Optimal two-part tari¤ licensing contracts with di¤erentiated goods and endogenous R&D
¤Ramon Fauli-Ollery, Xulia Gonzálezz, Joel Sandonisx
Abstract
In this paper we derive the optimal two-part tari¤ contract for the licensing of a cost- reducing innovation to a di¤erentiated goods industry of a general size. We analyze the cases where the patentee is an independent laboratory or an incumbent …rm. We show that regardless of the number of …rms, the degree of product di¤erentiation and the type of patentee, the innovation is licensed to all …rms. Moreover, we endogenize R&D investment and obtain that an internal patentee invests more (less) in R&D when the technological opportunity is low (high), which is supported by an empirical test using data on R&D expenditures of Spanish manufacturing …rms.
Key words: patent licensing, two-part tari¤ contracts, R&D, product di¤erentiation.
JEL codes: L11, L13, L14.
¤Financial support from Ministerio de Ciencia e Innovación and FEDER funds under SEJ 2007-62656, ECO2008-05771 and the IVIE is gratefully acknowledged.
yUniversity of Alicante. Economics Department, Campus de San Vicente del Raspeig, E-03071, Alicante, Spain. E-mail address: [email protected].
zUniversity of Vigo, Facultad de CC. Económicas, As Lagoas Marcosende s/n, 36210 Vigo, Spain. E-mail adress: [email protected].
xCorresponding author. University of Alicante. Economics Department, Campus de San Vicente del Raspeig, E-03071, Alicante, Spain. E-mail address: [email protected]. Telephone number: 34 96 5903400 (ext:
3232). Fax number: 34 96 5903898.
1 Introduction
Innovation is recognized by today’s economists as a crucial ingredient of growth. This recognition has motivated the development of a huge literature analyzing the determinants of innovation. In particular, one of the main issues is whether the market provides …rms with the right incentives to invest in research and development (R&D). Since Schumpeter’s seminal work, there have been many empirical and theoretical papers addressing the e¤ect of competition on R&D investment;
however, the results are inconclusive. One common characteristic of many of these papers is that they do not consider the existence of a market for technology. In other words, the return to R&D investment comes from the …nal goods market.
In this paper we add to the literature by allowing for the existence of a market for technology.
We consider the case of a research laboratory that owns a process innovation that would allow
…rms in a given industry to reduce their production costs. We consider two cases. If the laboratory remains independent as an external patentee, then it obtains revenues by licensing the patented innovation to the …rms producing in this industry. Another possibility is that the laboratory integrates vertically with one of the …rms in the industry, thus becoming an internal patentee. In this case, it may obtain revenues not only from licensing but also from participating in the …nal goods market.
As the literature on patent licensing has pointed out, the pro…tability of licensing depends on the type of licensing contracts that are available to the patentee. The earlier papers (Kamien and Tauman (1984,1986), Katz and Shapiro (1986) and Kamien et al. (1992)) show that licensing through a …xed fee or an auction is more pro…table for an independent laboratory than is licensing through a royalty. In practice, however, licensing contracts often include both a …xed fee and a royalty (for example, Rostocker (1984) and Yanagawa and Wada (2000) show that this is the case for about half of all licensing contracts).
In this paper, we obtain the optimal two-part tari¤ licensing contract for the general case of an n-…rm oligopolistic industry producing di¤erentiated goods. We analyze the case where the patentee is an independent laboratory and also the case of an incumbent patentee.1 We show that in both cases and regardless of the size of the innovation, the number of …rms in the industry, and the degree of product di¤erentiation the innovation is licensed to all …rms in the industry.
In the second part of the paper, we endogenize the size of the innovation by allowing the patentee to invest in cost-reducing R&D. This allows us to compare the optimal R&D investment of the laboratory when it is an outsider patentee and when it is vertically integrated with one of the …rms in the industry. We …nd that a vertically integrated laboratory invests more in R&D than an independent laboratory when the R&D investment is su¢ciently costly. This result has a nice empirical implication regarding the internal organization of leading innovative
…rms. When the technological opportunity in an industry is high, which means that the cost of achieving a given cost reduction is relatively low, we can expect that innovation is dominated by independent research laboratories. In contrast, when the technological opportunity is low, we expect vertically integrated …rms to be the leaders in innovation activities.
In the last part of the paper, we present some empirical evidence to test the result concerning our comparison of an internal versus an external patentee’s incentives to innovate. We use data from the Survey on Firm’s Innovation conducted by the Spanish Statistic Institute (INE), which are the o¢cial …gures on R&D in Spain. The results of the empirical analysis seem to support our theoretical prediction.
1There are some other papers in the literature that analyze patent licensing under di¤erentiated goods, such
as Muto (1993), Wang and Yang (1999), Mukherjee and Balasubramanian (2001), Poddar and Sinha (2004), Stamatopoulos and Tauman (2005). However they are con…ned either to pure upfront fee or pure royalty, or they consider only the duopoly case.
Apart from the present paper, the analysis of two-part tari¤ licensing contracts has been addressed, among others, by Fauli-Oller and Sandonis (2002), Sandonis and Fauli-Oller (2006), Sen and Tauman (2007), and Erutku and Richelle (2007). Fauli-Oller and Sandonis (2002) characterize situations where licensing a cost-reducing innovation to a rival …rm in a Bertrand setting reduces social welfare. Sandonis and Fauli-Oller (2006) analyze whether a research laboratory prefers to license a cost-reducing innovation as an outsider patentee or to merge with one of the industry …rms and license the innovation as an internal patentee. On the other hand, the papers by Sen and Tauman (2007) and Erutku and Richelle (2007) are closely related to this one. Yet they both focus on the case of homogenous goods, whereas we consider di¤erentiated goods. Erutku and Richelle (2007) study the optimal two-part tari¤ contract to license a cost- reducing innovation for the case of an independent laboratory. They show that, regardless of the number of …rms in the industry, the innovation is licensed to all …rms. In this paper we show that their result extends to the case of di¤erentiated goods and also to the case of a vertically integrated laboratory.
Sen and Tauman (2007) study the optimal auction plus royalty licensing policy for a general size oligopoly and for the cases of an internal and an external patentee. The di¤erence with respect to a two-part tari¤ contract is that, in an auction plus royalty contract, the …xed part is determined through an auction. The authors show that, in this case, the innovation is licensed to all …rms. They also study the incentives to innovate and show that the di¤erence between post- innovation and pre-innovation pro…ts is always higher for an external patentee. The intuition is that whereas an independent laboratory earns no pro…t in the absence of the innovation, an incumbent patentee earns the market pro…ts. Because the R&D investment is presumed to increase the probability of obtaining a given cost-reducing innovation, Sen and Tauman (2007) obtain that the external patentee invests more in R&D.
In this paper, we adopt a di¤erent modelling strategy because R&D, in our model, determines the cost reduction. Therefore, what matters in this setting is not the di¤erence between post- innovation and pre-innovation pro…ts but rather the marginal pro…tability of R&D investment.
We do not claim that our approach is more realistic than Sen and Tauman’s. We only want to stress that, in a framework where R&D reduces production costs in a continuous way, an incumbent patentee may have more incentive to innovate than an independent laboratory. We see our result as being complementary to the one in Sen and Tauman (2007).
The rest of the paper is organized as follows. Section 2 presents the model for the cases of an independent laboratory and an incumbent patentee. In Section 3, we compare their incentives to innovate and then test our conclusions empirically. Finally, we conclude in Section 4. All proofs that are omitted from the text are relegated to the Appendix.
2 The model
We consider symmetric …rms competing in quantities and selling di¤erentiated goods ( = 1 ). Firm sells good . The inverse demand of good is given by:
= ¡ ¡ X
6=
= 1
where is the quantity sold of good . All …rms produce with marginal cost .
We analyze two di¤erent settings. In the …rst, we assume there is an independent research laboratory that owns a patented process innovation. In the second setting, the laboratory is vertically integrated with one of the competing …rms in the industry. The innovation enables the …rms to reduce their cost of production to ¡ . We aim to derive the optimal two-part tari¤ licensing contract ( ) for both an external and an internal patentee, where denotes a
non negative …xed fee and a linear per-unit royalty.
Although most papers in the literature impose nonnegative royalties (exceptions include are Liao and Sen 2005; and Erutku and Richelle 2007), for simplicity we solve the model without taking this constraint into account.2
We start by analyzing the case of an external patentee.
2.1 The case of an external patentee
In this case, the structure of the game is as follows. In the …rst stage, the patentee o¤ers a two- part tari¤ contract ( ) to the competing …rms. In the second stage, …rms decide whether or not to accept the contract. The ones that do accept pay to the patentee. Finally, the …rms compete à la Cournot with a cost inherited from the licensing stage.
Assume that …rms have accepted a licensing contract ( ). In equilibrium, …rms that have not accepted the contract produce:
( ) = 8>
><
>>
:
( ¡ )(2 ¡ ) ¡ ( ¡ )
(2 ¡ )(2 + ( ¡ 1)) if ¡( ¡ )(2 ¡ )
,
0 otherwise.
Observe that, if is very low, then …rms that do not accept the contract are driven out of the market. Meanwhile, the …rms that accept the contract produce, in equilibrium,
( ) = 8>
><
>>
:
( ¡ )(2 ¡ ) ¡ (¡2 + (1 + ) ¡ )( ¡ )
(2 ¡ )(2 + ( ¡ 1)) if ¡( ¡ )(2 ¡ )
¡ + ¡
+ 1 otherwise.
Pro…ts of non-accepting and accepting …rms are given, respectively, by ¦( ) = (( ))2 and ¦( ) = (( ))2.
In the second stage, given that ¡ 1 …rms accept the contract, the kth …rm accepts the contract whenever · ¦( ) ¡ ¦( ¡ 1 ). Obviously, as the laboratory maximizes pro…ts,
2Our results do not change if we impose nonnegative royalties whenever is not very low.
in order for …rms to accept the contract,3 it will choose such that = ¦( )¡¦( ¡1 ).
This implies that the problem of choosing the optimal contract ( ) is equivalent to that of choosing ( ). Then, in the …rst stage, the external patentee solves the following problem subject to 1 · · and · :
(¦( ) ¡ ¦( ¡ 1 )) + ( ) (1) We proceed as follows. First, we prove that the research laboratory …nds it pro…table to license the innovation to all …rms in the industry then, we calculate the optimal royalty once is replaced by in (1). With regard to the …rst result, we know that, with a …xed fee contract, the input would be sold to only a subset of …rms in order to protect industry pro…ts from competition (Kamien and Tauman 1986). But with a two-part tari¤ contract, the laboratory can always license to one more …rm, without a¤ecting the level of competition, by choosing an appropriate royalty. The following lemma shows that it is always pro…table for the laboratory to license the innovation to all …rms regardless of how many there are in the industry. Assume that the laboratory licenses the innovation to …rms with a royalty . We seek to show that the laboratory can always increase pro…ts by licensing the innovation to all …rms through a higher royalty ( ) such that total industry output remains constant. In the particular case of homogeneous goods this claim is intuitive because it implies a constant …nal price4. With di¤erentiated goods, however, we can obtain the result via a technical condition.
Lemma 1 Assume that the laboratory licenses to …rms with a royalty . It can always increase pro…ts by licensing to all …rms with a royalty · such that ( ) = ( ¡ )( ) +
( )
3Since (¦()¡¦(¡1)) 0, this is the only equilibrium in the acceptance stage.
4This argument is used in Sen and Tauman (2007) to prove that, with an auction plus royalty contract, the input would be sold to all …rms. It is also used in Faulí-Oller and Sandonís (2009) in the context of an input market. In both papers, only the case of homogeneous goods is analyzed.
Proof. See the Appendix.
Lemma 1 shows that it is always feasible (and pro…table) for the laboratory to license the innovation to all …rms while increasing the royalty so that total output remains constant. Let us provide some intuition for this result. First, this strategy is always feasible (regardless of ) because of how the level of competition increases as the laboratory licenses the innovation to more …rms. Licensing to one more …rm increases the level of competition less when there are already many …rms using the innovation than when only a few of them do. This implies that, as more …rms buy a license, the laboratory needs progressively smaller increases in the royalty in order to keep the level of competition constant. Second, that this strategy is pro…table for the laboratory can be explained as follows. Its problem is one of maximizing total industry pro…ts less the outside option of …rms (i.e the pro…t of …rms that refuse the licensing contract and produce with the old technology). Licensing to all …rms leads, on the one hand, to an increase of total industry pro…ts due to an e¢ciency e¤ect. On the other hand, the e¤ect with respect to the outside option is ambiguous: licensing the innovation to more …rms for a given royalty reduces the outside option, but increasing the royalty leads to an increase in the outside option.
As we show in the proof, the overall e¤ect is negative and hence it is in the interest of the patentee to license the innovation to all …rms.
This result is central to the paper, so it would be interesting to know whether it holds for more general demand functions. In the Appendix we show that, for the case of homogeneous goods, it holds for concave demands satisfying a technical restriction concerning the third derivative of the inverse demand. We show that it also holds for the class of demands = ¡ , where
¸ 1.
Our next result uses Lemma 1 to derive the optimal two-part tari¤ contract when licensing to …rms.
Proposition 1 The laboratory optimally licenses the innovation to all …rms. The optimal roy- alty is: ¤() = 1 if 1 and ¤() = 2 otherwise, where
1 = ( ¡ 1)(( ¡ )(¡2 + ) + (4 + (¡6 + + 2)) 2(4 + (4(¡2 + ) + (6 + ( ¡ 1) + ( ¡ 6))))
2 = ( ¡ + )( ¡ 1) 2 + 2( ¡ 1)
1 = ( ¡ )(4 + (¡6 + ( ¡ 3)( ¡ 1) + 4))
( ¡ 1)(2 + ( ¡ 1))
Proof. See the Appendix.
Observe that the constraint · is never binding in equilibrium. Recall that the objective function of the patentee can be expressed as total industry pro…ts ¦( ) + ( ) minus the outside option of the licensees ¦( ¡ 1 ). Market pro…ts are increasing in up to ¤(), whereas reducing the outside option calls for a lower . The balance of the two e¤ects leads to an optimal royalty which is always lower than . Notice also that the optimal royalty increases with . The reason is that the greater the innovation’s e¤ect, the less the value of the licensees’
outside option for a given . This gives the patentee more ‡exibility to pursue the …rst incentive, that is, it increases the royalty in order to increase industry pro…ts.
2.2 The case of an internal patentee
Here, we consider the case where an innovation is owned by one of the …rms in the industry (say,
…rm 1). We must distinguish two cases. If the innovation is drastic (¡2+2+(¡+) 0)5, then the patentee earns the monopoly pro…ts in its market by not licensing the innovation; if the innovation is not drastic, monopolization never occurs. The timing of the game is the same as in Section 2.1. In this case there are three di¤erent cost levels in the market stage of the game: the owner of the innovation produces at ¡ , the licensees at cost ¡ + , and the
5Observe that this holds when the innovation is large enough, in particular when ¸( ¡ )(2 ¡ )
non-licensees at cost . As a consequence, the respective equilibrium outputs are given by the following expressions if the innovation is non-drastic:
( ) =
½ ¡2( ¡ ) + (2 ¡ ) + ( + ( ¡ ¡ 2) + )
(2 ¡ )(2 + ( ¡ 1)) if +¡2+2+(¡+)
( ¡ + )(2 ¡ ) +
(2 ¡ )(2 + ) otherwise.
( ) =
½ (2 ¡ )( ¡ + ) + ( + ( ¡ ¡ 2) ¡ ( ¡ ¡ 1))
(2 ¡ )(2 + ( ¡ 1)) if + ¡2+2+(¡+)
( ¡ ¡ )(2 ¡ ) ¡ 2
(2 ¡ )(2 + ) otherwise.
( ) =
½ ( ¡ )(2 ¡ ) + (¡( + 1) + )
(2 ¡ )(2 + ( ¡ 1)) if +¡2+2+(¡+)
0 otherwise.
If the innovation is drastic, non-licensees do not produce (( ) = 0). In this case, outputs in equilibrium for the patentee and licensees are given, respectively, by:
( ) =
½ ( ¡ + )
2 if ( ¡ + )(2 ¡ )
2 ( ¡ + )(2 ¡ ) +
(2 ¡ )(2 + ) otherwise.
( ) =
½ 0 if ( ¡ + )(2 ¡ )
2 ( ¡ ¡ )(2 ¡ ) ¡ 2
(2 ¡ )(2 + ) otherwise.
Here the subindex stands for patentee, subindex for non-licensees and superindex for the internal case. Market pro…ts of the patentee, the accepting …rms, and the non-accepting
…rms are given, respectively, by ¦( ) = ¡
( )¢2
, ¦( ) =¡
( )¢2
, and ¦( ) =
¡( )¢2
.
In the second stage, given that ¡ 1 …rms accept the contract, the kth …rm accepts the contract whenever · ¦( ) ¡ ¦( ¡ 1 ). Like the independent laboratory in Section 2.1,
the internal patentee also seeks to maximize licensing revenues6 and so will choose such that
= ¦( ) ¡ ¦( ¡ 1 ). This implies that the problem of choosing the optimal contract ( ) is equivalent to that of choosing ( ). Then, in the …rst stage, the patentee solves the following problem for 0 · · ¡ 1 and · .
¦( ) + ¡
¦( ) ¡ ¦( ¡ 1 ) + ( )¢
Much as we did for the case of an external patentee, before solving this maximization prob- lem we show that it is always pro…table for the patentee to license the innovation to all …rms regardless of the total number of …rms in the industry.
Lemma 2 Assume that the patentee licenses to …rms with a royalty . The patentee can always increase its pro…ts by licensing to all …rms with a royalty · such that
( ¡ 1)( ¡ 1 ) + ( ¡ 1 ) = ( ¡ ¡ 1) ( ) + ( ) + ( )
Proof. See the Appendix.
The intuition for this is similar to that for the case of an external patentee. In the Appendix we show that, for the case of homogeneous goods, Lemma 2 holds for concave demands satisfying the same technical restrictions as in the case of an external patentee.
Next, we derive the optimal royalty when licensing the innovation to ¡ 1 …rms.
Proposition 2 The laboratory optimally licenses the innovation to all …rms.
If 2
¡ 1, the optimal royalty is: ¤() = if · 1, ¤() = 1 if 1 2 and
¤() = 2 otherwise.
If ¸ 2
¡ 1, the optimal royalty is: ¤() = if 2 and ¤() = 2 otherwise; where
1 = ((¡2 + )(¡2 + ( ¡ 1)) + (¡4 + 2¡ (¡2 + )) + ( ¡ 1)(4 + (¡8 + + 2))) 2(4 + 4(¡2 + ) + 2(7 + (¡7 + )))
6Since (¦( ) ¡ ¦( ¡ 1 ))
0, this is the only equilibrium in the acceptance stage.
2 = ( ¡ + )(¡2 + )2( ¡ 1) 8 ¡ 2(8 + 3( ¡ 1) ¡ 4)
1 = ( ¡ )(¡2 + )(¡2 + ( ¡ 1)) 8 ¡ (¡(6 + ) ¡ 4( ¡ 3) + (4 + ))
2 = ( ¡ )(2 ¡ )(8 + 8( ¡ 2) + 22(5 ¡ 6 + 2) ¡ 3(2 ¡ 3 + 2)
(¡8 + 8 ¡ 22(2¡ 1) + 4(2 ¡ 3 + 2) ¡ 3(2 ¡ 3 + 2)) Proof. See the Appendix.
It is interesting to note that ¤() ¤(). This result is intuitive because an internal patentee obtains revenues not only from licensing but also from selling the good in the market.
Therefore, it is more interested in controlling competition by charging a higher royalty. Observe that, for the particular case = 2, the outside option does not depend on the royalty and, so the internal patentee maximizes industry pro…ts. This implies that 1 = 2.
3 Incentives to innovate
So far, we have analyzed licensing contracts while assuming that the innovation already existed.
The next step in the analysis is to endogenize the size of an innovation by allowing …rms to choose the amount of R&D. Our aim is to compare R&D investments of both an internal and an external patentee.
We assume that, previous to the licensing stage, the patentee chooses the level of R&D investment at the cost () = 2, where the parameter represents the expense of achieving a given cost reduction. In the licensing stage, the pro…ts of the external patentee, (()), net of R&D costs, can be written as:
() = 8>
>>
>>
>>
>>
>>
>>
>>
>>
<
>>
>>
>>
>>
>>
>>
>>
>>
>:
(2(¡2 + )2(2 + ( ¡ 1))2+ 24( ¡ 1)2+ 24( ¡ 1)2¡
¡2(2 + ( ¡ 1))(8 + 8( ¡ 2) + 3( ¡ 1) + 22(5 + ( ¡ 5))) +(2 + ( ¡ 1))(8 + 8( ¡ 2)
+2(¡4( ¡ 1)2+ 3( ¡ 1) + 22(5 + ( ¡ 5)))))
4(2+(¡1))2(4+(4(¡2)+(6+(¡1)+(¡6)))) if 1 ( ¡ + )2
4 + 4( ¡ 1) ...otherwise.
With an internal patentee, for simplicity we focus on the case ¸ ¡12 , which allows us to avoid corner solutions.7 In this case, the pro…ts of the internal patentee, net of R&D costs, are given by:
If () =
½¦( ) + ¡
¦( ) ¡ ¦( ¡ 1 ) + ( )¢
if · 2
( ¡ + )2(¡2+ (2 ¡ )2)
4(4 ¡ (8 + 3( ¡ 1) ¡ 4) otherwise.
Details on computing the actual value of () when · 2 are given in the Appendix.
Therefore, an external patentee will choose:
b = arg max f() ¡ ()g
and an internal patentee will choose:
b = arg max
f() ¡ ()g
The speci…c values ofb and b are calculated in the Appendix. The condition
4 guarantees concavity and interior solutions.
Next, we compare both investments. This comparison, although interesting, has not received much attention in the literature. The seemingly lone exception is Sen and Tauman (2007) who also compare the incentives to innovate of both an external and an internal patentee. They show that the di¤erence between the post-innovation and pre-innovation pro…ts is always higher
7Observe that this implies ¸ 3.
Figure 1: d = 10
for an external patentee. The reason is that whereas an external patentee earns no pro…t with no innovation, an internal patentee still earns market pro…ts. We adopt a di¤erent modelling strategy because, in our model, R&D is a continuous variable that determines the cost reduction and the post-innovation pro…ts. In our setting, then an investment’s incremental pro…ts are less important than its marginal pro…tability. We show that in this case an internal patentee may have more incentive to innovate. Although the complexity of the equilibrium investments precludes an explicit comparison, we can plot in a three-dimensional spaceb ¡b, for di¤erent values of , and . Figure 1 plots the di¤erence for = 10 , 2 [29 1], and 2 [10 30].
Ranges for and are chosen such that ¸ ¡12 and
4 are satis…ed. Figures 2 and 3 plot the di¤erence for values = 30 and = 50, respectively.
If we look at the three …gures, it is easy to see that the region where an internal patentee invests more in R&D increases in size as increases. This result has a nice empirical implication regarding the internal organization of leading innovative …rms. When the technological oppor-
Figure 2: d = 30
Figure 3: d = 50
tunity in an industry is high, which means that the cost of achieving a given cost reduction is relatively low, we can expect that innovation will be dominated by independent research labo- ratories. In contrast, when the technological opportunity is low, vertically integrated …rms are expected to be the leaders in innovation activities.
A second …nding is that an internal patentee invests more in R&D only when the goods are close enough substitutes. The reason could be that an internal patentee is in a better position to control for the level of competition because it is an active …rm in the …nal good’s industry.
In order to derive explicit results on the R&D comparison, it seems worthwhile to analyze the particular case where the goods are homogenous ( = 1). This will facilitate the comparison of our result with that of Sen and Tauman (2007), who also consider homogenous goods.
3.1 The case of homogeneous goods
In the case = 1, the constraint 12 guarantees that
b = ( ¡ )(1 + (2 ¡ 1))
¡(1 + ) + 4(1 + 3) and b = ( ¡ )(3 + (3 + (¡1 + (¡3 + 2))
¡3 ¡ (3 + (¡3 + )) + 4(1 + )2(3 + (¡3 + ))
are global maxima. As expected, the higher is the lower areb and b, but we also have that total expenditure in R&D (b2 and b2) is decreasing in . Therefore, 1 measures the degree of technological opportunity of the industry.
The literature has extensively studied the relationship between R&D investment and com- petition. It is possible to check thatb is always increasing in and that b increases with whenever R&D investment is expensive enough ( high enough). When is low, the trajectory ofb follows a U-shape with respect to the number of …rms.
Our next proposition compares both investments for the particular case of homogeneous goods.
Proposition 3 The internal patentee invests more in R&D than the external patentee when
¸ 3 and b(), where b() = 2(¡3 + (3 ¡ 4)) 2(1 + )(¡3 + ( ¡ 2)2)
This result con…rms the intuitions gained from the …gures.
3.2 Empirical evidence
In this section we present some empirical evidence that supports the main theoretical result of the previous section namely, that an external laboratory invests more (less) in R&D than an internal patentee when the technological opportunity of the industry is high (low). We use data from the "Encuesta sobre Innovación Tecnológica en las Empresas" (Survey on Techni- cal Innovation in Companies) conducted by the Spanish Statistic Institute.8 The survey o¤ers aggregate information on industries’ annual R&D investment, considering separately the expen- ditures from the …rms’ own R&D laboratories (internal R&D) as well as the amount they have spent contracting R&D outside the …rm (external R&D)9 as well as other R&D variables.
We use information about manufacturing …rms, expenditures on internal R&D is a direct measure of incentive to innovate a vertically integrated laboratory’s. Unfortunately, there is no comparable information about the R&D expenditures that external laboratories undertake in each industry. So, as a proxy of external laboratories’ R&D, we use the external R&D expenditures of the manufacturing …rms.10 We then use these two measures to de…ne the ratio of internal to external R&D. According to the theoretical model, this ratio should be higher in those industries with lower technological opportunities and, conversely, lower in industries with
8The data from this survey are the o¢cial …gures on R&D in Spain. We use information on 27 manufacturing industries from 2003 to 2007. The level of aggregation corresponds to the NACE 2- and 3-digit. The data is available from the INE website (www.ine.es).
9Organizing innovation in terms of internal versus external sourcing has been an issue in a number of empirical papers (see, e.g., Veugelers and Cassiman 1999; Piga and Vivarelli 2004). Firms can develop new products through their own R&D spending (internal R&D) or through contract with external laboratories (external R&D). In many cases, …rms conduct both types of R&D.
1 0External R&D includes R&D acquisitions from outside the …rm (usually through contracts); it does not include equipment acquisition or patent licenses. If one considers that external R&D is acquired mainly from external laboratories, then this measure, although incomplete, is a reasonable approximation of an external lab’s incentive to innovate.
higher technological opportunities.
We use the standard classi…cation of manufacturing …rms to categorize industries according to technological opportunities, distinguishing among four types.11 high-technology, medium- high-technology, medium-low-technology, and low-technology industries.12 As is common in this literature, we assume that …rms in high-tech industries (e.g., pharmaceutical, aircraft, spacecraft, communication equipment, computing) enjoy the highest level of technological opportunities.
At the other extreme, low-tech manufacturing industries (e.g. wood, food, textiles) exhibit the lowest level of technological opportunities.
1 1Technological opportunities are de…ned in the empirical literature as exogenous variations in the state of technology (see Ja¤e 1986).
1 2The category high-technology industries comprises: Aircraft and spacecraft; Pharmaceuticals; O¢ce, account- ing, and computing machinery; Radio, TV, and communication equipment; and Medical, precision, and optical instruments. Medium-high technology industries: Electrical machinery; Motor vehicles; Chemicals (excluding pharmaceuticals); Railroad equipment and transport equipment; and Machinery and equipment. Medium-low technology industries: Rubber and plastic products; Other nonmetallic mineral products; Building ships; Basic metals; and Fabricated metal products (excluding machinery and equipment). Low-technology industries: Recy- cling; Wood, paper, and printing and publishing; Food products, beverages, and tobacco; and Textiles, cloths, leather products, and footwear.
Table 1 : R&D indicators for Spanish manufacturing …rms, 2003-2007 Ratio
Int/Ext
R&D e¤ort
Firms with R&D
High-technology industries 5.4 7.5 40.9%
Medium-high-technology industries 5.8 2.4 24.7%
Medium-low-technology industries 9.79 1.7 10.2%
Low-technology industries 10.2 1.9 8.2%
Total 8.0 3.2 19.5%
Table 1 shows the main R&D indicators by industry type. The …rst column presents the average value of the ratio of internal to external R&D. All industries exhibit a ratio that exceeds 1, but there are signi…cative di¤erences among industry types. Low-tech industries have the highest ratio, with internal R&D expenditure that are 10 times greater than external ones.
Overall, the …gures suggest that external laboratories are more active in industries with higher technological opportunities (i.e., high and medium-high technology), and this result agrees with the theoretical model’s prediction. Moreover, high-tech industries have a larger proportion of
…rms with R&D activities, and the average R&D e¤ort is signi…cantly greater.13 In contrast, low-tech industries have a lower proportion of innovative …rms and less R&D e¤ort.
1 3Empirical studies have shown that technological opportunities induce a positive covariation with the total industry-level R&D expenditures and a higher proportion of …rms undertaking R&D activities.
Table 2: OLS regression results, ratio of internal to external R&D
(1) (2) (3)
Coef. T -ratio Coef. T -ratio Coef T -ratio
Constant 13.57 (5.3) 13.73 (5.7) 14.93 (5.3)
High-tech ¡3.80 (¡3.3)
R&D e¤ort ¡0.38 (¡6.2)
Proportion of R&D …rms ¡0.15 (¡3.2)
Trend ¡0.45 (¡1.1) ¡0.44 (¡1.1) ¡0.46 (¡1.1)
Proportion of large …rms ¡0.07 (¡3.4) ¡0.09 (¡3.5) ¡0.07 (¡3.2)
Observations 132 132 132
Adjusted R2 11.1% 9.5% 13.4%
Table 2 presents the result of di¤erent OLS regressions using aggregate industry data, where the dependent variable is the ratio between internal and external R&D. The explanatory vari- ables are three di¤erent measures of technological opportunities and two control variables: a temporal trend and the proportion of large …rms (more than 250 employees) in each industry.
The sources of technological opportunities are usually unobserved, so the empirical literature has typically used industry indicators. We use a dummy variable that takes the value 1 for those sectors included in high- and medium-high-technology classi…cations. Other variables include the average level of R&D e¤ort (aggregate R&D expenditures over total revenues) in a given industry and the proportion of …rms with R&D activities.
Column (1) of the table shows that the estimated parameter for the high-tech industry dummy is negative and signi…cant. The parameter value indicates that the ratio between internal and external R&D is, on average, almost a 30% lower in high-tech industries. Column (2) shows
that the ratio decreases as the R&D e¤ort of an industry increases, and this estimated parameter is also signi…cant. Finally, column (3) shows a negative correlation between the proportion of
…rms with R&D in a given industry and the ratio between internal and external R&D. The parameter of the temporal trend is negative, but it is estimated with low precision because of the short period of time analyzed. Finally …rms’ size also has a negative e¤ect on the ratio.
These results suggest that the main theoretical result of Section 2 explains what is actually observed in the Spanish economy.
4 Conclusions
In this paper we obtain the optimal two-part tari¤ licensing contract for the general case of an n-…rm oligopolistic industry producing di¤erentiated goods. We analyze the case where the patentee is an independent laboratory as well as the case where it is an incumbent patentee.
We show in both cases that regardless of the size of the innovation, the number of …rms in the industry, and the degree of product di¤erentiation, the innovation is licensed to all …rms in the industry.
This result has been previously obtained for the particular case of homogenous goods and an external patentee by Erutku and Richelle (2007). We show that it extends to the case of di¤erentiated goods, and even when the patentee is an incumbent …rm in the industry. This may seem counterintuitive, given that an internal patentee has market pro…ts to protect. However, we have shown that the patentee can always protect pro…ts i.e. control competition not by restricting the number of licensees but rather by increasing the royalty.
In the second part of the paper, we compare the incentive to innovate of an incumbent paten- tee with that of an independent laboratory. We …nd that this comparison depends on the level of technological opportunity. In particular, we …nd that when technological opportunity is low
(high), an internal (external) patentee optimally invests more in R&D. Our result is interesting compared it with that of Sen and Tauman (2007) who …nd that an external patentee has always more incentive to innovate. This di¤erence arises because R&D, in our model, determines the cost reduction whereas in Sen and Tauman increases the probability of developing a given in- novation. Therefore what matters, in our setting, is not the di¤erence between post-innovation and pre-innovation pro…ts but instead the marginal pro…tability of R&D investment. Finally, we test the theoretical prediction of our model concerning the comparison of the incentive to innovate of an internal versus an external patentee. We use data from the Spanish manufactur- ing industry that comes from the Survey on Technological Innovation in Companies conducted by the Spanish Statistic Institute (INE), which are the o¢cial …gures on R&D in Spain. The results of the empirical analysis seem to support our theoretical prediction.
All the results in this paper have been obtained under the assumption of Cournot competi- tion. It would be interesting to extend the results to the case of Bertrand competition. Although the formal analysis is cumbersome, we can provide some intuition at least for the case of an in- ternal patentee. Our main result -that the innovation is always licensed to all …rms- should remain true under Bertrand competition because an internal patentee may have less incentive to restrict the number of licensees under price competition than under quantity competition. The reason is that, under the former regime, the patentee has an additional means for moderating competition namely, using the royalty to establish a higher price. This e¤ect is the main result in Fauli-Oller and Sandonis (2002).
5 Appendix
Proof of Lemma 1
Let ( ) represent the laboratory’s pro…t if it licenses to …rms and sets a royalty · .
We have that
( ) = ( ) ¡ (( ¡ 1 ))2¡ ( ¡ ) (( ))2¡ ( ¡ )( )
where
( ) = (( ))2+ ( ¡ ) (( ))2+ ( ) + ( ¡ )( )
Observe that we have expressed the pro…ts of the incumbent patentee as the di¤erence between total industry pro…ts (( )) and the pro…ts of the remaining …rms. The e¢ciency term appears because we are computing industry pro…ts as if total output were produced using the new technology.
Let solve ( ) = ( ¡ )( ) + ( )
We next prove that licensing to all …rms with a royalty · is more pro…table than licensing to …rms with a royalty . This is equivalent to showing that the following expression is positive:
( ) ¡ ( ) = ( ) ¡ ( ) + (( ¡ 1 ))2+ ( ¡ ) (( ))2+ +( ¡ )( ) ¡ (( ¡ 1 ))2
It is convenient to proceed in two steps. We …rst prove that ( ) ¡ ( ) is positive and then prove that the remaining terms are also positive.
If ¡( ¡ )(2 ¡ )
· · , we have that:
( ) ¡ ( ) = (1 ¡ )( ¡ )( ¡ )2
(2 ¡ )2 ¸ 0
(( ¡ 1 ))2+ ( ¡ ) (( ))2¡ (( ¡ 1 ))2 = 2( ¡ )( ¡ )2
(2 ¡ )2(2 + ( ¡ 1))2 ¸ 0 If ¡( ¡ )(2 ¡ )
( ¡ 1) ¡( ¡ )(2 ¡ )
then
= ( ¡ + )(2 ¡ )( ¡ ) + (2 + ( ¡ 1))
(2 + ( ¡ 1))
where and ( ) = 0.
We must now distinguish two cases. First
if ((2+(¡1))(¡1)+(¡2+)(2+(+(¡2+)))¡(¡2+)(2+(+(¡2+))))
((2+(¡1))(¡1)) · ¡(¡)(2¡) , then ( ¡ 1 ) 0. It follows directly that
( ) ¡ ( ) = (1 ¡ )( ¡ )( ¡ + ¡ )2
(2 + ( ¡ 1))2 ¸ 0 (2)
Moreover, we have
( ¡ 1 ) ¡ ( ¡ 1 ) = ( ¡ )(2 + ( + ¡ 2))(( ¡ )(¡2 + ) + ( ¡ )) (2 ¡ )(2 + ( ¡ 1))(2 + ( ¡ 1)) (3) It is clear that (3) is decreasing in and that it is in the upper bound of the region, which implies
(( ¡ 1 ))2¡ (( ¡ 1 ))2 0
Second,
if ¡(¡)(2¡)(¡1) · ((2+(¡1))(¡1)+(¡2+)(2+(+(¡2+)))¡(¡2+)(2+(+(¡2+))))
((2+(¡1))(¡1))
then ( ¡ 1 ) = 0. The previous calculations applied to this case prove the result.
If · ¡ ( ¡ )(2 ¡ )
( ¡ 1) , we have that ( ¡ 1 ) = 0 and ( ¡ 1 ) = 0. Then
( ) ¡ ( ) = ( ) ¡ ( ), which is positive by (2).
Proof of Proposition 1
1 =arg max f(¦( ) ¡ ¦( ¡ 1 )) + ( )g
2=arg max f¦( ) + ( )g
If = 1, then 1 = 2. This implies that the optimal royalty is 1 when · 1 and 2 otherwise.
Proof of Lemma 2
Let ( ) represent the pro…t of the incumbent patentee if it licenses to …rms and sets a royalty . It is
useful to express this pro…t as
( ) = ( ) ¡ ¡
( ¡ 1 )¢2
¡ ( ¡ ¡ 1)¡
( )¢2
¡ ( ¡ ¡ 1) ( )
where
( ) = ¡
( )¢2
+ ( ¡ ¡ 1)¡
( )¢2
+¡
( )¢2
+ ( ) + ( ¡ ¡ 1)( )
Observe that we have expressed the pro…ts of the incumbent patentee as the di¤erence between total industry pro…ts (( )) and the pro…ts of the remaining …rms. The e¢ciency term appears because we are computing industry pro…ts as if total output were produced using the new technology.
Let solve ( ¡ 1)( ¡ 1 ) + ( ¡ 1 ) = ( ¡ ¡ 1) ( ) + ( ) + ( ).
We next prove that licensing to all …rms with a royalty · is more pro…table than licensing to …rms with a royalty . This is equivalent to showing that, next expression is positive:
( ) ¡ ( ) = ( ¡ 1 ) ¡ ( ) + ¡
( ¡ 1 )¢2
+ ( ¡ ¡ 1)¡
( )¢2
+( ¡ ¡ 1)( ) ¡ ( ¡ 1)¡
( ¡ 2 )¢2
It is convenient to proceed in two steps. We …rst prove that ( ¡ 1 ) ¡ ( ) is positive and then prove that the remaining terms are also positive. We begin by analyzing the non-drastic case, where (¡2 + 2 + ( ¡ + ) 0).
If ¸ +¡2 + 2 + ( ¡ + )
, we have · = ( ¡ ¡ 1) +
¡ 1 · . It is easy to check that
( ¡ 1 ) ¡ ( ) = (1 ¡ )( ¡ ¡ 1)( ¡ )2 (2 ¡ )2( ¡ 1) ¸ 0
¡
( ¡ 1 )¢2
+ ( ¡ ¡ 1)¡
( )¢2
¡ ( ¡ 1)¡
( ¡ 2 )¢2
= 2( ¡ ¡ 1)( ¡ )2
(2 ¡ )2( ¡ 1)(2 + ( ¡ 1))2 ¸ 0
If ( ¡ )(¡2 + ) +
( ¡ 1) · + ¡2 + 2 + ( ¡ + )
then
= ( ¡ + )(1 + ¡ )(¡2 + ) + (2 + ( ¡ 1))
(2 + )( ¡ 1)
where , and ( ) = 0.
We must now distinguish two cases. First if (2 + )( ¡ 1)( +¡2+2+(¡+)
(¡2) ¡ (¡+)(¡2+)(1+¡) (2+)(¡1) )
(2 + ( ¡ 1)) · +¡2+2+(¡+)
,
we have that ( ¡ 2 ) 0. It follows directly that:
( ¡ 1 ) ¡ ( ) = (1 ¡ )( ¡ ¡ 1)(( ¡ + )(¡2 + ) + 2)2
(2 ¡ )2(2 + )2( ¡ 1) ¸ 0 (4)
Moreover,
( ¡ 1 ) ¡ ( ¡ 1) ( ¡ 2 ) ¸ 0
which implies that
¡
( ¡ 1 )¢2
¡ ( ¡ 1)¡
( ¡ 2 )¢2
0
Second, if( ¡ )(¡2 + ) +
( ¡ 1) · (2 + )( ¡ 1)( +¡2+2+(¡+)
(¡2) ¡(¡+)(¡2+)(1+¡) (2+)(¡1) )
(2 + ( ¡ 1))
then ( ¡ 2 ) = 0. The previous calculations applied to this case prove the result.
If · ( ¡ )(¡2 + ) +
( ¡ 1) , we have that ( ¡ 2 ) = 0 and ( ¡ 1 ) = 0. Then
( ) ¡ ( ) = ( ¡ 1 ) ¡ ( ), which is positive by (4).
Next we analyze the case of a drastic innovation, where (¡2 + 2 + ( ¡ + ) 0).
In this case, non-licensees do not produce and so we have ( ) ¡ ( ) = ( ¡ 1 ) ¡ ( ).
If ( ¡ + )(2 ¡ )
2 · · , then only the incumbent patentee produces, therefore, = and ( ) ¡ ( ) = ( ¡ 1 ) ¡ ( ) = 0.
If ( ¡ + )(2 ¡ )
2 , the licensees and the patentee are active. As a result,
( ¡ 1 ) ¡ ( ) = ( ¡ 1 ) ¡ ( ) = (1¡)(¡¡1)((¡+)(¡2+)+2)2 (2¡)2(2+)2(¡1) ¸ 0
Proof of Proposition 2
1 =arg max f¦( ¡ 1 ) + ( ¡ 1)¡
¦( ¡ 1 ) ¡ ¦( ¡ 2 )) + ( ¡ 1)( ¡ 1 )¢ g
2 =arg max f¦( ¡ 1 ) + ( ¡ 1)¦( ¡ 1 ) + ( ¡ 1)( ¡ 1 )g
If = 2, then 1 = 2. This implies that the optimal royalty is 1 when · 2 and 2 otherwise. On the other hand, 1 and 2
¡ 1, 1 , hence the participation constraint is binding and the optimal royalty is .
The value of
()when
· 2³ 1
(¡2+)2(2+(¡1))2
´
(
((¡2 + )(2 + ( ¡ 1))( ¡ 1)((¡2 + )(¡2 + ( ¡ 1)) +(¡4 + 2¡ (¡2 + )) + ( ¡ 1)(4 + (¡8 + + 2)))(( ¡ + )
(¡2 + ) +
((¡2 + )(¡2 + ( ¡ 1)) + (¡4 + 2¡(¡2 + ))+
+( ¡ 1)(4 + (¡8 + + 2)))
4+4(¡2)+2(7+(¡7)) ))
(2(4 + 4( ¡ 2) + 2(7 + ( ¡ 7)))) ¡ (2 + ( ¡ 2))( ¡ 1)
( ¡
((¡2 + )(¡2 + ( ¡ 1)) + (¡4 + 2¡(¡2 + )) + ( ¡ 1) (4 + (¡8 + + 2)))
2(4+4(¡2)+2(7+(¡7))) )
(2(¡2 + ) ¡ 2(¡2 + ) + (¡2 + +
((¡2 + )(¡2 + ( ¡ 1)) + (¡4 + 2¡(¡2 + )) + ( ¡ 1) (4 + (¡8 + + 2)))
4+4(¡2)+2(7+(¡7)) ¡
2(¡2+)
((¡2 + )(¡2 + ( ¡ 1)) + (¡4 + 2¡(¡2 + )) + ( ¡ 1) (4 + (¡8 + + 2)))
2(4+4(¡2)+2(7+(¡7))) )+
((( ¡ + )(¡2 + ) 0 BB BB BB BB BB
@
2(¡2+)
((¡2 + )(¡2 + ( ¡ 1)) + (¡4 + 2¡(¡2 + ))+
( ¡ 1)(4 + (¡8 + + 2)))
2(4+4(¡2)+2(7+(¡7)))
1 CC CC CC CC CC A
)2) )
The values of the optimal R&D investment.
b = ( ¡ )((8 + 8( ¡ 2) + 3( ¡ 1) + 22(5 + ( ¡ 5))))
((2 + ( ¡ 1))(¡(¡2 + )2 + 4(4 + (4( ¡ 2) + (6 + ( ¡ 1) + ( ¡ 6))))));
b =
¡( ¡ )((¡2 + )(5( ¡ 1)3¡32
+24( ¡ 1)2(3 + ( ¡ 5)) ¡ 16(1 + (3 ¡ 7))
¡43( ¡ 2)(¡2 + (10 + ( ¡ 8))) ¡ 82(¡3 + (19 + (3 ¡ 16)))))
(6( ¡ 1)3+45( ¡ 4)( ¡ 1)3+64 + 64( ¡ 3) + 162(16 + ( ¡ 11))
¡163(¡2 + ( ¡ 3)(3 ¡ 5)) ¡ 4(¡2 + )2(2 + ( ¡ 1))2
(4 + 4( ¡ 2) + 2(7 + ( ¡ 7))) ¡ 44(11 + (¡41 + (43 + ( ¡ 15)))))
Proof of Lemmas 1 and 2 for a general demand and homogenous goods
Assume we have …rms and market demand is given by (), where 0() 0 and () is concave. Firms have constant marginal costs. Denote by the sum of marginal costs. Then, in an interior equilibrium, we have:
() ¡ + 0() = 0 (5)
The pro…ts of a …rm with marginal cost are given by
(
_
) = ( (()) ¡ )2
¡0(()) ,
where
_
= P
6= and () is implicitly de…ned in (5). We have that (
_
)
_
0.
We start with Lemma 1. In the case of homogenous goods, ( ) ¡ ( ) = 0 Then, a su¢cient condition for the laboratory to sell to all …rms is
¡
( ¤) ¡ ( ¢
) ¡ ( ¡ )¡
( ) ¡ ( ¢
) ¸ 0, (6)
where ¤ = (¡)+(¡1)(¡+), = (¡1)(¡+), and = (¡¡1)+(¡+).
We also have that ¤¡ = (¡)(¡) and ¡ = (¡) . Hence, (6) can be rewritten as
( ¤) ¡ ( )
( ) ¡ ( ) = R¤
(
_
)
_
R
(
_
)
_
¸ ¡
A su¢cient condition for this to hold is that 2(
_
)
_
2
0 Then R¤
(
_
)
_
R
(
_
)
_
(¤¡ )(
_
)
_
( ¡ )(
_
)
_
= ¡
For the case of an internal patentee (Lemma 2), a similar line of argument applies. Here, we have that ( ¡ 1 ) ¡ ( ) = 0 Then, a su¢cient condition for the patentee to sell to all
…rms is:
¡
( ¤) ¡ ( ¢
) ¡ ( ¡ ¡ 1)¡
( ) ¡ ( ¢
) ¸ 0 (7)
where ¤ = ¡ + ( ¡ 1)( ¡ + ) + ( ¡ ¡ 1), = ¡ + ( ¡ 2)( ¡ + ), and
= ¡+(¡+)+(¡¡2). We also have that ¤¡ =(¡¡1)(¡)
¡1 and ¡ = (¡)¡1 . Now (6) can be rewritten as
( ¤) ¡ ( )
( ) ¡ ( ) = R¤
(
_
)
_
R
(
_
)
_
¸ ¡ ¡ 1
A su¢cient condition for this to hold is that 2(
_
)
_
2
0 Then, R¤
(
_
)
_
R
(
_
)
_
(¤¡ )(
_
)
_
( ¡ )(
_
)
_
= ¡ ¡ 1
It is tedious but direct to show that 2(
_
)
_
2
0 if ¡0() is log-concave and 000() · 0.
We show that this also holds for the class of demands = ¡ , where ¸ 1.
