Comparison between different licensing schemes in a Stackelberg model when the follower is the innovator

Abstract

In the present paper we consider a differentiated-good Stackelberg model, when the follower firm engages in an R $\&$ D process that gives an endogenous cost-reducing innovation. We assume that there can exist a technology transfer between innovator and non-innovator firm. The aim of this paper is three-fold: to do a comparison of the fixed-fee and royalty licensing cases, of the two-part tariff and royalty licensing cases, respectively of the two-part tariff and fixed-fee licensing cases, in order to state in which case is indicated for the innovator firm to license its technology. This comparison allows us to identify which licensing scheme is more profitable for the innovator (follower firm).

Keywords

Industrial organization game theory licensing Stackelberg competition differentiated goods innovation size

1. Introduction

Technology transfer, also called Transfer of Technology or Technology Commercialization, is the process of skill transferring, knowledge, technologies, methods of manufacturing, samples of manufacturing and facilities among governments or universities and other institutions to ensure that scientific and technological developments are accessible to a wide range of users who can then further develop and exploit the technology into new products, processes, applications, materials or services. It is closely related to knowledge transfer.

Technology transfer between firms can be done with many significant methods. One of these methods is licensing. Over time, licensing activity has been the subject of much theoretical inquiry [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. For example, Kitagawa et al. [12] investigate how the type of innovation, either for product or process, influences the licensing scheme. Kabiraj and Kabiraj [13] consider an international Cournot model competition, and they show that a tariff on foreign products can influence the licensing strategy of the foreign firm. They also show that a tariff can be chosen so as to induce fixed-fee licensing. Hong et al. [14] developed a duopoly game model in which the innovating firm has three options for licensing its innovation: fixed-fee licensing, royalty licensing and two-part tariff licensing, by assuming that the R&D outcome is stochastic.

The technology transfer between firms can be done by means of a per-unit royalty (the licensee firm pays a royalty rate per unit of production), a fixed-fee payment (the licensee firm pays a lump sum independent of the production output) or a two-part tariff (fixed-fee plus per-unit royalty) licensing contract. Martin and Saracho [15] also consider patent licensing by ad valorem royalties, and show that the patentee may prefer ad valorem royalties to per-unir royalties. Fan et al. [16] show that per-unit royalty licensing is more profitable if the licensor is more efficiente in using the innovation, whereas ad valorem licensing is more profitable if the licensee is more efficient. Hsu et al. [17] compared, in a Cournot duopoly model, two licensing forms between competitors of different productivity, ad valorem and per-unit royalty licensing. They founds that ad valorem royalty licensing is superior to per-unit royalty licensing for the patent-holding firm when the cost-reducing innovation is non-drastic. Yan and Yang [18] investigated the licensing behavior in a differentiated Bertrand model by considering uncertain R&D outcomes and technology spillover. They showed that, in the case of a non-drastic innovation, fixed-fee licensing is better than royalty licensing when product substitution and technology spillover are both small, while it is royalty licensing otherwise. Furthermore, allowing a two-part tariff licensing, this is superior (equivalent) to royalty licensing when technology spillover is small (large), but always better than fixed-fee licensing for any degree of product substitution and technology spillover. Zou and Chen [19] examined product innovation licensing in both exclusive and non-exclusive schemes each under unit/revenue royalty and fixed fee in a vertically differentiated Cournot oligopoly, where a quality-leading firm is an internal licensor. They found that, under a non-exclusive licensing, royalty licensing is the optimal policy choice for the licensor if quality difference within firms is small, regardless of whether a unit or revenue royalty scheme is offered. In the case of exclusive licensing, a two-part tariff is optimal. Wang et al. [20] studied the relationship between privatization and licensing (by public or private firms) with the consideration of either a domestic or a foreign private firm. They showed that, in the case of a domestic private firm, public licensing facilitates privatization, but private licensing hinders privatization. Furthermore, in the case of a foreign private firm, both public and private licensing facilitate privatization.

These type of contracts cover a wide range of well-known situations. As it was emphasized in [3], the technology transfer between firms by different licensing schemes can be done in different duopoly models, as Cournot, Bertrand or Stackelberg model.

Furthermore, we recall the basic framework of our study, as it was given in [3]:

We consider a duopoly model where two firms, denoted by $F_{1}$ and $F_{2}$ , produce a differentiated good. The inverse demand functions are given by $p_{i}=1-q_{i}-dq_{j},$ where:

•
$p_{i}$ represents the price of firm $F_{i}$ , $i=1,2$ ;
•
$q_{i}$ and $q_{j}$ represent the outputs of firms $F_{i}$ and $F_{j}$ , $i,j=1,2$ , $i\neq j$ ;
•
$d$ represents the degree of the differentiation of the goods, $d\in(0,1)$ .

The duopoly market is modeled as a Stackelberg competition: the leader chooses its output level and then the follower is free to choose its optimal output taking into account the leader’s output. So, the firms do not decide simultaneously the level of their outputs. Initially, both firms have identical unit production cost $c_{i}=c$ , with $i=1,2$ and $0<c<1$ .

We consider the case when only the follower firm $F_{2}$ can engage in an R&D process in order to improve its technology1
¹
Throughout the paper we use the notation superscript $F$ to refer to this case.

. This allows a reduction of its production costs by an amount that we call innovation size. The cost-reducing innovation creates a new technology that reduces innovating firm’s unit cost by the amount of $k$ , while the amount invested in R&D is $k^{2}/2$ . So, the innovation size is endogenous, firm $F_{2}$ (the follower) is the licensor and, in case the technology transfer occurs, firm $F_{1}$ (the leader) is the licensee. The innovation is drastic, if the non-innovator firm will produce non-positive output in case of not accepting licensing contract (this means that the non-innovator firm will be out of the market, and the innovator firm will become a monopolist). The innovation is non-drastic, if the non-innovator firm competes on the market even if it decides to use its own technology, getting positive profits.

We work under the assumption, on one hand, that there is no technology transfer between the two firms, and, on the other hand, that there will be a technology transfer between the two firms based on a per-unit royalty licensing contract. Therefore, we consider the following five stages game: in the first stage, the innovator firm $F_{2}$ decides the value of the innovation size (or, equivalently, the amount to invest in R&D). In the second stage, the innovator firm $F_{2}$ decides whether to license the technology or not, because licensing reduces the marginal cost of the leader firm $F_{1}$ . If it decides to license the new technology, then it charges a payment from the licensee. In the third stage, firm $F_{1}$ decides whether to accept or reject the offer made by firm $F_{2}$ . Then, both firms represent the players of a Stackelberg game. So, in the fourth stage firm $F_{1}$ decides its output; and in the last stage firm $F_{2}$ , being aware of the leader’s output, chooses the output to produce.

Furthermore, we consider that there exists a technology transfer between the two firms based on a fixed-fee, per-unit royalty or two-part tariff licensing contract. We note that the innovation can be either non-drastic or drastic, depending on the value of the differentiated parameter $d$ . Letting $d_{1}$ , $0<d_{1}<1$ , be such that $2d_{1}^{3}-3d_{1}^{2}-4d_{1}+4=0,$ we have the following:2
²
We note that $d_{1}\simeq$ 0.781.

(A)
if $0<d<d_{1}$ , then we work under the non-drastic innovation case;
(B)
if $d_{1}<d<1$ , then we work under the drastic innovation.

The aim of the present paper is three-fold: in order to state in which case is indicated for the innovator firm to license its technology, on one hand, we do a comparison of the fixed-fee and royalty licensing cases. On the other hand, we do a comparison of the two-part tariff and royalty licensing cases, respectively of the two-part tariff and fixed-fee licensing cases. The main conclusions are relieved. We emphasize that the degree $d$ of the differentiation of the goods plays a great importance in the results.

A preliminar version of this paper was presented at the International Conference of Numerical Analysis and Applied Mathematics 2017, at Rhodes, Greece.
2. Comparison between the different licensing schemes

In [3] we studied, on one hand, the case when the licensing holds by means of a per-unit royalty contract, in a differentiated-good Stackelberg duopoly when the follower firm is the innovator. Letting $r$ be the per-unit royalty, the production costs of firm $F_{1}$ and firm $F_{2}$ are, respectively, given by $c-k_{r}$ and $c-k_{r}+r$ , where $k_{r}$ is the reduction on the production cost of the innovator firm. So, the profits of firms $F_{1}$ and $F_{2}$ are, respectively, defined by3

³
Throughout the paper we use the notation subscript $r$ to refer to the per-unit royalty licensing case.

$\displaystyle\pi_{1,r}=(1-q_{1,r}-dq_{2,r}-c+k_{r}-r)q_{1,r}$ (1)

and

$\displaystyle\pi_{2,r}=(1-q_{2,r}-dq_{1,r}-c+k_{r})q_{2,r}-(k_{r})^{2}/2+rq_{1% ,r}.$ (2)

We recall that standard computations yield the profits of the firms given by

$\displaystyle\pi_{1,r}^{F}=\frac{8(1-c)^{2}(1-d)(d^{3}-d^{2}-2d+2)}{(9d^{2}-8d% -4)^{2}},\,\,\,\pi_{2,r}^{F}=\frac{(1-c)^{2}(d^{2}+8d-12)}{2(9d^{2}-8d-4)},$ (3)

where $c$ is the unit production cost, common to both firms, and $d$ is the parameter that measures the differentiation of the goods produced by both firms.

In [21] we studied the cases when there is a technology transfer between the innovator and the non-innovator firm based on a fixed-fee licensing contract, respectively based on a two-part tariff licensing contract.

When the technology transfer occurs based on a fixed-fee licensing contract, we define the profit functions $\pi_{1,f}^{F}$ and $\pi_{2,f}^{F}$ for the leader and follower firms as follows:4

⁴

Throughout the paper we use the notation subscript $f$ to refer to the fixed-fee licensing case.

$\displaystyle\pi_{1,f}=(1-q_{1,f}-dq_{2,f}-c+k_{f})q_{1,f}-f$ (4)

and

$\displaystyle\pi_{2,f}=(1-q_{2,f}-dq_{1,f}-c+k_{f})q_{2,f}-(k_{f})^{2}/2+f,$ (5)

where $k_{f}$ is the reduction on the production cost of the innovator firm and $f$ is the value of the fixed-fee.

We emphasize that in the case of fixed-fee licensing scheme, non-drastic innovation, the firms’ profits are, respectively, given by

$\displaystyle\pi_{1,f}^{F}=\frac{2(1-c)^{2}(2-d^{2})(4d^{6}-12d^{5}-7d^{4}+40d% ^{3}-8d^{2}-32d+16)}{(7d^{4}-24d^{2}+16)^{2}}$ (6)

and

$\displaystyle\pi_{2,f}^{F}=\frac{(1-c)^{2}h_{1}(d)}{2d(3d-4)(3d^{2}-8)(7d^{4}-% 24d^{2}+16)^{2}},$ (7)

where $h_{1}(d)=95d^{12}-36d^{11}-404d^{10}-912d^{9}+504d^{8}+7968d^{7}-2240d^{6}-225% 28d^{5}+9856d^{4}+27136d^{3}-15360d^{2}-12288d+8192.$

Furthermore, in the case of fixed-fee licensing scheme, drastic innovation case, the follower firm’s profit is given by5

⁵

Throughout the paper we use the notation superscript $\sim$ to refer to the drastic innovation case.

$\displaystyle\tilde{\pi}_{2,f}^{F}=\frac{(1-c)^{2}(7d^{7}+120d^{6}-384d^{5}-38% 4d^{4}+1824d^{3}-384d^{2}-2304d+1536)}{2d(9d^{3}-12d^{2}-24d+32)}.$ (8)

In the case of two-part tariff licensing scheme, firm $F_{1}\,^{\prime}$ s profit function $\pi_{1,rf}$ is given by6

⁶

Throughout the paper we use the notation subscript $r f$ to refer to the two-part tariff licensing case.

$\displaystyle\pi_{1,rf}=(1-q_{1,rf}-dq_{2,rf}-c+k_{rf}-r_{1})q_{1,rf}-f_{1}.$ (9)

and firm $F_{2}\,^{\prime}$ s total profit $\pi_{2,rf}$ in this case will be its own profit in the product market due to competition plus the fixed-fee it charges and the royalties it receives, i.e.

$\displaystyle\pi_{2,rf}=(1-q_{2,rf}-dq_{1,rf}-c+k_{rf})q_{2,rf}-(k_{rf})^{2}/2% +f_{1}+r_{1}q_{1,rf},$ (10)

where $k_{rf}$ is the reduction on the production cost of the innovator firm.

We highlight that in the case of two-part tariff licensing scheme, non-drastic innovation, the profits for the leader and follower firms are, respectively, given by

$\displaystyle\pi_{1,rf}^{F}=\frac{2(1-c)^{2}(2-d^{2})h_{2}(d)}{(7d^{4}-24d^{2}% +16)^{2}},\,\,\,\pi_{2,rf}^{F}=\frac{(1-c)^{2}h_{3}(d)}{2(7d^{4}-24d^{2}+16)^{% 2}h_{4}(d)}.$ (11)

where $h_{2}(d)=4d^{6}-12d^{5}-7d^{4}+40d^{3}-8d^{2}-32d+16,$ $h_{3}(d)=55d^{14}+1124d^{13}-1948d^{12}-11664d^{11}+10328d^{10}+68576d^{9}-439% 04d^{8}-217088d^{7}+143872d^{6}+355328d^{5}-257536d^{4}-286720d^{3}+223232d^{2% }+90112d-73728$ and $h_{4}(d)=31d^{6}+4d^{5}-256d^{4}+192d^{3}+272d^{2}-192d-64.$

Furthermore, in the case of two-part tariff licensing scheme, drastic innovation case, the follower firm’s profit is given by

$\displaystyle\tilde{\pi}_{2,rf}^{F}=\frac{(1-c)^{2}(-9d^{6}+52d^{5}+24d^{4}-35% 2d^{3}+272d^{2}+320d-320)}{2(31d^{6}+4d^{5}-256d^{4}+192d^{3}+272d^{2}-192d-64% )}.$ (12)

In the next sections, we compare the profits that both innovator and non-innovator firms earn under different licensing schemes, in order to conclude which contract is more profitable to the licensor. Furthermore, we also do a static analysis on the differences of the profits in order to conclude if the incentive to choose one or the other type of contract increases or decreases with the differentiation of the goods.

2.1 Comparison between fixed-fee and royalty contracts

In this section, we compare the profits that both innovator and non-innovator firms earn under fixed-fee and under royalty contracts, in order to conclude which contract is more profitable to the licensor.

2.1.1 Non-drastic innovation (i.e. $d\in(0,d_{1})$ )

Let $h_{5}(d)$ be such that $h_{5}(d)=29246d^{16}-157419d^{15}+97656d^{14}+889864d^{13}-1574040d^{12}-16320% 48d^{11}+5809216d^{10}-621824d^{9}-9987456d^{8}+6729472d^{7}+7696384d^{6}-9369% 600d^{5}-1361920d^{4}+4976640d^{3}-835584d^{2}-1048576d+360448$ .

From Eqs (3), (6) and (7), it is easy to see that

$\displaystyle\pi_{2,f}^{F}-\pi_{2,r}^{F}>0,\forall\,d\in(0,d_{2}),∼{}∼{}∼{}\pi% _{2,f}^{F}-\pi_{2,r}^{F}<0,\forall\,d\in(d_{2},d_{1}),$ (13) $\displaystyle\frac{\partial(\pi_{2,f}^{F}-\pi_{2,r}^{F})}{\partial d}<0,\,% \forall\,d\in(0,d_{2}),∼{}∼{}∼{}\frac{\partial(\pi_{2,r}^{F}-\pi_{2,f}^{F})}{% \partial d}>0,\,\forall\,d\in(d_{2},d_{1}),$ (14)

and

$\displaystyle\pi_{1,f}^{F}-\pi_{1,r}^{F}<0,\,\forall\,d\in(0,d_{1}),$ (15) $\displaystyle\frac{\partial(\pi_{1,f}^{F}-\pi_{1,r}^{F})}{\partial d}>0,\,% \forall\,d\in(0,d_{3})\cup(d_{4},d_{1}),∼{}∼{}∼{}\frac{\partial(\pi_{1,f}^{F}-% \pi_{1,r}^{F})}{\partial d}<0,\,\forall\,d\in(d_{3},d_{4}),$ (16)

where $d_{2},d_{3},d_{4}$ , $0<d_{2},d_{3},d_{4}<1$ are, respectively, such that $207d_{2}^{14}-2012d_{2}^{13}+5234d_{2}^{12}+8056d_{2}^{11}-48508d_{2}^{10}+183% 84d_{2}^{9}+143088d_{2}^{8}-143424d_{2}^{7}-167360d_{2}^{6}+267520d_{2}^{5}+40% 704d_{2}^{4}-193536d_{2}^{3}+47104d_{2}^{2}+40960d_{2}-16384=0,$ and $h_{5}(d_{3})=0,h_{5}(d_{4})=0.$ 7

⁷

We note that $d_{2}\simeq 0.631,$ $d_{3}\simeq 0.632$ and $d_{4}\simeq 0.741$ .

Therefore, we have the following result.

Theorem 1. (i) If the goods are sufficiently differentiated ( $d\in(0,d_{2})$ ), then the innovator firm prefers to license its technology by a fixed-fee contract than by a royalty one. Furthermore, this incentive increases with the differentiation of the goods;

(ii) If the goods are neither sufficiently differentiated nor sufficiently homogenous ( $d\in(d_{2},d_{1})$ ), then the innovator firm prefers to license its technology by a royalty contract than by a fixed-fee one. Furthermore, this incentive decreases with the differentiation of the goods.

We observe that, for the non-innovator firm it is always better a royalty contract than a fixed-fee one. Furthermore, the incentive of the non-innovator firm to accept the new technology by a royalty contract instead of a fixed-fee one either decreases ( $d\in(0,d_{3})\cup(d_{4},d_{1})$ ) or increases ( $d\in(d_{3},d_{4})$ ) with the differentiation of the goods.

2.1.2 Drastic innovation (i.e.

d\in[d_{1},1)

)

From Eqs (3), (8) and the fact that $\tilde{\pi}_{2,r}^{F}=\pi_{2,r}^{F},\,\forall\,d\in[d_{1},1),$ it is easy to see that

$\displaystyle\tilde{\pi}_{2,f}^{F}-\tilde{\pi}_{2,r}^{F}>0,\,\forall\,d\in[d_{% 1},1),∼{}∼{}∼{}\text{and}∼{}∼{}∼{}\frac{\partial(\tilde{\pi}_{2,f}^{F}-\tilde{% \pi}_{2,r}^{F})}{\partial d}<0,\,\forall\,d\in[d_{1},1).$ (17)

Therefore, we have the following result.

Theorem 2. If the innovation is drastic, then the innovator firm prefers to license its technology by a fixed-fee contract than by a royalty one. Furthermore, this incentive increases with the differentiation of the goods.

2.2 Comparison between two-part tariff and royalty contracts

Now, we compare the profits that both innovator and non-innovator firms earn under two-part tariff and under royalty contracts, in order to conclude which contract is more profitable to the licensor.

2.2.1 Non-drastic innovation (i.e. $d\in(0,d_{1})$ )

For the follower firm, from Eqs (3) and (11), standard computations yield that

$\displaystyle\pi_{2,rf}^{F}-\pi_{2,r}^{F}>0,\,\forall\,d\in(0,d_{1}),∼{}∼{}∼{}% \text{and}∼{}∼{}∼{}\frac{\partial(\pi_{2,rf}^{F}-\pi_{2,r}^{F})}{\partial d}<0% ,\,\forall\,d\in(0,d_{1}).$ (18)

For the leader firm, based on Eqs (3) and (11), we obtain that $\pi_{1,rf}^{F}-\pi_{1,r}^{F}<0,\,\forall\,d\in(0,d_{1}),$ and

$\displaystyle\frac{\partial(\pi_{1,rf}^{F}-\pi_{1,r}^{F})}{\partial d}>0,\,% \forall\,d\in(0,d_{3})\cup(d_{4},d_{1}),∼{}∼{}∼{}\frac{\partial(\pi_{1,rf}^{F}% -\pi_{1,r}^{F})}{\partial d}<0,\,\forall\,d\in(d_{3},d_{4}).$ (19)

Therefore, we have the following result.

Theorem 3. If the innovation is non-drastic, then the innovator firm prefers to license its technology by a two-part tariff contract than by a royalty one. Furthermore, this incentive increases with the differentiation of the goods.

We remark that, for the non-innovator firm it is better a royalty contract than a two-part tariff one, if the innovation is non-drastic. However, the incentive of the non-innovator firm to accept the new technology by a royalty contract instead of a two-part tariff one either increases ( $d\in(d_{3},d_{4})$ ) or decreases ( $d\in(0,d_{3})\cup(d_{4},d_{1})$ ) with the differentiation of the goods.

2.2.2 Drastic innovation (i.e.

d\in[d_{1},1)

)

For the follower firm, based on Eqs (3), (12) and the fact that $\tilde{\pi}_{2,r}^{F}=\pi_{2,r}^{F},\,\forall\,d\in[d_{1},1),$ standard computations yield that

$\displaystyle\tilde{\pi}_{2,rf}^{F}-\tilde{\pi}_{2,r}^{F}>0,\,\forall\,d\in[d_% {1},1),∼{}∼{}∼{}\text{and}∼{}∼{}∼{}\frac{\partial(\tilde{\pi}_{2,rf}^{F}-% \tilde{\pi}_{2,r}^{F})}{\partial d}<0,\,\forall\,d\in[d_{1},1).$ (20)

Hence, we have the following result.

Theorem 4. If the innovation is drastic, then the innovator firm prefers to license its technology by a two-part tariff contract than by a royalty one. Furthermore, this incentive increases with the differentiation of the goods.

We recall that in the drastic innovation case, the profit $\tilde{\pi}_{1,rf}$ is null, and so for the non-innovator firm, also as in the non-drastic innovation case, it is better a royalty contract than a two-part tariff one.

2.3 Comparison between two-part tariff and fixed-fee contracts

Here, we compare the profits that both innovator and non-innovator firms earn under two-part tariff and under fixed-fee contracts, in order to conclude which contract is more profitable to the licensor.

2.3.1 Non-drastic innovation (i.e. $d\in(0,d_{1})$ )

Let $d_{5}$ , $0<d_{5}<1$ , be such that $25d_{5}^{10}-104d_{5}^{9}+124d_{5}^{8}+352d_{5}^{7}-1408d_{5}^{6}+704d_{5}^{5}% +2880d_{5}^{4}-3328d_{5}^{3}-1280d_{5}^{2}+3072d_{5}-1024=0.$ 8

⁸
We note that $d_{5}\simeq 0.569.$

From Eqs (7) and (11), standard computations yield that

$\displaystyle\pi_{2,rf}^{F}-\pi_{2,f}^{F}<0,\,\forall\,d\in(0,d_{5}),∼{}∼{}∼{}% \pi_{2,rf}^{F}-\pi_{2,f}^{F}>0,\,\forall\,d\in(d_{5},d_{1}),$ (21)

and

$\displaystyle\frac{\partial(\pi_{2,f}^{F}-\pi_{2,rf}^{F})}{\partial d}<0,\,% \forall\,d\in(0,d_{5}),∼{}∼{}∼{}\frac{\partial(\pi_{2,rf}^{F}-\pi_{2,f}^{F})}{% \partial d}>0,\,\forall\,d\in(d_{5},d_{1}).$ (22)

For the leader firm, based on Eqs (6) and (11), we obtain that

$\displaystyle\pi_{1,rf}^{F}-\pi_{1,f}^{F}<0,\,\forall\,d\in(0,d_{1}),$ (23)

and

$\displaystyle\frac{\partial(\pi_{1,rf}^{F}-\pi_{1,f}^{F})}{\partial d}>0,\,% \forall\,d\in(0,d_{3})\cup(d_{4},d_{1}),∼{}∼{}∼{}\frac{\partial(\pi_{1,rf}^{F}% -\pi_{1,f}^{F})}{\partial d}<0,\,\forall\,d\in(d_{3},d_{4}).$ (24)

Hence, we have the following result.

Theorem 5. (i) If the goods are sufficiently differentiated ( $d\in(0,d_{5})$ ), then the innovator firm prefers to license its technology by a fixed-fee contract than by a two-part tariff one. Furthermore, this incentive increases with the differentiation of the goods;

(ii) If the goods are neither sufficiently differentiated nor sufficiently homogenous ( $d\in(d_{5},d_{1})$ ), then the innovator firm prefers to license its technology by a two-part tariff contract than by a fixed-fee one. Furthermore, this incentive decreases with the differentiation of the goods.

2.3.2 Drastic innovation (i.e.

d\in[d_{1},1)

)

For the follower firm, based on Eqs (8) and (12), standard computations yield that

$\displaystyle\tilde{\pi}_{2,rf}^{F}-\tilde{\pi}_{2,f}^{F}<0,\,\forall\,d\in[d_% {1},1),∼{}∼{}∼{}\text{and}∼{}∼{}∼{}\frac{\partial(\tilde{\pi}_{2,rf}^{F}-% \tilde{\pi}_{2,f}^{F})}{\partial d}>0,\,\forall\,d\in[d_{1},1).$ (25)

Therefore, we have the following result.

Theorem 6. If the innovation is drastic, then the innovator firm prefers to license its technology by a fixed-fee contract than by a two-part tariff one. Furthermore, this incentive decreases with the differentiation of the goods.

We recall that in the drastic innovation case, the profits $\tilde{\pi}_{1,f}^{F}$ and $\tilde{\pi}_{1,rf}^{F}$ are null, and so the comparison in this case does not make sense.

3. Conclusions

By considering a differentiated-good Stackelberg duopoly, the present paper emphasizes the cases in which it is indicated for the innovator firm (follower firm) to license its technology to the non-innovator firm (leader firm). We point out the main results in both non-drastic or drastic innovation cases (from Li and Ji [22] we note that the innovation can be either non-drastic or drastic). We computed explicitly the profits of these duopoly models. We conclude that the degree of the differentiation of the goods has a great significance in the results.

Footnotes

Acknowledgments

We thank the anonymous reviewers for their helpful and constructive comments that greatly contributed to improving the final version of the paper. Author F. Ferreira thanks to UNIAG, R&D unit funded by FCT - Portuguese Foundation for the Development of Science and Technology, Ministry of Science, Technology and Higher Education, under the Projects UID/GES/04752/2019 and UIDB/04752/2020.

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Comparison between different licensing schemes in a Stackelberg model when the follower is the innovator

Abstract

Keywords

1. Introduction

3 Throughout the paper we use the notation subscript r to refer to the per-unit royalty licensing case.

2.1.1 Non-drastic innovation (i.e. d ∈ ( 0 , d 1 ) )

2.2.1 Non-drastic innovation (i.e. d ∈ ( 0 , d 1 ) )

2.3.1 Non-drastic innovation (i.e. d ∈ ( 0 , d 1 ) )

8 We note that d 5 ≃ 0.569 .

Footnotes

Acknowledgments

References

³
Throughout the paper we use the notation subscript $r$ to refer to the per-unit royalty licensing case.

2.1.1 Non-drastic innovation (i.e. $d\in(0,d_{1})$ )

2.2.1 Non-drastic innovation (i.e. $d\in(0,d_{1})$ )

2.3.1 Non-drastic innovation (i.e. $d\in(0,d_{1})$ )

⁸
We note that $d_{5}\simeq 0.569.$