # 2008 Elsevier B.V. All rights reserved. JEL classification: D45; D43; L13 Keywords: Licensing; New technology; Obsolescent technology
1. Introduction Licensing represents an important form of technology transfer between two or more firms. The existing literature focuses on either optimal licensing contracts1 or on welfare implications.2 These papers mainly analyze cost-reducing innovations, and seldom consider vertical product innovations. However, competition between vertically differentiated firms is common in many countries. For example, multinational firms compete with local firms in many developing countries, and products produced by multinational firms are often superior in quality to those produced by local firms. Despite the practical significance in the real world, the literature overlooks the licensing behaviors between vertically differentiated firms. In addition, the previous studies assume away licensing choices between technologies by modeling that the innovating * Corresponding author at: Institute of Economics, Nankai University, 94 Weijin Road, Tianjin 300071, PR China. Tel.: +86 22 81216012. 1 One strand of this literature is to model an innovating firm as a nonproducing firm, for example, Kabiraj (2004), Kamien and Tauman (1986) and Li and Geng (2008). Another strand assumes an innovating firm as a producer, such as Erkal (2005), Kamien and Tauman (2002), Katz and Shapiro (1985), Poddar and Sinha (2004), Wang (1998) and Wang and Yang (1999). 2 For instance, Erkal (2005), Fauli-Oller and Sandonis (2002, 2003), Kabiraj (2005), Liao and Sen (2005), Lin (1996), Mukherjee (2005) and Mukherjee and Mukherjee (2005).
0922-1425/$ – see front matter # 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.japwor.2008.04.002
firm has a single innovation.3 Although, this assumption may help to simplify the model analysis, it is important to recognize that a firm that produces a high-quality product may have both a new technology and an obsolescent technology. Therefore, the firm has to make a choice between licensing the obsolescent technology and licensing the new technology. Licensing the obsolescent technology, while keeping the new one for its own use, may allow the patent-holding firm to obtain some licensing revenue, as well as maintain its advantage in competing with its competitors. In contrast, licensing the new technology enables the patent holder to realize higher licensing income, but intensifies market competition. The patent-holding firm has to balance these effects when making its licensing decision. It is widely observed that patent-holding firms are reluctant to transfer their advanced technologies to their competitors, and, instead, they may have incentives to sell their old technologies. For instance, a German firm declined to transfer its magnetic levitation technology when it negotiated with Chinese government over the Maglev train project. It is also reported that multinational firms in China have strong incentives to prevent their new technologies from being 3 An exception is Rockett (1990) who extends the licensing literature to permit firms to choose the age of their technology. However, she focuses on cost-reducing innovations by assuming that the licensor and the licensee are homogenous duopolists in the output market.
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C. Li, J. Song / Japan and the World Economy 21 (2009) 183–190
transferred to their Chinese rivals. This paper studies a vertically differentiated duopoly where one firm produces a high-quality product and the other produces a low-quality product. The two firms compete in choosing outputs.4 The highquality firm has both a new innovation and an obsolescent innovation. The purpose of the present work is to investigate the choice between licensing the obsolescent technology and licensing the new technology. In this paper, we concentrate the analysis on Cournot competition rather than on Bertrand competition. This deserves some comment. As we are interested in the licensing choice between the obsolescent technology and the new technology, this is a natural assumption. From a theoretical perspective, two papers, by De Fraja (1996) and Avenel and Caprice (2006), support the assumption of quantity competition by indicating that Bertrand competition eliminates head-to-head competition. Casual observation shows that many firms compete headon with their rival firms. For example, in China, Gome Corporation and Suning Group sell identical household appliances, in different cities. Another example is China Unicom and China Mobile, which also provide similar services. Therefore, both theoretical analysis and casual evidence suggest that our quantity assumption is very reasonable. In contrast to the conventional wisdom, it is shown that, irrespective of the licensing contract, licensing the new technology is always superior to licensing the obsolescent technology, from the perspective of the innovating firm. Although licensing the new innovation increases market competition and reduces the licensor’s competitive advantage, it permits the patent-holding firm to extract more licensing revenue from its licensee. The increased benefit exceeds the loss from fiercer competition. Another interesting result is that the optimal licensing contract involves output royalty alone, even if both royalties and fixed-fees are feasible. However, consumer surplus and social welfare are higher under fixed-fee licensing. The remainder of the paper is organized as follows: In the next section we provide the basic model. In Section 3, we study the licensing decision and derive the main results, by considering fixed-fee, royalty and two-part tariff contracts. In Section 4, we conclude the paper with some discussions. Some proofs are displayed in Appendix A.
vertical product differentiation. A larger b implies closer substitutability between the two products. A smaller b indicates a larger quality difference. Firm 1 also has an obsolescent technology, which may enable it to produce a product with quality s1. Where s1 = ts3 = t and s2 = ls1 = lt b. t 2 (0,1) and l 2 (0,1) reflect the degrees of product differentiation. Consumers buy, at most, one unit of the vertically differentiated product. The utility function is usi pi if he buys a good with qualitysi U¼ 0 if he does not buy where si reflects the quality of the product, i = 1, 2, 3. u 2 [0,1] is a taste parameter with uniform distribution. The density function is one. pi is the price. The population is normalized to one. For the sake of simplicity, we further stipulate that the marginal costs of both firms are zero. It is worth mentioning that, under this assumption, firms have no incentive to supply low-quality products if they have an advanced technology. Hence, each firm produces only one product, whether licensing occurs or not. The licensing game involves three stages. In the first stage, firm 1 decides whether to license its technology and chooses a technology to license. If firm 1 has an incentive to license its technology, it makes firm 2 a ‘‘take-it-or-leave-it’’ offer, by setting either a fixed-fee or a royalty rate, or both. In the second stage, firm 2 decides whether to accept the offer. In the third stage, the two firms engage in quantity competition.5 3. Model analysis 3.1. Pre-licensing equilibrium We begin our analysis by considering the case where licensing does not occur. In this case, the quality levels of the two firms are s3 = 1 and s2 = b, respectively. The marginal consumer is determined by u p1 ¼ bu p2 ;
u1 ¼
p1 p 2 : 1b
(1)
The demands are q1 ¼
Z
1
du ¼ 1
u1
p1 p2 1b
(2)
2. The model and There are two firms, 1 and 2, in a market. Firm 1 is highquality producer and firm 2 is low-quality producer. Each produces a single product with quality si, i = 3, 2. Where s2 = bs3, b 2 (0,1). To simplify the notation, s3 is fixed at s3 = 1, and hence, s2 = b. The parameter b captures the degree of
q2 ¼
Z
u1
du ¼ p2 =b
p1 p2 p2 : 1b b
(3)
Solving the inverse demands yields6: p1 ¼ 1 q1 bq2
(4)
4
If the two firms compete in pricing, they have strong incentives to maintain a high level of differentiation, thus licensing cannot occur. To be precise, it is easy to understand that licensing new technology cannot take place. We can also show that licensing obsolescent technology is not profitable for the patentholding firm. The proofs are available from the authors, upon request.
5
We will come back to this assumption in Section 4. Motta (1993) and Avenel and Caprice (2006) also take this approach, in order to model output competition between vertically differentiated duopolies. 6
C. Li, J. Song / Japan and the World Economy 21 (2009) 183–190
and p2 ¼ bð1 q1 q2 Þ:
(5)
The profits are p1 ¼ ð1 q1 bq2 Þq1 and p2 ¼ bð1 q1 q2 Þq2 :
(6)
Straightforward calculations give the best response functions: q1 ¼
1 bq2 1 q1 and q2 ¼ : 2 2
It is worth mentioning that, since the quantities supplied by the two firms cannot exceed the aggregate demand, q1 + q2 1, as suggested by Martin (2002, pp. 57–59), this condition must be imposed when each firm maximizes its profit. When this is taken into account, we find that firm 1’s reaction function has a kink. 7 The demands, prices and profits are given by q1 ¼
2b ; 4b
q2 ¼
b p2 ¼ ; 4b
1 ; 4b
p1 ¼
p1 ¼
ð2 bÞ2 ð4 bÞ
2b ; 4b
and p2 ¼
; 2
b ð4 bÞ
: 2
to consumers who purchase the product. An increase in b also enhances firm 2’s profit. These two effects are useful in increasing the welfare. On the other hand, a higher b leads to lower profit for firm 1, and makes some consumers, who previously consumed the low-quality product, no longer purchase it. This is deleterious to both consumer surplus and social welfare. In this case, the former effect dominates, thus with the increase in b, both consumer surplus and social welfare increase. 3.2. Fixed-fee licensing In this subsection, we consider the case where the patentholding firm licenses its innovation by using a fixed-fee contract. We first investigate the situation where firm 1 transfers its obsolescent technology. In this case, firm 2 has to pay firm 1 a fixed fee, F O, in order to improve its product quality from s2 to s1 = ts3 =t. The marginal consumer is determined by u p1 = tu p2, u2 = ( p1 p2)/(1 t). The demands are q1 ¼
¼
Z
p2 =b
2=ð4bÞ
bu
1=ð4bÞ
¼
2
b 2ð4 bÞ
Z 1 b 2b u du þ du 4b 4b 2=ð4bÞ
2
þ
ð4 b Þ 2
2ð4 bÞ
¼
4þbb
du ¼ 1
p1 p2 1t
and (10)
u2
p p2 p2 : q2 ¼ du ¼ 1 1t t p2 =t Inverse demands are p1 ¼ 1 q1 tq2
p2 ¼ tð1 q1 q2 Þ:
and
(11)
Profit functions are pFO 1 ¼ ð1 q1 tq2 Þq1 þ F O
and
pFO 2 ¼ tð1 q1 q2 Þq2 F O : Solving first-order conditions, we have the outputs, prices and profits: 2t ; 4t
p2 ¼
2
2ð4 bÞ2
1
Z
q1 ¼
u1
Z
u2
(7) Obviously, firm 1’s output, price and profit are higher than those of firm 2, and they are decreasing in b. Because firm 1 owns superior technology, it has both quality and price advantages when competing with firm 2, thus its output and profit are higher. With the increase in b, the products become closer substitutes, which reduces firm 1’s profitability by lowering its competitive advantage. Thus firm 1’s output, price and profit decrease as b rises. Conversely, an increase in b reduces the qualitative difference, thereby improving firm 2’s ability to compete. Hence firm 2’s output, price and profit increase as b rises. Consumer surplus is Z u1 Z 1 CS ¼ ðbu p2 Þdu þ ðu p1 Þdu
185
:
q2 ¼
t ; 4t
pFO 2 ¼
1 ; 4t
pFO 1 ¼
t ð4 tÞ2
p1 ¼
ð2 tÞ2 ð4 tÞ2
2t ; 4t
þ FO
and
FO :
(8) In equilibrium, firm 1 can set a fixed fee high enough to extract the increased profit of firm 2. Thus, the optimal fixed fee is
Social welfare is W ¼ p1 þ p2 þ CS ¼
12 5b þ b2 2ð4 bÞ2
:
(9)
Fo ¼
3
and @W/ Since @CS/@b = (12 7b)/2(4 b) > 0 @b = (4 + 3b)/2(4 b)3 > 0, both consumer surplus and social welfare increase in b. The intuition is clear. On one hand, a rise in b improves the low-type product quality, which is beneficial 7
Thanks to the referee for suggesting this point.
t 2
ð4 tÞ
b ð4 bÞ2
:
(12)
However, the patent holder has an incentive to transfer its old FO technology if and only if pFO 1 p1 . Solving p1 p1 yields 4ð43tÞ b 125t . Note that given b < t, we have 4/5 < t < 1.8 8
We thank the referee for pointing out this condition.
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C. Li, J. Song / Japan and the World Economy 21 (2009) 183–190
Again, firm 1 will charge a fixed fee high enough so that it leaves firm 2 with zero increased profit. Thus, firm 1 will set the fixed fee: 1 b : FN ¼ 9 ð4 bÞ2
(15)
To ensure that the innovating firm has an incentive to transfer its new technology, we require that pFN 1 p1 . Solving this inequality yields 4/7 b < 1. Fig. 1. The incentive of firm 1 to transfer its obsolescent (new) technology, under a fixed-fee contract.
Lemma 1. Under a fixed-fee contract, firm 1 has an incentive to transfer its obsolescent technology if and only if 4/5 < t < 1 and b 4(4 3t)/(12 5t). Lemma 1 is illustrated by Fig. 1. The shaded area reflects the conditions under which firm 1 has an incentive to license its obsolescent technology. Fixed-fee licensing enables firm 1 to gain some licensing income, but it intensifies market competition. Whether or not firm 1 is willing to transfer its obsolescent innovation depends on the net effect. Since fixedfee licensing allows the licensor to obtain the licensee’s increased profit, the patent holder has an incentive to license its technology if and only if the industry profit is higher after licensing. If the degree of product differentiation is high (small b), the firms’ profits are higher when they continue selling differentiated products and consumers find products more suited to their needs. Such a form of discrimination helps to raise aggregate profit. Therefore, selling a license is not profitable. However, if the degree of product differentiation is low (large b), then licensing the old technology does not affect competition much, which would explain why licensing is profitable. Note that with b < t, a large b implies a large t. Therefore, under a fixed-fee contract, licensing the obsolescent technology is profitable if and only if 4/5 < t < 1 and b 4(4 3t)/(12 5t). It follows, then, that we explore the case where firm 1 transfers its new innovation through fixed-fee licensing. If firm 1 licenses its new technology, then both firms produce high quality, s3 = 1. The marginal consumer, who is indifferent as to whether R 1 to buy the good or not, is u3 = p. The market demand is q ¼ p du ¼ 1 p. The price is p = 1 q1 q2. The profit functions are given by pFN 1 ¼ ð1 q1 q2 Þq1 þ F N
and
(13)
pFN 2 ¼ ð1 q1 q2 Þq2 F N : Equilibrium outputs, prices and profits are 1 1 q 1 ¼ q2 ¼ ; p ¼ ; 3 3 1 pFN 2 ¼ FN : 9
pFN 1 ¼
1 þ FN; 9
Lemma 2. Under a fixed-fee contract, the patent-holding firm has an incentive to transfer its new technology if and only if 4/7 b < 1. The intuition behind this lemma is similar to that of Lemma 1. Licensing new technology provides the patent holder with some licensing revenue, but it heavily intensifies competition. When the degree of production differentiation is small (large b), the licensing revenue exceeds the loss from intense competition. The patent-holding firm would like to transfer its new technology. When the degree of production differentiation is large (small b), the licensing income is outweighed by the loss from increased market competition. The patent holder will not transfer its new technology. Which technology should be transferred, though, from the viewpoint of the patent holder? To address this question, we need to compare firm 1’s profits when it licenses its obsolescent technology with that which it receives when it licenses its new technology. We offer the following statement. Proposition 1. Under a fixed-fee contract, if 4/5 < t < 1 and b 4(4 3t)/(12 5t), firm 1 will transfer its advanced technology. The proof appears in Appendix A. Fig. 1 depicts this proposition. The shaded region represents the conditions under which the patent-holding firm has an incentive to transfer its advanced technology. It is widely believed that the innovating firm prefers to transfer its obsolescent innovation while keeping the new one for its own use. However, Proposition 1 implies that, under a fixed-fee licensing agreement, if the products are close substitutive goods prior to licensing, then it is optimal for the patent holder to transfer its advanced technology. This result is in sharp contrast to the general practice in the real world. When the two products are not very differentiated, the licensing of the new technology does not impact competition greatly and it generates higher licensing revenue, which more than offsets the loss from fierce competition. Thus, it is better for the patent-holding firm to transfer its new innovation, when the goods are close substitutes prior to licensing. When licensing occurs under a fixed-fee contract, consumer surplus is
and (14)
CSFN ¼
Z
1
ðu pÞdu ¼ u3
1 2 u du ¼ : 3 9 1=3
Z
1
(16)
C. Li, J. Song / Japan and the World Economy 21 (2009) 183–190
Social welfare is
Since
4 FN FN W FN ¼ pFN ¼ : 1 þ p2 þ CS 9
(17)
pRO 1 p1 ¼ ¼
3.3. Royalty licensing In this subsection, we analyze licensing by means of royalty contract. Under this contract, firm 1 transfers its technology to firm 2 at a royalty rate r. To ensure that the licensee produces a positive output, we make a restriction that 0 < r < t/2. As before, we first consider licensing the old innovation. In this case, the inverse demands are the same as in Eq. (11). Firms’ profit functions become: pRO 1 ¼ ð1 q1 tq2 Þq1 þ rq2
and
(18)
pRO 2 ¼ ½tð1 q1 q2 Þ rq2 :
Straightforward algebra leads to the following equilibrium values: q1 ¼
2tþr ; 4t
p2 ¼ pRO 2
q2 ¼
t 2r ; ð4 tÞt
p1 ¼
2tþr ; 4t
ð2 tÞr þ t RO ð2 t þ rÞ2 rðt 2rÞ ; p1 ¼ ; þ 4t ð4 tÞt ð4 tÞ2
¼
ðt 2rÞ2 2
ð4 tÞ t
and
:
4ð4 bÞ2 3ð1 lÞbt 4ð4 bÞ2
ð2 bÞ2 ð4 bÞ2
>0
We can state: Lemma 3. Under royalty licensing, the patent-holding firm always has an incentive to transfer its old technology. Under a royalty contract, licensing the old technology makes the goods become closer substitutes, which generates fiercer market competition. This has a negative effect on the patentholding firm’s profit. However, royalty licensing increases the licensee’s marginal cost, which switches the licensor’s competitive advantage from quality to cost. Hence, it is profitable for the innovating firm to transfer the old technology. We proceed to analyze the other licensing scenario, where firm 1 transfers its new technology through a royalty contract. To ensure that the licensee produces a positive output, we need another constraint, 0 < r < 1/2. In this case, the inverse demand can be written as p = 1 q1 q2. The profit functions are as follows: and
(21)
pRN 2 ¼ ð1 q1 q2 Þq2 rq2 :
ð2 t þ rÞ2
ð4 tÞ2 p2 0 1 0 < r < t: 2 r
Standard calculations yield the following equilibrium values: q1 ¼
rðt 2rÞ þ ð4 tÞt
@pRO 1 @r
the constraint 0 < r < t/2, we have ¼ > 0. That is, pRO increases in r. To maximize profit, 1 the patent holder will set the royalty rate as high as possible. Solving:
Under
ð83tÞðt2rÞ ð4tÞ2 t
ðt 2rÞ2
b 0 ð4 tÞ2 t ð4 bÞ2 pffiffiffi 1 ð4 tÞ l 0
rþ1 ; 3
pRN 1 ¼
:
s:t: pRO 2
pRO 2 p2 ¼
16 16b þ btð3 þ lÞ
pRN 1 ¼ ð1 q1 q2 Þq1 þ rq2
To determine the optimal royalty rate, firm 1 has to solve the following problem: max pRO 1 ¼
187
yields
q2 ¼
1 2r ; 3
1 þ 5r 5r 2 ; 9
and
p¼
rþ1 ; 3
pRN 2 ¼
ð1 2rÞ2 : 9
To determine the optimal royalty rate, the patent-holding firm solves: 1 þ 5r 5r 2 r 9 : p 0 s:t:pRN 2 2 1 0
Thus, firm 1 will choose the following royalty rate to extract firm 2’s increased profit: pffiffiffi 1 ð4 tÞ l t: (19) r ¼ 1 2 4b
The constraint 0 < r < 1/2 implies that @pRN 1 =@r ¼ 5ð1 2rÞ=9 > 0. Thus pRN is strictly increasing in royalty 1 rate r. The patent holder will set r as high as possible, to realize more profit. Solving pRN 2 p2 ¼ 0 yields the optimal royalty rate: pffiffiffi 1 3 b 1 r¼ : (22) 2 4b
Firm 1’s equilibrium profit is given by
Hence firm 1’s profit is
pRO 1 ¼
16 16b þ btð3 þ lÞ 4ð4 bÞ
2
:
(20)
pRN 1 ¼
16 13b þ b2 4ð4 bÞ2
:
(23)
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C. Li, J. Song / Japan and the World Economy 21 (2009) 183–190
Since: pRN 1 p1 ¼
16 13b þ b2 4ð4 bÞ
2
ð2 bÞ2 ð4 bÞ
2
¼
3ð1 bÞb 4ð4 bÞ2
> 0;
Thus we have: Lemma 4. Under royalty licensing, the patent-holding firm always has an incentive to transfer its new technology. Lemma 4 indicates that, under a royalty contract, it is always profitable to license the new innovation, from the viewpoint of the patent-holding firm. The intuition is similar to that of Lemma 3. From the perspective of the patent holder, however, is it better to transfer the new innovation? Comparing firm 1’s profits, we obtain: RO pRN 1 p1 ¼
¼
16 13b þ b2 4ð4 bÞ 3ð1 tÞb 4ð4 bÞ2
2
16 16b þ btð3 þ lÞ
> 0:
u4
pffiffiffi 1 b 1 u du pffiffi 2 4 b ½1 b=ð4bÞ=2 pffiffiffi 2 ð4 þ b bÞ : ¼ 8ð4 bÞ2 1
W
¼
þ
pRN 2
RN
þ CS
¼
ð4 þ
(24)
pffiffiffi pffiffiffi b bÞð12 b 3bÞ 8ð4 bÞ2
q1 ¼
:
(25) pffiffiffi FN RN 2 Since CS CS ¼ ð112 65b þ 7b 18ð4 bÞ bÞ= 9 72ð4 bÞ2p> and W FN W RN ¼ ð80 31b þ 5b2 ffiffiffi 0 2 18ð4 bÞ bÞ= 72ð4 bÞ > 0;10thus, under fixed-fee licensing, consumer surplus and social welfare are higher. However, 2 2 FN because pRN 1 p1 ¼ ð16 17b þ b Þ=36ð4 bÞ > 0, the 9 It is easy to show that the numerator is decreasing in b and achieves zero at b = 1. 10 Straightforward algebra shows that the numerator is decreasing in b and achieves zero at b = 1.
2tþr ; 4t
p2 ¼
q2 ¼
t 2r ; ð4 tÞt
p1 ¼
2tþr ; 4t
ð2 tÞr þ t ; 4t
pTO 1 ¼ pTO 2 ¼
ð2 t þ rÞ2 ð4 tÞ2 ðt 2rÞ2 ð4 tÞ2 t
þ
rðt 2rÞ þ f; ð4 tÞt
and
f
where f ¼ ðt 2rÞ2 =ð4 tÞ2 t b=ð4 bÞ2 is the fixed fee. Under the constraint 0 < r < t/2, we have 2 @pTO =@r ¼ ð4 3tÞðt 2rÞ=ð4 tÞ t > 0. That is, pTO 1 1 increases in r. Thus, firm 1 has an incentive to maximize its royalty rate, and does not charge a fixed fee. Solving: f ¼
Social welfare is pRN 1
and
Standard calculations generate the equilibrium outputs, prices and profits.
This proposition says that, under a royalty contract, it is always superior to license the new innovation, from the viewpoint of the innovating firm. This finding also stands in sharp contrast to the conventional wisdom. When the patent holder transfers pffiffiffi its advanced technology, the market price is p ¼ ½1p ffiffiffi b=ð4 bÞ=2. The marginal consumer is u4 ¼ p ¼ ½1 b=ð4 bÞ=2. Consumer surplus is Z 1 RN CS ¼ ðu pÞ du
RN
Under a two-part tariff contract (r, f), firm 2 has to pay firm 1 a fixed fee f and a unit-output fee r. To ensure that the licensee has a positive output, we must place a restriction that 0 < r < t/ 2. When firm 1 transfers its old technology, the inverse demands are the same as in Eq (11). The firms’ profit functions are
pTO 2 ¼ ½tð1 q1 q2 Þ rq2 f :
Proposition 2. Under royalty licensing, it is optimal for the patent-holding firm to transfer its advanced technology.
¼
3.4. Two-part tariff licensing
pTO 1 ¼ ð1 q1 tq2 Þq1 þ rq2 þ f
4ð4 bÞ2
Thus we have the following result:
Z
patent-holding firm prefers royalty licensing. Compared to fixed-fee licensing, a royalty contract enables the licensor to manipulate the marginal cost of the licensee, and therefore allows the licensor to maintain its competitive advantage over the licensee. Royalty licensing increases the marginal cost of the licensee and distorts the output markets, however, thereby reducing the consumer surplus and social welfare.
ðt 2rÞ2 2
ð4 tÞ t
b ð4 bÞ2
¼0
yields the optimal royalty: pffiffiffi 1 ð4 tÞ l r ¼ 1 t: 2 4b
(26)
Hence, firm 1’s profit is pTO 1 ¼
16 16b þ btð3 þ lÞ 4ð4 bÞ2
:
Interestingly, the solution under a two-part tariff contract is the same as that under a royalty contract. In contrast to Rostoker (1984), who reports that a two-part tariff is the most popular contract in technology licensing, we find that when the patent holder transfers its old innovation, a royalty contract is optimal. Although both royalty and fixed-fee contracts can increase the patent holder’s profit, only a royalty contract can weaken competition, by increasing the rival company’s operating
C. Li, J. Song / Japan and the World Economy 21 (2009) 183–190
costs. To maximize profit, the licensor has a strong incentive to set the royalty rate as high as possible and does not charge a fixed fee. A royalty contract permits the licensor to transform its strategic advantage from quality to cost, and achieve maximal profit. Our result is similar to Rockett (1990), who finds that, in the case of no imitation, the optimal licensing contract is a royalty alone. But our result differs from Mukherjee and Pennings (2006), who argue that, in the context of international technology licensing with strategic trade consideration, the optimal contract involves an up-front fixed-fee alone. Similarly, we can show that, when firm 1 transfers its advanced technology by using a two-part tariff, the optimal contract involves a royalty alone. Comparing the profits, we know that the patent-holding firm prefers to license its new technology. 4. Conclusions and discussions In this paper, we develop a duopolistic model, where a firm that produces a high-quality product competes with another firm that produces a low-quality product. The high-quality producer has a new technology and an obsolescent technology. The patent-holding firm decides first whether to transfer its technology, and then chooses which technology to transfer. Transferring the obsolescent technology, while keeping the new one for its own use, enables the patent holder to obtain some licensing income, as well as maintain its competitive advantage. Licensing the new technology leads to a higher level of licensing revenue, but generates intense competition. Casual observation reveals that the innovating firm prefers to transfer its old innovation, while keeping the new one as a means of comparative advantage. In contrast to the general belief, we show that, regardless of the licensing contract, it is optimal to transfer the new technology, from the viewpoint of the patent-holding firm. Licensing the new technology permits the licensor to obtain more licensing profit, which dominates the loss from intense competition. Thus licensing the new technology is always superior to licensing the obsolescent one. The second interesting result is that the optimal licensing contract for the licensor is a royalty alone. It is worth mentioning that our results depend on the assumption of Cournot competition. Our results will not hold under Bertrand competition. If the firms engage in price competition, then licensing will not occur. Licensing the obsolescent technology harms the innovating firm, since the loss from intense competition outweighs the gain from licensing income. Licensing the new technology generates zero profits for both firms. Thus licensing will not occur under Bertrand competition. In addition, this paper focuses its analysis on the innovation that is licensed. For the sake of simplicity, we follow the general practice in the literature, see, for instance, Fauli-Oller and Sandonis (2003), and ignore imitation.11 We should keep in 11 As argued by Rockett (1990), it is reasonable to assume that imitation is impossible when patent protection is very strong.
189
mind that technology licensing may help the licensees to develop their own technologies, through learning by doing. Taking into account the externalities, the patent-holding firm may change its preference for licensing. Acknowledgements This paper has benefited greatly from comments and suggestions from Yasushi Hamao (editor) and two anonymous referees. We thank Jingwen Yang, Hongyan Fu and Jiao Sun for their excellent research assistance. Changying Li gratefully acknowledges financial support from the Program for New Century Excellent Talents in Universities (NCET07-0449) and the MOE Project of Key Research Institute of Humanities and Social Sciences at Universities (07JJD790137). The responsibility for errors obviously rests with us. Appendix A. Proof of Proposition 1 Proof. Under a fixed-fee contract, Lemma 1 implies that firm 1 has an incentive to license its obsolescent technology if and only if 4/5 < t < 1 and b 4(4 3t)/(12 5t). Since 4(4 3t)/ d/dt[4(4 3t)/(12 5t)] = 64/(12 5t)2 < 0, (12 5t) is decreasing in t, thus b 4(4 3t)/(12 5t) 4(4 3t)/(12 5t)jt = 1 = 4/7. From Lemma 2, we know that if 4/7 b < 1, then firm 1 has an incentive to license its new innovation. Thus, if 4/5 < t < 1 and b 4(4 3t)/ (12 5t), either licensing the obsolescent technology or licensing the new technology is profitable. Note that 4/ FO 5 < t < 1 implies that pFN 1 p1 ¼ ð1 tÞð7t 4Þ=9 2 ð4 tÞ 0. Therefore, we complete the proof. References Avenel, E., Caprice, S., 2006. Upstream market power and product line differentiation in retailing. International Journal of Industrial Organization 24, 319–334. De Fraja, G., 1996. Product line competition in vertically differentiated markets. International Journal of Industrial Organization 14, 389–414. Erkal, N., 2005. Optimal licensing policy in differentiated industries. The Economic Record 81 (252), 51–64. Fauli-Oller, R., Sandonis, J., 2002. Welfare reducing licensing. Games and Economic Behavior 41 (2), 192–205. Fauli-Oller, R., Sandonis, J., 2003. To merger or to license: implications for competition policy. International Journal of Industrial Organization 21, 655–672. Kabiraj, T., 2004. Patent licensing in a leadership structure. The Manchester School 72 (2), 188–205. Kabiraj, T., 2005. Technology transfer in a Stackelberg structure: licensing contracts and welfare. The Manchester School 73 (1), 1–28. Kamien, M., Tauman, Y., 1986. Fees versus royalties and the private value of a patent. Quarterly Journal of Economics 101, 471–491. Kamien, M., Tauman, Y., 2002. Patent licensing: the inside story. The Manchester School 70 (1), 7–15. Katz, M.L., Shapiro, C., 1985. On the licensing of innovation. The Rand Journal of Economics 16 (4), 504–520. Li, C., Geng, X., 2008. Licensing to a durable-good monopoly. Economic Modelling 25, 876–884.
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Liao, C., Sen, D., 2005. Subsidy in licensing: optimality and welfare implications. The Manchester School 73 (3), 281–299. Lin, P., 1996. Fixed-fee licensing of innovations and collusion. The Journal of Industrial Economics 44 (4), 443–449. Martin, S., 2002. Advanced Industrial Economics, Second Edition. Blackwell Publisher Inc.. Motta, M., 1993. Endogenous quality choice: price vs. quantity competition. The Journal of Industrial Economics 41 (2), 113–131. Mukherjee, A., 2005. Innovation, licensing and welfare. The Manchester School 73 (1), 29–39. Mukherjee, A., Mukherjee, S., 2005. Foreign competition with licensing. The Manchester School 73 (6), 653–663.
Mukherjee, A., Pennings, E., 2006. Tariffs, licensing and market structure. European Economic Review 50, 1699–1707. Poddar, S., Sinha, U.B., 2004. On the patent licensing in spatial competition. The Economic Record 80(249), 208–218. Rockett, K., 1990. The quality of licensed technology. International Journal of Industrial Organization 8, 559–574. Rostoker, M., 1984. A survey of licensing. IDEA 24, 59–92. Wang, X.H., 1998. Fee versus royalty licensing in a Cournot duopoly model. Economics Letters 60, 55–62. Wang, X.H., Yang, B., 1999. On licensing under Bertrand competition. Australian Economic Papers 38, 106–119.
Japan and the World Economy 21 (2009) 183–190 www.elsevier.com/locate/jwe
Technology licensing in a vertically differentiated duopoly Changying Li a,b,*, Juan Song c a
Center for Transnational Studies, Nankai University, Tianjin 300071, PR China b Institute of Economics, Nankai University, Tianjin 300071, PR China c School of Law, Linyi Normal University, Linyi, Shandong Province 276000, PR China Received 2 October 2007; received in revised form 10 April 2008; accepted 14 April 2008
Abstract In this paper, we develop a vertically differentiated duopoly model where a high-quality producer competes against a low-quality producer, a la Cournot competition. The high-quality firm has both a new technology and an obsolescent technology. After first deciding whether to license, the firm then chooses which of the two technologies to license. We show that, irrespective of the licensing contract, licensing the new technology is always superior to licensing the obsolescent technology. This finding poses a sharp contrast to the conventional wisdom. # 2008 Elsevier B.V. All rights reserved. JEL classification: D45; D43; L13 Keywords: Licensing; New technology; Obsolescent technology
1. Introduction Licensing represents an important form of technology transfer between two or more firms. The existing literature focuses on either optimal licensing contracts1 or on welfare implications.2 These papers mainly analyze cost-reducing innovations, and seldom consider vertical product innovations. However, competition between vertically differentiated firms is common in many countries. For example, multinational firms compete with local firms in many developing countries, and products produced by multinational firms are often superior in quality to those produced by local firms. Despite the practical significance in the real world, the literature overlooks the licensing behaviors between vertically differentiated firms. In addition, the previous studies assume away licensing choices between technologies by modeling that the innovating * Corresponding author at: Institute of Economics, Nankai University, 94 Weijin Road, Tianjin 300071, PR China. Tel.: +86 22 81216012. 1 One strand of this literature is to model an innovating firm as a nonproducing firm, for example, Kabiraj (2004), Kamien and Tauman (1986) and Li and Geng (2008). Another strand assumes an innovating firm as a producer, such as Erkal (2005), Kamien and Tauman (2002), Katz and Shapiro (1985), Poddar and Sinha (2004), Wang (1998) and Wang and Yang (1999). 2 For instance, Erkal (2005), Fauli-Oller and Sandonis (2002, 2003), Kabiraj (2005), Liao and Sen (2005), Lin (1996), Mukherjee (2005) and Mukherjee and Mukherjee (2005).
0922-1425/$ – see front matter # 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.japwor.2008.04.002
firm has a single innovation.3 Although, this assumption may help to simplify the model analysis, it is important to recognize that a firm that produces a high-quality product may have both a new technology and an obsolescent technology. Therefore, the firm has to make a choice between licensing the obsolescent technology and licensing the new technology. Licensing the obsolescent technology, while keeping the new one for its own use, may allow the patent-holding firm to obtain some licensing revenue, as well as maintain its advantage in competing with its competitors. In contrast, licensing the new technology enables the patent holder to realize higher licensing income, but intensifies market competition. The patent-holding firm has to balance these effects when making its licensing decision. It is widely observed that patent-holding firms are reluctant to transfer their advanced technologies to their competitors, and, instead, they may have incentives to sell their old technologies. For instance, a German firm declined to transfer its magnetic levitation technology when it negotiated with Chinese government over the Maglev train project. It is also reported that multinational firms in China have strong incentives to prevent their new technologies from being 3 An exception is Rockett (1990) who extends the licensing literature to permit firms to choose the age of their technology. However, she focuses on cost-reducing innovations by assuming that the licensor and the licensee are homogenous duopolists in the output market.
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C. Li, J. Song / Japan and the World Economy 21 (2009) 183–190
transferred to their Chinese rivals. This paper studies a vertically differentiated duopoly where one firm produces a high-quality product and the other produces a low-quality product. The two firms compete in choosing outputs.4 The highquality firm has both a new innovation and an obsolescent innovation. The purpose of the present work is to investigate the choice between licensing the obsolescent technology and licensing the new technology. In this paper, we concentrate the analysis on Cournot competition rather than on Bertrand competition. This deserves some comment. As we are interested in the licensing choice between the obsolescent technology and the new technology, this is a natural assumption. From a theoretical perspective, two papers, by De Fraja (1996) and Avenel and Caprice (2006), support the assumption of quantity competition by indicating that Bertrand competition eliminates head-to-head competition. Casual observation shows that many firms compete headon with their rival firms. For example, in China, Gome Corporation and Suning Group sell identical household appliances, in different cities. Another example is China Unicom and China Mobile, which also provide similar services. Therefore, both theoretical analysis and casual evidence suggest that our quantity assumption is very reasonable. In contrast to the conventional wisdom, it is shown that, irrespective of the licensing contract, licensing the new technology is always superior to licensing the obsolescent technology, from the perspective of the innovating firm. Although licensing the new innovation increases market competition and reduces the licensor’s competitive advantage, it permits the patent-holding firm to extract more licensing revenue from its licensee. The increased benefit exceeds the loss from fiercer competition. Another interesting result is that the optimal licensing contract involves output royalty alone, even if both royalties and fixed-fees are feasible. However, consumer surplus and social welfare are higher under fixed-fee licensing. The remainder of the paper is organized as follows: In the next section we provide the basic model. In Section 3, we study the licensing decision and derive the main results, by considering fixed-fee, royalty and two-part tariff contracts. In Section 4, we conclude the paper with some discussions. Some proofs are displayed in Appendix A.
vertical product differentiation. A larger b implies closer substitutability between the two products. A smaller b indicates a larger quality difference. Firm 1 also has an obsolescent technology, which may enable it to produce a product with quality s1. Where s1 = ts3 = t and s2 = ls1 = lt b. t 2 (0,1) and l 2 (0,1) reflect the degrees of product differentiation. Consumers buy, at most, one unit of the vertically differentiated product. The utility function is usi pi if he buys a good with qualitysi U¼ 0 if he does not buy where si reflects the quality of the product, i = 1, 2, 3. u 2 [0,1] is a taste parameter with uniform distribution. The density function is one. pi is the price. The population is normalized to one. For the sake of simplicity, we further stipulate that the marginal costs of both firms are zero. It is worth mentioning that, under this assumption, firms have no incentive to supply low-quality products if they have an advanced technology. Hence, each firm produces only one product, whether licensing occurs or not. The licensing game involves three stages. In the first stage, firm 1 decides whether to license its technology and chooses a technology to license. If firm 1 has an incentive to license its technology, it makes firm 2 a ‘‘take-it-or-leave-it’’ offer, by setting either a fixed-fee or a royalty rate, or both. In the second stage, firm 2 decides whether to accept the offer. In the third stage, the two firms engage in quantity competition.5 3. Model analysis 3.1. Pre-licensing equilibrium We begin our analysis by considering the case where licensing does not occur. In this case, the quality levels of the two firms are s3 = 1 and s2 = b, respectively. The marginal consumer is determined by u p1 ¼ bu p2 ;
u1 ¼
p1 p 2 : 1b
(1)
The demands are q1 ¼
Z
1
du ¼ 1
u1
p1 p2 1b
(2)
2. The model and There are two firms, 1 and 2, in a market. Firm 1 is highquality producer and firm 2 is low-quality producer. Each produces a single product with quality si, i = 3, 2. Where s2 = bs3, b 2 (0,1). To simplify the notation, s3 is fixed at s3 = 1, and hence, s2 = b. The parameter b captures the degree of
q2 ¼
Z
u1
du ¼ p2 =b
p1 p2 p2 : 1b b
(3)
Solving the inverse demands yields6: p1 ¼ 1 q1 bq2
(4)
4
If the two firms compete in pricing, they have strong incentives to maintain a high level of differentiation, thus licensing cannot occur. To be precise, it is easy to understand that licensing new technology cannot take place. We can also show that licensing obsolescent technology is not profitable for the patentholding firm. The proofs are available from the authors, upon request.
5
We will come back to this assumption in Section 4. Motta (1993) and Avenel and Caprice (2006) also take this approach, in order to model output competition between vertically differentiated duopolies. 6
C. Li, J. Song / Japan and the World Economy 21 (2009) 183–190
and p2 ¼ bð1 q1 q2 Þ:
(5)
The profits are p1 ¼ ð1 q1 bq2 Þq1 and p2 ¼ bð1 q1 q2 Þq2 :
(6)
Straightforward calculations give the best response functions: q1 ¼
1 bq2 1 q1 and q2 ¼ : 2 2
It is worth mentioning that, since the quantities supplied by the two firms cannot exceed the aggregate demand, q1 + q2 1, as suggested by Martin (2002, pp. 57–59), this condition must be imposed when each firm maximizes its profit. When this is taken into account, we find that firm 1’s reaction function has a kink. 7 The demands, prices and profits are given by q1 ¼
2b ; 4b
q2 ¼
b p2 ¼ ; 4b
1 ; 4b
p1 ¼
p1 ¼
ð2 bÞ2 ð4 bÞ
2b ; 4b
and p2 ¼
; 2
b ð4 bÞ
: 2
to consumers who purchase the product. An increase in b also enhances firm 2’s profit. These two effects are useful in increasing the welfare. On the other hand, a higher b leads to lower profit for firm 1, and makes some consumers, who previously consumed the low-quality product, no longer purchase it. This is deleterious to both consumer surplus and social welfare. In this case, the former effect dominates, thus with the increase in b, both consumer surplus and social welfare increase. 3.2. Fixed-fee licensing In this subsection, we consider the case where the patentholding firm licenses its innovation by using a fixed-fee contract. We first investigate the situation where firm 1 transfers its obsolescent technology. In this case, firm 2 has to pay firm 1 a fixed fee, F O, in order to improve its product quality from s2 to s1 = ts3 =t. The marginal consumer is determined by u p1 = tu p2, u2 = ( p1 p2)/(1 t). The demands are q1 ¼
¼
Z
p2 =b
2=ð4bÞ
bu
1=ð4bÞ
¼
2
b 2ð4 bÞ
Z 1 b 2b u du þ du 4b 4b 2=ð4bÞ
2
þ
ð4 b Þ 2
2ð4 bÞ
¼
4þbb
du ¼ 1
p1 p2 1t
and (10)
u2
p p2 p2 : q2 ¼ du ¼ 1 1t t p2 =t Inverse demands are p1 ¼ 1 q1 tq2
p2 ¼ tð1 q1 q2 Þ:
and
(11)
Profit functions are pFO 1 ¼ ð1 q1 tq2 Þq1 þ F O
and
pFO 2 ¼ tð1 q1 q2 Þq2 F O : Solving first-order conditions, we have the outputs, prices and profits: 2t ; 4t
p2 ¼
2
2ð4 bÞ2
1
Z
q1 ¼
u1
Z
u2
(7) Obviously, firm 1’s output, price and profit are higher than those of firm 2, and they are decreasing in b. Because firm 1 owns superior technology, it has both quality and price advantages when competing with firm 2, thus its output and profit are higher. With the increase in b, the products become closer substitutes, which reduces firm 1’s profitability by lowering its competitive advantage. Thus firm 1’s output, price and profit decrease as b rises. Conversely, an increase in b reduces the qualitative difference, thereby improving firm 2’s ability to compete. Hence firm 2’s output, price and profit increase as b rises. Consumer surplus is Z u1 Z 1 CS ¼ ðbu p2 Þdu þ ðu p1 Þdu
185
:
q2 ¼
t ; 4t
pFO 2 ¼
1 ; 4t
pFO 1 ¼
t ð4 tÞ2
p1 ¼
ð2 tÞ2 ð4 tÞ2
2t ; 4t
þ FO
and
FO :
(8) In equilibrium, firm 1 can set a fixed fee high enough to extract the increased profit of firm 2. Thus, the optimal fixed fee is
Social welfare is W ¼ p1 þ p2 þ CS ¼
12 5b þ b2 2ð4 bÞ2
:
(9)
Fo ¼
3
and @W/ Since @CS/@b = (12 7b)/2(4 b) > 0 @b = (4 + 3b)/2(4 b)3 > 0, both consumer surplus and social welfare increase in b. The intuition is clear. On one hand, a rise in b improves the low-type product quality, which is beneficial 7
Thanks to the referee for suggesting this point.
t 2
ð4 tÞ
b ð4 bÞ2
:
(12)
However, the patent holder has an incentive to transfer its old FO technology if and only if pFO 1 p1 . Solving p1 p1 yields 4ð43tÞ b 125t . Note that given b < t, we have 4/5 < t < 1.8 8
We thank the referee for pointing out this condition.
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C. Li, J. Song / Japan and the World Economy 21 (2009) 183–190
Again, firm 1 will charge a fixed fee high enough so that it leaves firm 2 with zero increased profit. Thus, firm 1 will set the fixed fee: 1 b : FN ¼ 9 ð4 bÞ2
(15)
To ensure that the innovating firm has an incentive to transfer its new technology, we require that pFN 1 p1 . Solving this inequality yields 4/7 b < 1. Fig. 1. The incentive of firm 1 to transfer its obsolescent (new) technology, under a fixed-fee contract.
Lemma 1. Under a fixed-fee contract, firm 1 has an incentive to transfer its obsolescent technology if and only if 4/5 < t < 1 and b 4(4 3t)/(12 5t). Lemma 1 is illustrated by Fig. 1. The shaded area reflects the conditions under which firm 1 has an incentive to license its obsolescent technology. Fixed-fee licensing enables firm 1 to gain some licensing income, but it intensifies market competition. Whether or not firm 1 is willing to transfer its obsolescent innovation depends on the net effect. Since fixedfee licensing allows the licensor to obtain the licensee’s increased profit, the patent holder has an incentive to license its technology if and only if the industry profit is higher after licensing. If the degree of product differentiation is high (small b), the firms’ profits are higher when they continue selling differentiated products and consumers find products more suited to their needs. Such a form of discrimination helps to raise aggregate profit. Therefore, selling a license is not profitable. However, if the degree of product differentiation is low (large b), then licensing the old technology does not affect competition much, which would explain why licensing is profitable. Note that with b < t, a large b implies a large t. Therefore, under a fixed-fee contract, licensing the obsolescent technology is profitable if and only if 4/5 < t < 1 and b 4(4 3t)/(12 5t). It follows, then, that we explore the case where firm 1 transfers its new innovation through fixed-fee licensing. If firm 1 licenses its new technology, then both firms produce high quality, s3 = 1. The marginal consumer, who is indifferent as to whether R 1 to buy the good or not, is u3 = p. The market demand is q ¼ p du ¼ 1 p. The price is p = 1 q1 q2. The profit functions are given by pFN 1 ¼ ð1 q1 q2 Þq1 þ F N
and
(13)
pFN 2 ¼ ð1 q1 q2 Þq2 F N : Equilibrium outputs, prices and profits are 1 1 q 1 ¼ q2 ¼ ; p ¼ ; 3 3 1 pFN 2 ¼ FN : 9
pFN 1 ¼
1 þ FN; 9
Lemma 2. Under a fixed-fee contract, the patent-holding firm has an incentive to transfer its new technology if and only if 4/7 b < 1. The intuition behind this lemma is similar to that of Lemma 1. Licensing new technology provides the patent holder with some licensing revenue, but it heavily intensifies competition. When the degree of production differentiation is small (large b), the licensing revenue exceeds the loss from intense competition. The patent-holding firm would like to transfer its new technology. When the degree of production differentiation is large (small b), the licensing income is outweighed by the loss from increased market competition. The patent holder will not transfer its new technology. Which technology should be transferred, though, from the viewpoint of the patent holder? To address this question, we need to compare firm 1’s profits when it licenses its obsolescent technology with that which it receives when it licenses its new technology. We offer the following statement. Proposition 1. Under a fixed-fee contract, if 4/5 < t < 1 and b 4(4 3t)/(12 5t), firm 1 will transfer its advanced technology. The proof appears in Appendix A. Fig. 1 depicts this proposition. The shaded region represents the conditions under which the patent-holding firm has an incentive to transfer its advanced technology. It is widely believed that the innovating firm prefers to transfer its obsolescent innovation while keeping the new one for its own use. However, Proposition 1 implies that, under a fixed-fee licensing agreement, if the products are close substitutive goods prior to licensing, then it is optimal for the patent holder to transfer its advanced technology. This result is in sharp contrast to the general practice in the real world. When the two products are not very differentiated, the licensing of the new technology does not impact competition greatly and it generates higher licensing revenue, which more than offsets the loss from fierce competition. Thus, it is better for the patent-holding firm to transfer its new innovation, when the goods are close substitutes prior to licensing. When licensing occurs under a fixed-fee contract, consumer surplus is
and (14)
CSFN ¼
Z
1
ðu pÞdu ¼ u3
1 2 u du ¼ : 3 9 1=3
Z
1
(16)
C. Li, J. Song / Japan and the World Economy 21 (2009) 183–190
Social welfare is
Since
4 FN FN W FN ¼ pFN ¼ : 1 þ p2 þ CS 9
(17)
pRO 1 p1 ¼ ¼
3.3. Royalty licensing In this subsection, we analyze licensing by means of royalty contract. Under this contract, firm 1 transfers its technology to firm 2 at a royalty rate r. To ensure that the licensee produces a positive output, we make a restriction that 0 < r < t/2. As before, we first consider licensing the old innovation. In this case, the inverse demands are the same as in Eq. (11). Firms’ profit functions become: pRO 1 ¼ ð1 q1 tq2 Þq1 þ rq2
and
(18)
pRO 2 ¼ ½tð1 q1 q2 Þ rq2 :
Straightforward algebra leads to the following equilibrium values: q1 ¼
2tþr ; 4t
p2 ¼ pRO 2
q2 ¼
t 2r ; ð4 tÞt
p1 ¼
2tþr ; 4t
ð2 tÞr þ t RO ð2 t þ rÞ2 rðt 2rÞ ; p1 ¼ ; þ 4t ð4 tÞt ð4 tÞ2
¼
ðt 2rÞ2 2
ð4 tÞ t
and
:
4ð4 bÞ2 3ð1 lÞbt 4ð4 bÞ2
ð2 bÞ2 ð4 bÞ2
>0
We can state: Lemma 3. Under royalty licensing, the patent-holding firm always has an incentive to transfer its old technology. Under a royalty contract, licensing the old technology makes the goods become closer substitutes, which generates fiercer market competition. This has a negative effect on the patentholding firm’s profit. However, royalty licensing increases the licensee’s marginal cost, which switches the licensor’s competitive advantage from quality to cost. Hence, it is profitable for the innovating firm to transfer the old technology. We proceed to analyze the other licensing scenario, where firm 1 transfers its new technology through a royalty contract. To ensure that the licensee produces a positive output, we need another constraint, 0 < r < 1/2. In this case, the inverse demand can be written as p = 1 q1 q2. The profit functions are as follows: and
(21)
pRN 2 ¼ ð1 q1 q2 Þq2 rq2 :
ð2 t þ rÞ2
ð4 tÞ2 p2 0 1 0 < r < t: 2 r
Standard calculations yield the following equilibrium values: q1 ¼
rðt 2rÞ þ ð4 tÞt
@pRO 1 @r
the constraint 0 < r < t/2, we have ¼ > 0. That is, pRO increases in r. To maximize profit, 1 the patent holder will set the royalty rate as high as possible. Solving:
Under
ð83tÞðt2rÞ ð4tÞ2 t
ðt 2rÞ2
b 0 ð4 tÞ2 t ð4 bÞ2 pffiffiffi 1 ð4 tÞ l 0
rþ1 ; 3
pRN 1 ¼
:
s:t: pRO 2
pRO 2 p2 ¼
16 16b þ btð3 þ lÞ
pRN 1 ¼ ð1 q1 q2 Þq1 þ rq2
To determine the optimal royalty rate, firm 1 has to solve the following problem: max pRO 1 ¼
187
yields
q2 ¼
1 2r ; 3
1 þ 5r 5r 2 ; 9
and
p¼
rþ1 ; 3
pRN 2 ¼
ð1 2rÞ2 : 9
To determine the optimal royalty rate, the patent-holding firm solves: 1 þ 5r 5r 2 r 9 : p 0 s:t:pRN 2 2 1 0
Thus, firm 1 will choose the following royalty rate to extract firm 2’s increased profit: pffiffiffi 1 ð4 tÞ l t: (19) r ¼ 1 2 4b
The constraint 0 < r < 1/2 implies that @pRN 1 =@r ¼ 5ð1 2rÞ=9 > 0. Thus pRN is strictly increasing in royalty 1 rate r. The patent holder will set r as high as possible, to realize more profit. Solving pRN 2 p2 ¼ 0 yields the optimal royalty rate: pffiffiffi 1 3 b 1 r¼ : (22) 2 4b
Firm 1’s equilibrium profit is given by
Hence firm 1’s profit is
pRO 1 ¼
16 16b þ btð3 þ lÞ 4ð4 bÞ
2
:
(20)
pRN 1 ¼
16 13b þ b2 4ð4 bÞ2
:
(23)
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C. Li, J. Song / Japan and the World Economy 21 (2009) 183–190
Since: pRN 1 p1 ¼
16 13b þ b2 4ð4 bÞ
2
ð2 bÞ2 ð4 bÞ
2
¼
3ð1 bÞb 4ð4 bÞ2
> 0;
Thus we have: Lemma 4. Under royalty licensing, the patent-holding firm always has an incentive to transfer its new technology. Lemma 4 indicates that, under a royalty contract, it is always profitable to license the new innovation, from the viewpoint of the patent-holding firm. The intuition is similar to that of Lemma 3. From the perspective of the patent holder, however, is it better to transfer the new innovation? Comparing firm 1’s profits, we obtain: RO pRN 1 p1 ¼
¼
16 13b þ b2 4ð4 bÞ 3ð1 tÞb 4ð4 bÞ2
2
16 16b þ btð3 þ lÞ
> 0:
u4
pffiffiffi 1 b 1 u du pffiffi 2 4 b ½1 b=ð4bÞ=2 pffiffiffi 2 ð4 þ b bÞ : ¼ 8ð4 bÞ2 1
W
¼
þ
pRN 2
RN
þ CS
¼
ð4 þ
(24)
pffiffiffi pffiffiffi b bÞð12 b 3bÞ 8ð4 bÞ2
q1 ¼
:
(25) pffiffiffi FN RN 2 Since CS CS ¼ ð112 65b þ 7b 18ð4 bÞ bÞ= 9 72ð4 bÞ2p> and W FN W RN ¼ ð80 31b þ 5b2 ffiffiffi 0 2 18ð4 bÞ bÞ= 72ð4 bÞ > 0;10thus, under fixed-fee licensing, consumer surplus and social welfare are higher. However, 2 2 FN because pRN 1 p1 ¼ ð16 17b þ b Þ=36ð4 bÞ > 0, the 9 It is easy to show that the numerator is decreasing in b and achieves zero at b = 1. 10 Straightforward algebra shows that the numerator is decreasing in b and achieves zero at b = 1.
2tþr ; 4t
p2 ¼
q2 ¼
t 2r ; ð4 tÞt
p1 ¼
2tþr ; 4t
ð2 tÞr þ t ; 4t
pTO 1 ¼ pTO 2 ¼
ð2 t þ rÞ2 ð4 tÞ2 ðt 2rÞ2 ð4 tÞ2 t
þ
rðt 2rÞ þ f; ð4 tÞt
and
f
where f ¼ ðt 2rÞ2 =ð4 tÞ2 t b=ð4 bÞ2 is the fixed fee. Under the constraint 0 < r < t/2, we have 2 @pTO =@r ¼ ð4 3tÞðt 2rÞ=ð4 tÞ t > 0. That is, pTO 1 1 increases in r. Thus, firm 1 has an incentive to maximize its royalty rate, and does not charge a fixed fee. Solving: f ¼
Social welfare is pRN 1
and
Standard calculations generate the equilibrium outputs, prices and profits.
This proposition says that, under a royalty contract, it is always superior to license the new innovation, from the viewpoint of the innovating firm. This finding also stands in sharp contrast to the conventional wisdom. When the patent holder transfers pffiffiffi its advanced technology, the market price is p ¼ ½1p ffiffiffi b=ð4 bÞ=2. The marginal consumer is u4 ¼ p ¼ ½1 b=ð4 bÞ=2. Consumer surplus is Z 1 RN CS ¼ ðu pÞ du
RN
Under a two-part tariff contract (r, f), firm 2 has to pay firm 1 a fixed fee f and a unit-output fee r. To ensure that the licensee has a positive output, we must place a restriction that 0 < r < t/ 2. When firm 1 transfers its old technology, the inverse demands are the same as in Eq (11). The firms’ profit functions are
pTO 2 ¼ ½tð1 q1 q2 Þ rq2 f :
Proposition 2. Under royalty licensing, it is optimal for the patent-holding firm to transfer its advanced technology.
¼
3.4. Two-part tariff licensing
pTO 1 ¼ ð1 q1 tq2 Þq1 þ rq2 þ f
4ð4 bÞ2
Thus we have the following result:
Z
patent-holding firm prefers royalty licensing. Compared to fixed-fee licensing, a royalty contract enables the licensor to manipulate the marginal cost of the licensee, and therefore allows the licensor to maintain its competitive advantage over the licensee. Royalty licensing increases the marginal cost of the licensee and distorts the output markets, however, thereby reducing the consumer surplus and social welfare.
ðt 2rÞ2 2
ð4 tÞ t
b ð4 bÞ2
¼0
yields the optimal royalty: pffiffiffi 1 ð4 tÞ l r ¼ 1 t: 2 4b
(26)
Hence, firm 1’s profit is pTO 1 ¼
16 16b þ btð3 þ lÞ 4ð4 bÞ2
:
Interestingly, the solution under a two-part tariff contract is the same as that under a royalty contract. In contrast to Rostoker (1984), who reports that a two-part tariff is the most popular contract in technology licensing, we find that when the patent holder transfers its old innovation, a royalty contract is optimal. Although both royalty and fixed-fee contracts can increase the patent holder’s profit, only a royalty contract can weaken competition, by increasing the rival company’s operating
C. Li, J. Song / Japan and the World Economy 21 (2009) 183–190
costs. To maximize profit, the licensor has a strong incentive to set the royalty rate as high as possible and does not charge a fixed fee. A royalty contract permits the licensor to transform its strategic advantage from quality to cost, and achieve maximal profit. Our result is similar to Rockett (1990), who finds that, in the case of no imitation, the optimal licensing contract is a royalty alone. But our result differs from Mukherjee and Pennings (2006), who argue that, in the context of international technology licensing with strategic trade consideration, the optimal contract involves an up-front fixed-fee alone. Similarly, we can show that, when firm 1 transfers its advanced technology by using a two-part tariff, the optimal contract involves a royalty alone. Comparing the profits, we know that the patent-holding firm prefers to license its new technology. 4. Conclusions and discussions In this paper, we develop a duopolistic model, where a firm that produces a high-quality product competes with another firm that produces a low-quality product. The high-quality producer has a new technology and an obsolescent technology. The patent-holding firm decides first whether to transfer its technology, and then chooses which technology to transfer. Transferring the obsolescent technology, while keeping the new one for its own use, enables the patent holder to obtain some licensing income, as well as maintain its competitive advantage. Licensing the new technology leads to a higher level of licensing revenue, but generates intense competition. Casual observation reveals that the innovating firm prefers to transfer its old innovation, while keeping the new one as a means of comparative advantage. In contrast to the general belief, we show that, regardless of the licensing contract, it is optimal to transfer the new technology, from the viewpoint of the patent-holding firm. Licensing the new technology permits the licensor to obtain more licensing profit, which dominates the loss from intense competition. Thus licensing the new technology is always superior to licensing the obsolescent one. The second interesting result is that the optimal licensing contract for the licensor is a royalty alone. It is worth mentioning that our results depend on the assumption of Cournot competition. Our results will not hold under Bertrand competition. If the firms engage in price competition, then licensing will not occur. Licensing the obsolescent technology harms the innovating firm, since the loss from intense competition outweighs the gain from licensing income. Licensing the new technology generates zero profits for both firms. Thus licensing will not occur under Bertrand competition. In addition, this paper focuses its analysis on the innovation that is licensed. For the sake of simplicity, we follow the general practice in the literature, see, for instance, Fauli-Oller and Sandonis (2003), and ignore imitation.11 We should keep in 11 As argued by Rockett (1990), it is reasonable to assume that imitation is impossible when patent protection is very strong.
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mind that technology licensing may help the licensees to develop their own technologies, through learning by doing. Taking into account the externalities, the patent-holding firm may change its preference for licensing. Acknowledgements This paper has benefited greatly from comments and suggestions from Yasushi Hamao (editor) and two anonymous referees. We thank Jingwen Yang, Hongyan Fu and Jiao Sun for their excellent research assistance. Changying Li gratefully acknowledges financial support from the Program for New Century Excellent Talents in Universities (NCET07-0449) and the MOE Project of Key Research Institute of Humanities and Social Sciences at Universities (07JJD790137). The responsibility for errors obviously rests with us. Appendix A. Proof of Proposition 1 Proof. Under a fixed-fee contract, Lemma 1 implies that firm 1 has an incentive to license its obsolescent technology if and only if 4/5 < t < 1 and b 4(4 3t)/(12 5t). Since 4(4 3t)/ d/dt[4(4 3t)/(12 5t)] = 64/(12 5t)2 < 0, (12 5t) is decreasing in t, thus b 4(4 3t)/(12 5t) 4(4 3t)/(12 5t)jt = 1 = 4/7. From Lemma 2, we know that if 4/7 b < 1, then firm 1 has an incentive to license its new innovation. Thus, if 4/5 < t < 1 and b 4(4 3t)/ (12 5t), either licensing the obsolescent technology or licensing the new technology is profitable. Note that 4/ FO 5 < t < 1 implies that pFN 1 p1 ¼ ð1 tÞð7t 4Þ=9 2 ð4 tÞ 0. Therefore, we complete the proof. References Avenel, E., Caprice, S., 2006. Upstream market power and product line differentiation in retailing. International Journal of Industrial Organization 24, 319–334. De Fraja, G., 1996. Product line competition in vertically differentiated markets. International Journal of Industrial Organization 14, 389–414. Erkal, N., 2005. Optimal licensing policy in differentiated industries. The Economic Record 81 (252), 51–64. Fauli-Oller, R., Sandonis, J., 2002. Welfare reducing licensing. Games and Economic Behavior 41 (2), 192–205. Fauli-Oller, R., Sandonis, J., 2003. To merger or to license: implications for competition policy. International Journal of Industrial Organization 21, 655–672. Kabiraj, T., 2004. Patent licensing in a leadership structure. The Manchester School 72 (2), 188–205. Kabiraj, T., 2005. Technology transfer in a Stackelberg structure: licensing contracts and welfare. The Manchester School 73 (1), 1–28. Kamien, M., Tauman, Y., 1986. Fees versus royalties and the private value of a patent. Quarterly Journal of Economics 101, 471–491. Kamien, M., Tauman, Y., 2002. Patent licensing: the inside story. The Manchester School 70 (1), 7–15. Katz, M.L., Shapiro, C., 1985. On the licensing of innovation. The Rand Journal of Economics 16 (4), 504–520. Li, C., Geng, X., 2008. Licensing to a durable-good monopoly. Economic Modelling 25, 876–884.
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