Technology Transfer and Imitation in a Cournot Oligopoly

Abstract

I examine and compare patent licensing by fixed fee and unit royalty under Cournot competition. I consider licensing by an incumbent patent holder to one or two other competing firms that can obtain a patented technological improvement through technology transfer or imitation. Assuming that imitation is perfect, certain, instantaneous, and non-infringing, I analyze the effects of licensing on market structure, firms’ individual profits, and consumer surplus. This provides a theoretical framework that explains when technology licensing is superior to imitation for both firms and consumers, what is the optimal licensing choice for firms, and how imitation affects firms’ licensing behavior and competition in a highly concentrated industry. In particular, I show that licensing through a unit royalty is preferable to licensing through a fixed fee for a patent holder, while licensing through a fixed fee is at least as beneficial as licensing through a unit royalty for consumers. Moreover, the patent holder can use licensing to prevent imitation, but cannot use it selectively to affect competition, at least before the patent expires and when one of the competing firms can imitate. I contribute to the literature that considers the patent holder as a producer by showing how technology licensing can affect competition and improve consumer surplus in oligopolistic industries. This is important for policy makers to identify when technology licensing is used strategically to transfer surplus from consumers to producers.

Introduction

Patents emerged as a system that grants privileges to inventors by providing territorial protection for a certain period of time. Essentially, a patent is a legal title enforceable in court and granted for an invention with new, non-obvious and useful technical features with potential industrial application (Taylor & Silberston, 1973). The patent system can be seen as an institution of ownership for inventions or technological improvements and a source of scientific information. Patenting thus serves as a mechanism that facilitates the diffusion of technology and promotes progress. Typically, there are two conventional sources of profit for a patent holder: working with the patent itself and licensing the patent rights for a royalty (Nordhaus, 1969; Harhoff et al., 2007; Pepall et al., 2008, Chapter13). Patent licensing is an important source of profitability for innovative firms and a valuable source of information for an industry.¹ In addition, firms use patent protection strategically to gain a competitive advantage over their rivals. For example, a common reason firms license their patents is to prevent competitors from developing similar or superior technologies. Typically, firms can use patent licensing to deter entry, prevent imitation, or influence competition (Rockett, 1990; Denicolo & Franzoni, 2003). However, technology transfer is also a form of collaboration between firms focused on value creation that benefits all members of the ecosystem (Primario et al., 2024). Overall, licensing is an important, inseparable extension of patents, considering that most technology transfer agreements are made through the patent system.

While several studies consider an external patent holder, a duopoly setup, or the effects of imitation independently, I analyze an incumbent patent holder under the threat of imitation competing in an oligopolistic industry. I provide a framework to show how imitation affects the market equilibrium (Wang, 1998; Sen & Tauman, 2007; Colombo & Filippini, 2015). The objective of this paper is to understand the factors that increase firms’ profits and consumer surplus by improving production efficiency through technology transfer. This includes identifying surplus thresholds at which consumers are indifferent to licensing or imitation and the effect on consumer surplus is positive, as well as thresholds that lead to the emergence of monopolies. Analyzing the combined effect of factors such as cost reduction in production, imitation, and licensing choice on firms’ licensing policies has been neglected in previous studies, making this analysis novel and interesting. The results of this paper will help to understand this research gap by developing a theoretical framework that analyzes the incentives and strategic motivations of innovating firms to license new technologies. In general, this theoretical framework can explain why firms in knowledge-based industries, such as information and communication technology (ICT), semiconductor, electronics, and other highly concentrated industries, enter into technology licensing agreements (Grindley & Teece, 1997; Cohen et al., 2000; Torrisi, 2011).

Utilizing Wang (1998)’s model, which examines patent licensing in a duopoly context, I consider an oligopolistic industry composed of firms with asymmetric unit production costs. One of the firms can imitate a superior patented technology. Specifically, I compare fixed-fee and per-unit royalty licensing of a technological improvement that reduces unit costs in an industry with two and three incumbents producing a homogeneous product.² I assume that the firms compete simultaneously in quantities, i.e. a Cournot competition (Cournot, 1838). Suppose one of the firms develops and patents a cost-reducing technology, and at least one of the other competing firms can obtain the protected invention through technology transfer or imitation. I assume that the imitation is perfect, occurs with certainty, and carries no risk of infringement. The patent holder could offer an exclusive license to either or both competitors.³ This decision also depends on the size of the invention or the cost reduction in production due to the protected technology. I consider both non-drastic and drastic technological improvements and show when each licensing choice dominates. For example, the patent holder of a drastic invention prefers to become a monopolist rather than a licensor.⁴ Recognizing that firms license their proprietary technologies to increase their profits, the patent holder will ensure that the licensing agreement maximizes its profit.

I contribute to the literature on technology licensing and competition in several ways. First, I show when technology transfer occurs in equilibrium and imitation is dominant. Second, I determine the optimal licensing choice from the perspective of the patent holder and consumers in an oligopoly setting. Third, I analyze the effects of imitation on firms’ licensing behavior and competition. In particular, I show that royalty licensing is at least as beneficial to the patent holder as fixed-fee licensing, while it is at most as beneficial to consumers as fixed-fee licensing. I also show that per-unit royalty licensing, whether exclusive or non-exclusive, occurs in equilibrium in several situations. In addition, I find that the patent holder can use licensing to prevent imitation. These results are consistent with propositions in the existing literature (Wang, 1998; Denicolo & Franzoni, 2003). Finally, licensing under the threat of imitation does not appear to be selective; thus, at least before patent expiration, the patent holder cannot use it to affect competition such as by offering an exclusive license to a weaker competitor to undermine a stronger competitor.

The analysis has implications for policy makers and practitioners, as it identifies factors that facilitate technology licensing and affect competition and consumer surplus. In addition, this theoretical framework can explain technology transfer in highly concentrated industries among competing firms and identify surplus thresholds that indicate when cooperation may lead to monopolies that may be challenged by competition authorities (Baumol, 1992; Rey & Tirole, 2013; Lindblom et al., 2024). Any such intervention will affect innovation and learning processes, which should be incorporated into existing policies to promote industrial development (Lundvall, 2007). Understanding the pro- and anti-competitive aspects of cooperation through technology licensing can assist policy makers in revising their existing policies (Weerasinghe et al., 2024).

The rest of this paper presents an analysis of the issues outlined above. Section 2 provides a review of selected research on patent licensing. In Section 3, I present the problem in the context of a Cournot duopoly. Section 4 presents an oligopoly model of (non-cooperative) licensing for both non-drastic and drastic inventions, considering a three-firm industry with asymmetric unit costs. Section 5 concludes and suggests avenues for future research. All proofs are given in the Appendix.

Related Research

Technology transfer through patent licensing has become a key component for firms (and countries) to gain competitive advantage and stay at the frontier (Hu et al., 2023; Yoon & Kwon, 2023). Licensing allows a patent holder to retain control over an invention and to impose conditions on its use. In essence, a patent holder has an enforceable right to decide who may use a patented technological improvement and who may not. One of the earliest studies of patent law is Smith (1890), who argues that it is only through the patent system that the human passion for improvement and innovation can acquire a marketable value. Arrow (1962) put patents into an economic perspective by formally analysing the profits that a patent holder can realize from licensing a cost-reducing technology to a perfectly competitive industry.⁵ In a similar context, Taylor and Silberston (1973) analyze the economic impact of the patent system by focusing on patent holders that are incumbents in an industry. The authors suggest that the patent system may not provide firms with a perfect appropriability of the returns to their investments in research and development because patent protection is not absolute and thus cannot prevent imitation.⁶

In the 1980s, studies by Gilbert and Newbery (1982), Gallini (1984), Kamien and Tauman (1986), Shapiro (1985), Katz and Shapiro (1985, 1986), Shepard (1987), and Kamien et al. (1988) laid the foundation for a game-theoretic approach to optimal patent licensing, which later became the basis for subsequent work on technology transfer in oligopolistic industries.⁷ The game-theoretic research analyzes and compares the most common licensing choices of a patented technology, i.e. auction licensing, fixed-fee licensing, per-unit royalty licensing, ad valorem royalty licensing, and a combination of an up-front fixed fee and a per-unit royalty.⁸ For example, Kamien and Tauman (1986) compare the performance of fixed fees and royalties and find that fixed-fee licensing is preferable to a per-unit royalty from the perspective of both the patent holder and consumers. Marjit (1990) considers technology transfer between two incumbents in a Cournot industry and suggests that the smaller the technology gap between the two firms, the more feasible technology transfer is. Wang (1998) compares licensing modes in a homogeneous-good Cournot duopoly with an incumbent patent holder. The author concludes that royalty licensing dominates fixed-fee licensing in equilibrium due to the unit cost advantage; a per-unit royalty always increases the licensee’s unit cost of production by the amount of the royalty. Extending this analysis to a differentiated Cournot duopoly, Wang (2002) reaffirms the superiority of royalty licensing. Mukhopadhyay et al. (1999) study technology transfer through patents in a generalized oligopolistic industry. The authors suggest that cost asymmetry may be a key determinant of technology transfer. Kamien and Tauman (2002) use Wang (1998)’s model to characterize a general Cournot oligopoly market and find that an incumbent patent holder operating in an industry composed of a large number of firms is always better off licensing through a per-unit royalty than through a fixed fee.

Fauli-Oller and Sandonis (2002) study two-part tariff contracts to license a cost-reducing invention to a rival firm and find that in a differentiated-goods Bertrand (Bertrand, 1883) and Cournot duopoly, a positive royalty is always part of an optimal licensing contract.⁹ Filippini (2002) performs the same analysis in a Stackelberg (Stackelberg, 2011) duopoly and confirms that the patent holder’s profit under royalty licensing exceeds that under fixed-fee licensing. Filippini (2005) shows that per-unit royalties still prevail in equilibrium; however, royalty licensing may reduce social welfare and does not benefit consumers. Sen (2002) proposes a simple royalty licensing contract that allows an incumbent firm with a cost-reducing technology to earn a monopoly profit in a Cournot oligopoly market with at least three firms. In a follow-up study, Sen (2005) shows that royalties can dominate fixed-fee and auction arrangements in a general Cournot oligopoly with an outsider inventor.¹⁰ Sen and Tauman (2007) extend the existing work on two-part tariff contracts by considering the licensing of a cost-reducing technology in a Cournot oligopoly of general size for both incumbent and outside inventors. The authors show that relatively important inventions are licensed through a unit royalty, while less important inventions are not licensed through a unit royalty. In a similar context, I show how the licensing choice depends on the size of the invention and when technology transfer does not occur for improvements of a certain size.

In an empirical setting, Mansfield et al. (1981) and Mansfield (1985) examine the effects of imitation speed and cost on the incentives to innovate and patent. Their results suggest that the disclosure of valuable technical information and the ease of imitation may explain the increase in imitation activity and imperfect appropriability. I also analyze the behavior of innovating firms by focusing on the effects of the level of innovation and imitation costs on technology transfer. Gallini (2002) suggests that licensing may reduce the number of costly infringement disputes because it discourages imitation. On the one hand, strong enforcement of property rights increases imitation costs and thus may improve the appropriability problem (Panagopoulos, 2011). On the other hand, it leads to entry barriers and thus may hinder innovation (Bessen & Hunt, 2007). Kitagawa et al. (2014) analyze the case where an incumbent firm can license a new technology to a competitor who can imitate the protected invention without risking patent infringement. Unlike most existing research in this area, the authors acknowledge that imitation often produces imperfect substitutes for the new product, and point to the important influence of technology development costs on optimal patent licensing. Other studies have also discussed the effects of imitation on competition and interactions between competing firms in the same industry (Gambardella & McGahan, 2010; Frankenberger & Stam, 2020). In this context, Laudien and Fernandez (2023) focus on the effect of market entry, especially by large firms, on industrial structure. They emphasize the importance of the size effect, which is related to innovation, on the competitiveness of small incumbents. Durmaz and Polat (2020) empirically examine research and development spillovers in the computer industry across countries and argue that the growth of domestic output depends not only on domestic research capital and imports of capital and goods, but also on technology diffusion, learning progress, and imitation-enhancing activities (see, also, Coe et al., 2009; Chui et al., 2023). Gossart et al. (2020) address the technological dimension of patent litigation, which, similar to imitation, discourages innovation.

The Problem Statement

Consider an industry consisting of two identical incumbent firms that is, Firm i for i=$\{1, 2\}$. The inverse demand function for the good is P=a-Q, where P is the price of the good, Q is the output of the industry, and a is an industry parameter characterizing the demand for the good. Let $q_{i}$ for $i=\{1, 2\}$ be the quantity of homogeneous goods produced by Firm i at a unit production cost of $c_{i}$ for $i=\{1, 2\}$, where $0<c_{i}<a$. Then let $\pi _{i}$ for $i=\{1, 2\}$ be the profit of Firm i:

$$\begin{aligned} \pi _{i} = (P-c_{i})q_{i} \text{, } \text{ for } i={1, 2}\text{. } \end{aligned}$$

(1)

Let the superscript $*$ denote the value of a variable at equilibrium. Then, from the first-order condition of each firm’s profit maximization, I derive the common Nash-Cournot equilibrium outputs, profits, and market price as follows:¹¹

$$\begin{aligned} \begin{array}{llll} q^{*}_{i} & =& \dfrac{1}{3}(a-2c_{i}+c_{j}) \text{, } \text{ for } i \ne j={1, 2}\text{, }\\ \\ \pi ^{*}_{i} & =& \dfrac{1}{9}{(a-2c_{i}+c_{j})}^{2} \text{, } \text{ for } i \ne j={1, 2}\text{, }\\ \\ P^{*} & =& \frac{1}{3}(a-c_{1}+c_{2}). \end{array} \end{aligned}$$

(2)

Suppose that Firm 1 develops a new technology that reduces the per-unit cost of producing a homogeneous good relative to the common per-unit cost c by an amount $\epsilon$, where $0<\epsilon <c$.¹² In other words, $\epsilon$ reflects the exogenously determined size of the cost reduction or innovation. Now consider whether Firm 1 should offer to transfer technology to Firm 2 in exchange for a royalty. In this case, Firm 2 should decide whether to accept the offer; not accepting the offer implies that Firm 2 will either produce using the existing technology or imitate it.¹³

First, suppose that no technology transfer takes place because Firm 1 does not offer a license or Firm 2 does not accept the offer. In this case, Firm 1 uses the cost-reducing technology and has a unit cost of $c-\epsilon$. Firm 2, however, can choose to use the existing technology or to imitate at a lump-sum cost of C, where $C>0$. If Firm 2 chooses to use the existing technology and thus not to imitate, then a direct substitution of $c_1=c-\epsilon$ and $c_2=c$ in Eq. 2 yields the following Nash-Cournot profits of Firms 1 and 2:

$$\begin{aligned} \begin{array}{llllll} {\pi ^{ND}_{1}}^{*} & =& \dfrac{1}{9}{(a-c+2\epsilon )}^{2} & \text{ for } \epsilon< a-c,\\ \\ & =& \dfrac{1}{4}(a-c+\epsilon )^{2} & \text{ otherwise },\\ \\ {\pi ^{ND}_{2}}^{*} & =& \dfrac{1}{9}{(a-c-\epsilon )}^{2} & \text{ for } \epsilon < a-c,\\ \\ & =& 0 & \text{ otherwise }, \end{array} \end{aligned}$$

(3)

where $\pi ^{ND}_{i}$ for $i=\{1, 2\}$ is the profit of Firm i if there is no imitation. Note that the profit of Firm 1 is always positive; however, Firm 2 produces a quantity greater than zero only if $\epsilon <a-c$. Firm 2 produces a quantity equal to zero if $\epsilon \ge a-c$, in which case Firm 1 becomes a monopoly. According to the existing literature, a cost-reducing invention is drastic if the price charged by a monopolist using the new technology is less than or equal to the unit production cost associated with the existing technology (Arrow, 1962). In this context, a drastic invention is one that is large enough to drive competitors out of the industry. In particular, if $\epsilon <a-c$, then the invention is non-drastic, and if $\epsilon \ge a-c$, then the invention is drastic.

Now suppose that Firm 2 chooses to imitate, and let $\pi ^{D}_{i}$ for $i=\{1, 2\}$ be the profit of Firm i if imitation has occurred. Assume that imitation, which is instantaneous and has a probability of success of one, results in a cost reduction of $\epsilon$. Then Firm 2’s unit cost is $c_{2}=c-\epsilon$. Considering Firm 2’s imitation costs and substituting $c_{1}=c_{2}=c-\epsilon$ in Eq. 2, I find the following profits in equilibrium:

$$\begin{aligned} \begin{array}{llll} {\pi ^{D}_{1}}^{*} & =& \dfrac{1}{9}{(a-c+\epsilon )}^{2},\\ \\ {\pi ^{D}_{2}}^{*} & =& \dfrac{1}{9}{(a-c+\epsilon )}^{2}-C. \end{array} \end{aligned}$$

(4)

It is clear that Firm 2 will never imitate if $C>\frac{1}{9}{(a-c+\epsilon )}^{2}$. It is also clear that Firm 2 will choose to imitate if it gains at least as much as if it chooses not to imitate. Comparing ${\pi ^{ND}_{2}}^{*}$ and ${\pi ^{D}_{2}}^{*}$, I find that Firm 2 will imitate a non-drastic technological improvement if $C\le \frac{4}{9}(a-c)\epsilon$ and a drastic one if $C\le \frac{1}{9}{(a-c+\epsilon )}^{2}$. I assume that imitation is preferable to using the existing technology if Firm 2 is indifferent between these choices.

A Duopoly Model of Technology Transfer

I now consider the case where technology transfer occurs. A technology transfer is an agreement that will occur if each firm is better off accepting the agreement than rejecting it. Let F for $F > 0$ be a fixed fee. In this case, both firms use the cost-reducing technology and have the same unit cost, $c_{1}=c_{2}=c-\epsilon$. In addition, let $\pi ^{F}_{i}$ for $i=\{1, 2\}$ be the profit of Firm i if a fixed-fee technology transfer occurs. Considering the fee transfer and substituting the unit costs in Eq. 2, we obtain $\pi ^{F}_{1}=\frac{1}{9}{(a-c+\epsilon )}^{2}+F$ and $\pi ^{F}_{2}=\frac{1}{9}{(a-c+\epsilon )}^{2}-F$. Clearly, Firm 2 will accept a licensing offer if the profit it will gain is equal to or greater than its profit from using the existing technology or imitation. In other words, if $\pi ^{F}_{2} \ge \text {max}$ $({\pi ^{D}_{2}}^{*}, {\pi ^{ND}_{2}}^{*})$, Firm 2 is better off with a technology transfer. Therefore, the maximum fee that Firm 1 can charge Firm 2 in this case is:

$$\begin{aligned} F^{*}= & \left\{ \begin{array}{ll} \text{ min } \text{( }\frac{4}{9}[a-c]\epsilon \text{, } \text{ C) } & \text{ for } \epsilon < a-c ,\\ \\ \text{ min } \text{( }\frac{1}{9}[a-c+\epsilon ]^2\text{, } \text{ C) } & \text{ otherwise }. \end{array} \right. \end{aligned}$$

(5)

According to the Eq. 5, if the technological improvement is non-drastic, the patent holder will charge a fixed fee equal to the minimum of $\frac{4}{9}(a-c)\epsilon$ and cost of imitation. However, if the invention is drastic, the maximum fixed fee is equal to the imitation cost, considering that Firm 2 will never imitate if $C > \frac{1}{9}(a-c+\epsilon )^2$. It follows that the equilibrium profits are:

$$\begin{aligned} \begin{array}{ll} {\pi ^{F}_{1}}^{*}=\dfrac{1}{9}{(a-c+\epsilon )}^{2}+F^{*},\\ \\ {\pi ^{F}_{2}}^{*}=\dfrac{1}{9}{(a-c+\epsilon )}^{2}-F^{*}, \end{array} \end{aligned}$$

(6)

where the Eq. 5 defines $F^{*}$.

I can now determine when technology transfer occurs in equilibrium. First, suppose that using the existing technology is better than imitation for Firm 2. Comparing ${\pi _{1}^{F}}^{*}$ and ${\pi _{1}^{ND}}^{*}$, I find that Firm 1 will always transfer the patented technology if $\epsilon < \frac{2}{3}(a-c)$. However, if $\frac{2}{3}(a-c) \le \epsilon < a-c$, no technology transfer will occur in equilibrium. In this case, both firms will produce a quantity greater than zero, in contrast to the case of drastic invention, where Firm 2 becomes a monopolist. Now suppose that imitation is at least as beneficial to Firm 2 as not imitating. Then technology transfer will always occur in equilibrium, regardless of the nature of the technological improvement. I formulate the following proposition:

Proposition 1

Technology transfer through a fixed-fee license will occur if and only if

• 1. $0<\epsilon <\dfrac{2}{3}(a-c)$ or

• 2. $\dfrac{2}{3}(a-c) \le \epsilon < a-c$ and $C \le \dfrac{4}{9}(a-c)\epsilon$, or

• 3. $\epsilon \ge a-c$ and $C \le \dfrac{1}{9}(a-c+\epsilon )^{2}$.

Fig. 1

Illustration of equilibrium under fixed-fee licensing

Figure 1 below illustrates Proposition 1. The region above the $\epsilon$-axis corresponds to the case where using the existing technology is better than imitating for Firm 2, while the region below the $\epsilon$-axis corresponds to the case where imitating is at least as beneficial as choosing not to imitate. According to the Fig. 1, technology transfer will always occur in the region above the $\epsilon$-axis and to the left of the dotted line, as well as in the region below the $\epsilon$-axis.

Now consider the case of technology transfer through royalty licensing. Let r, for $r>0$ denote the per-unit royalty rate that Firm 1 will charge Firm 2 in return for transferring the patented technology. In this case, the unit cost is $c_1=c-\epsilon$ for Firm 1 and $c_2=c-\epsilon +r$ for Firm 2. In addition, let $\pi _{i}^{R}$ for $i=\{1, 2\}$ be the profit of Firm i if technology transfer occurs through a per-unit royalty. Then, considering that the patent holder’s profit depends on the quantity that Firm 2 will produce using the patented technology, and substituting the unit costs in Eq. 2, it follows that $\pi ^{R}_{1}=\frac{1}{9}{(a-c+\epsilon +r)}^{2}+\frac{1}{3}(a-c+\epsilon -2r)r$ and $\pi ^{R}_{2}=\frac{1}{9}{(a-c+\epsilon -2r)}^{2}$. Clearly, the patent holder will charge Firm 2 a unit royalty that maximizes its profit, while Firm 2 will accept the offer if $\pi ^{R}_{2} \ge \text {max}$ $({\pi ^{ND}_{2}}^{*}, {\pi ^{D}_{2}}^{*})$. Solving the maximization problem of Firm 1, I find that the unit royalty that satisfies these conditions is:

$$\begin{aligned} r^{*}= & \left\{ \begin{array}{ll} \text{ min } \text{( }\epsilon \text{, } \frac{1}{2}[a-c+\epsilon ]-\frac{1}{2}\sqrt{(a-c+\epsilon )^2-9C}\text{) } & \text{ for } \epsilon < a-c,\\ \\ \text{ min } \text{( }\frac{1}{2}[a-c+\epsilon ]\text{, } \frac{1}{2}[a-c+\epsilon ]-\frac{1}{2}\sqrt{(a-c+\epsilon )^2-9C}\text{) } & \text{ otherwise }. \end{array} \right. \end{aligned}$$

(7)

Proof

See the Appendix.

According to the Eq. 7, if the technological improvement is non-drastic, the optimal per-unit royalty is equal to the minimum of the size of the innovation and the second term in parentheses. In the case of a drastic invention, however, the optimal royalty is equal to $\frac{1}{2}(a-c+\epsilon )-\frac{1}{2}\sqrt{(a-c+\epsilon )^2-9C}$, i.e. the maximum royalty in this case is equal to $\frac{1}{2}(a-c+\epsilon )$. Recall that when $C>\frac{1}{9}(a-c+\epsilon )^{2}$, imitation never occurs and the expression under the square root in Eq. 7 becomes negative. Therefore, in this case, I define the second term in parentheses only for non-negative values.¹⁴

Obviously, the optimal unit royalty cannot be larger than the size of the innovation in any case. The equilibrium profits of the firms in this case are:

$$\begin{aligned} \begin{array}{llll} {\pi ^{R}_{1}}^{*} & =& \dfrac{1}{9}{(a-c+\epsilon +r^{*})}^{2}+\dfrac{1}{3}(a-c+\epsilon -2r^{*})r^{*},\\ \\ {\pi ^{R}_{2}}^{*} & =& \dfrac{1}{9}{(a-c+\epsilon -2r^{*})}^{2}, \end{array} \end{aligned}$$

(8)

where the Eq. 7 defines $r^{*}$. I can now determine when technology transfer occurs in equilibrium. Suppose that using the existing technology is better than imitation for Firm 2. Then, by comparing the equilibrium profits of the patent holder, I find that technology transfer will always occur in equilibrium if the technological improvement is non-drastic. However, in the case of a drastic invention, Firm 1 is indifferent between licensing and not licensing. In this case, Firm 1 will always earn the monopoly rent and Firm 2 will earn no profit with or without technology transfer. I assume that Firm 1 becomes a monopolist in this case. Suppose that imitation is at least as beneficial to Firm 2 as not imitating. Then, with either a drastic or non-drastic invention, technology transfer will always occur in equilibrium. I state the following proposition:

Proposition 2

Technology transfer through a per-unit royalty license will occur if and only if:

• 1. $0<\epsilon <a-c$ or

• 2. $\epsilon \ge a-c$ and $C \le \dfrac{1}{9}(a-c+\epsilon )^{2}$.

Figure 2 illustrates Proposition 2. The region above the $\epsilon$-axis corresponds to the case where existing technology is better than imitation for Firm 2, while the region below the $\epsilon$-axis corresponds to the case where imitation is at least as beneficial as choosing not to imitate. According to the Fig. 2, licensing will occur in the region above the $\epsilon$-axis and to the left of the dotted line, as well as in the region below the $\epsilon$-axis.

Fig. 2

Illustration of equilibrium under royalty licensing

Finally, I will compare the patent holder’s equilibrium profits under the two licensing choices and analyze the impact of technology transfer on consumer surplus (Singh & Vives, 1984; Fauli-Oller & Sandonis, 2002; Tian, 2016). First, consider the case where using the existing technology is better than imitation for Firm 2; that is, ${\pi _{2}^{ND}}^{*} > {\pi _{2}^{D}}^{*}$. If $0<\epsilon <\frac{2}{3}(a-c)$, technology transfer could occur through both a fixed fee and a unit royalty. However, a comparison shows that royalty licensing is better than fixed-fee licensing from the patent holder’s perspective because ${\pi ^{R}_{1}}^{*}>{\pi ^{F}_{1}}^{*}$. In addition, let $q_{i}^{F}$ and $q_{i}^{R}$ be the quantity of Firm i for $i=\{1, 2\}$ under fixed-fee and royalty licensing, respectively. Then, from the consumer’s point of view, fixed-fee licensing is more attractive than per-unit royalty licensing because ${q^{F}_{1}}^{*}+{q^{F}_{2}}^{*} > {q^{R}_{1}}^{*}+{q^{R}_{2}}^{*}$. However, if $\frac{2}{3}(a-c) \le \epsilon < a-c$, technology transfer occurs only through a unit royalty. Therefore, royalty licensing is also better than fixed-fee licensing for the patent holder in this interval of innovation size. Let $q^{ND}_{i}$ be the quantity of Firm i for $i=\{1, 2\}$ if Firm 2 chooses not to imitate. Then consumers are the same with or without technology transfer since ${q^{ND}_{1}}^{*}+{q^{ND}_{2}}^{*}={q^{R}_{1}}^{*}+{q^{R}_{2}}^{*}$. Finally, if the invention is drastic, no technology transfer will occur in equilibrium under either licensing choice. Now consider the case where imitation is at least as beneficial to Firm 2 as using the existing technology; that is, ${\pi _{2}^{D}}^{*} \ge {\pi _{2}^{ND}}^{*}$. In this case, technology transfer occurs through both a fixed-fee and a unit royalty. However, comparing the patent holder’s profits under the two licensing regimes shows that, regardless of the type of technological improvement, royalty licensing is better than fixed-fee licensing because ${\pi ^{R}_{1}}^{*}>{\pi ^{F}_{1}}^{*}$ $\forall \epsilon$. Furthermore, licensing through a fixed fee is always better than licensing through a unit royalty for consumers, since ${q^{F}_{1}}^{*}+{q^{F}_{2}}^{*}>{q^{R}_{1}}^{*}+{q^{R}_{2}}^{*}$ $\forall \epsilon$. I establish the following result:

Proposition 3

The equilibrium of the duopoly game when Firm 1 has a licensing choice between a fixed fee and a unit royalty is characterized as follows.

• 1. Technology transfer, whether or not the technological improvement is drastic, through a per-unit royalty license is at least as beneficial to the patent holder as technology transfer through a fixed-fee license.

• 2. Technology transfer through a fixed-fee license is at least as beneficial to consumers as a technology transfer through a unit royalty.

• 3. No technology transfer will occur under either a fixed-fee or a per-unit royalty license if and only if the technological improvement is drastic and the imitation costs are sufficiently high, especially if $\epsilon \ge a-c$ and $C > \frac{1}{9}(a-c+\epsilon )^{2}$. In this case, Firm 1 becomes a monopoly.

Proposition 3 suggests that, in equilibrium, royalty licensing is at least as beneficial to the patent holder as fixed-fee licensing, and at most as beneficial to consumers as fixed-fee licensing. The patent holder’s cost-advantage explains the superiority of the unit royalty over a fixed fee. I showed that allowing imitation does not change Wang (1998)’s results about fixed-fee and royalty licensing in a homogeneous-good Cournot duopoly.

An Oligopoly Model of Technology Transfer

Consider a non-cooperative licensing game with three players with asymmetric costs: a patent holder, a strong competitor that can imitate the protected technology, and a weak competitor that has the highest unit cost in the industry and cannot imitate the new technology. Moreover, the game is one of complete information, and imitation is instantaneous and certain. The results from this specification suggest that royalty licensing may be at least as desirable as fixed-fee licensing. In addition, a patent holder can use licensing to prevent imitation, but not to choose competition during the patent term.

Let $q_{i}$ and $\pi _{i}$ for $i=\{1,2,3\}$ be the quantity and profit of Firm i for $i=\{1,2,3\}$, respectively. In addition, let the inverse demand function for a homogeneous good be $P=a-Q$, where a is an industry parameter characterizing the market demand for the product, and Q is the output of the industry. Obviously, industry output is equal to the sum of the quantities produced by each firm in the industry, so $Q=\sum _{i=1}^{3}q_{i}$. Let also $c_{i}$ for $i=\{1,2,3\}$, where $0<c_{i}<a$, be the unit cost for Firm i. I use the same procedure as in the duopoly game to obtain the firm’s Nash-Cournot quantity and profit in equilibrium:

$$\begin{aligned} \begin{array}{llll} q^{*}_{i} & =& \dfrac{1}{4}(a-3c_{i}+c_{j}+c_{z}) \text{, } \text{ for } i \ne j \ne z = {1, 2, 3}\text{, }\\ \\ \pi ^{*}_{i} & =& \dfrac{1}{16}{(a-3c_{i}+c_{j}+c_{z})}^{2} \text{, } \text{ for } i \ne j \ne z = {1, 2, 3}\text{. } \end{array} \end{aligned}$$

(9)

Suppose the cost per unit is the same for Firms 1 and 2, specifically $c_{1}=c_{2}=c$, while the unit cost for Firm 3 is $c_{3}=c'$, where $c'>c>0$. Clearly, Firm 3 has the highest unit cost in the industry. Therefore, I can call Firm 3 the weak competitor. Suppose Firm 1 develops and patents a technological improvement that reduces the unit cost by $\epsilon$, where $0<\epsilon <c$. In this case, Firm 1 is the patent holder and Firm 2 can be considered the strong competitor. Assume also that the weak competitor can only obtain the patented invention through technology transfer.¹⁵ However, the strong competitor can obtain the invention either through technology transfer or through imitation by incurring an up-front positive cost C. Define $\pi _{i}^{ND}$ for $i=\{1,2,3\}$ to be the profit of Firm i if Firm 2 uses the existing technology. Then substituting $c_{1}=c-\epsilon$, $c_{2}=c$, and $c_{3}=c'$ in 9 gives the profits of the firms as follows: $\pi ^{ND}_{1}=\frac{1}{16}{(a-2c+3\epsilon +c')}^{2}$, $\pi ^{ND}_{2}=\frac{1}{16}{(a-2c-\epsilon +c')}^{2}$, and $\pi ^{ND}_{3}=\frac{1}{16}{(a+2c-\epsilon -3c')}^{2}$. Firm 2’ quantity and therefore its profit, is positive only if $\epsilon < a-2c+c'$. That is, if $\epsilon \ge a-2c+c'$, the technological improvement will drive the strong competitor and obviously the weak competitor out of the industry. In this case, the invention is drastic, and Firm 1 will earn a monopoly profit, while Firms 2 and 3 will earn no profit. Furthermore, Firm 3’s profit is positive if $\epsilon < a+2c-3c'$, since $c'<\frac{1}{3}(a+2c)$.¹⁶ However, if $\epsilon \ge a+2c-3c'$, then the weak competitor will earn no profit, and thus the industry can become either a duopoly or a monopoly, depending on the relative levels of $\epsilon$ and $a-2c+c'$. Specifically, the industry becomes a duopoly if $a+2c-3c' \le \epsilon < a-2c+c'$ and a monopoly if $\epsilon \ge a-2c+c'$. In this case, the equilibrium profits of the firms are:

$$\begin{aligned} \begin{array}{llllll} {\pi ^{ND}_{1}}^{*} & =& \dfrac{1}{16}{(a-2c+3\epsilon +c')}^{2} & \text{ for } \epsilon< a+2c-3c',\\ \\ & =& \dfrac{1}{9}{(a-c+2\epsilon )}^{2} & \text{ for } a+2c-3c' \le \epsilon< a-2c+c',\\ \\ & =& \dfrac{1}{4}{(a-c+\epsilon )}^{2} & \text{ otherwise },\\ \\ {\pi ^{ND}_{2}}^{*} & =& \dfrac{1}{16}{(a-2c-\epsilon +c')}^{2} & \text{ for } \epsilon< a+2c-3c',\\ \\ & =& \dfrac{1}{9}{(a-c-\epsilon )}^{2} & \text{ for } a+2c-3c' \le \epsilon< a-2c+c',\\ \\ & =& 0 & \text{ otherwise },\\ \\ {\pi ^{ND}_{3}}^{*} & =& \dfrac{1}{16}{(a+2c-\epsilon -3c')}^{2} & \text{ for } \epsilon < a+2c-3c',\\ \\ & =& 0 & \text{ otherwise }. \end{array} \end{aligned}$$

(10)

Suppose that Firm 2 chooses to imitate, and define $\pi ^{D}_{i}$ for $i=\{1, 2, 3\}$ as the profit of Firm i in this case. Similar to the duopoly game, imitation reduces the unit cost by $\epsilon$. Considering the imitation cost of C incurred by Firm 2, and substituting $c_{1}=c_{2}=c-\epsilon$ and $c_{3}=c'$ in Eq. 9, I find the firms’ profits as follows: ${\pi ^{D}_{1}}=\frac{1}{16}{(a-2[c-\epsilon ]+c')}^{2}$, ${\pi ^{D}_{2}}=\frac{1}{16}{(a-2[c-\epsilon ]+c')}^{2}-C$, and ${\pi ^{D}_{3}}=\frac{1}{16}{(a+2[c-\epsilon ]-3c')}^{2}$. That is, Firm 2 will consider imitation if $D<\frac{1}{16}{(a-2[c-\epsilon ]+c')}^{2}$, while Firm 3 will earn a profit greater than zero if $\epsilon <\frac{1}{2}(a+2c-3c')$, since $c'<\frac{1}{3}(a+2c)$. The equilibrium profits in this case are then:

$$\begin{aligned} \begin{array}{llllll} {\pi ^{D}_{1}}^{*} & =& \dfrac{1}{16}{(a-2[c-\epsilon ]+c')}^{2} & \text{ for } \epsilon<\dfrac{1}{2}(a+2c-3c'),\\ \\ & =& \dfrac{1}{9}{(a-c+\epsilon )}^{2} & \text{ otherwise },\\ \\ {\pi ^{D}_{2}}^{*} & =& \dfrac{1}{16}{(a-2[c-\epsilon ]+c')}^{2}-C & \text{ for } \epsilon<\dfrac{1}{2}(a+2c-3c'),\\ \\ & =& \dfrac{1}{9}{(a-c+\epsilon )}^{2}-C & \text{ otherwise },\\ \\ {\pi ^{D}_{3}}^{*} & =& \dfrac{1}{16}{(a+2[c-\epsilon ]-3c')}^{2} & \text{ for } \epsilon <\dfrac{1}{2}(a+2c-3c'),\\ \\ & =& 0 & \text{ otherwise }. \end{array} \end{aligned}$$

(11)

I can now determine the equilibrium of the game when there is no technology transfer. I find the following results: for $\epsilon < \frac{1}{2}(a+2c-3c')$, Firm 2 will choose to imitate if $C \le \frac{3}{16}(2a-4c+\epsilon +2c')\epsilon$; for $\frac{1}{2}(a+2c-3c') \le \epsilon < a+2c-3c'$, imitation will occur if $C \le \frac{1}{9}{(a-c+\epsilon )}^{2}-\frac{1}{16}{(a-2c-\epsilon +c')}^{2}$; for $a+2c-3c' \le \epsilon < a-2c+c'$, imitation will occur if $C \le \frac{4}{9}(a-c)\epsilon$; and in the case of a drastic invention, Firm 2 will imitate if $C \le \frac{1}{9}{(a-c+\epsilon )}^{2}$. Suppose that Firm 2 prefers to imitate when it is indifferent between its choices.

Technology Transfer Through Fixed-fee Licensing

Suppose that the proprietary technology owned by Firm 1 is transferable through a patent licensing agreement. In particular, Firm 1 could offer an exclusive license to the weak competitor, an exclusive license to the strong competitor, or licenses to both competitors. In addition, Firm 1 could license through either a fixed fee or a unit royalty. I will first consider fixed-fee licensing and then turn to royalty licensing. I will consider each fixed-fee licensing option in turn.

License to Firm 2: In this case, the unit cost is $c_{1}=c_{2}=c-\epsilon$ for Firms 1 and 2, and $c_3=c'$ for Firm 3. Let $F_{2}$, where $F_{2} > 0$ and $\pi ^{F}_{i}$ for $i=\{1,2,3\}$ be the fixed fee and the profit of Firm i, respectively. Then, considering the license fee and substituting the unit costs in Eq. 9, I obtain the profits of the firms as follows: $\pi ^{F}_{1}=\frac{1}{16}{(a-2[c-\epsilon ]+c')}^{2}+F_{2}$, $\pi ^{F}_{2}=\frac{1}{16}{(a-2[c-\epsilon ]+c')}^{2}-F_{2}$, and $\pi ^{F}_{3}=\frac{1}{16}{(a+2[c-\epsilon ]-3c')}^{2}$. If $\epsilon \ge \frac{1}{2}(a+2c-3c')$, it is optimal for Firm 3 not to produce, in which case the firms’ profits are $\pi ^{F}_{1}=\frac{1}{9}{(a-c+\epsilon )}^{2}+F_{2}$, $\pi ^{F}_{2}=\frac{1}{9}{(a-c+\epsilon )}^{2}-F_{2}$, and $\pi ^{F}_{3}=0$. Firm 2 will accept a licensing offer only if its profit as a licensee is equal to or greater than the maximum of its profit from using the existing technology and from imitation. In particular, considering that $\pi ^{F}_{2} \ge \text {max}$ $({\pi ^{ND}_{2}}^{*}, {\pi ^{D}_{2}}^{*})$ I find that the maximum fee that Firm 1 can charge Firm 2 is:

$$\begin{aligned} F_{2}^{*}= & \left\{ \begin{array}{ll} \text{ min } \text{( }\frac{3}{16}[2a-4c+\epsilon +2c']\epsilon \text{, } \text{ C) } & \text{ for } \epsilon< \frac{1}{2}(a+2c-3c'),\\ \\ \text{ min } \text{( }\frac{1}{9}{[a-c+\epsilon ]}^2-\frac{1}{16}{[a-2c-\epsilon +c']}^{2}\text{, } \text{ C) } & \text{ for } \frac{1}{2}(a+2c-3c') \le \epsilon< a+2c-3c',\\ \\ \text{ min } \text{( }\frac{4}{9}[a-c]\epsilon \text{, } \text{ C) } & \text{ for } a+2c-3c' \le \epsilon < a-2c+c',\\ \\ \text{ min } \text{( }\frac{1}{9}{[a-c+\epsilon ]}^2\text{, } \text{ C) } & \text{ otherwise }. \end{array} \right. \end{aligned}$$

(12)

According to the Eq. 12, and given that imitation has occurred, the optimal fee the patent holder can charge is $\frac{3}{16}(2a-4c+\epsilon +2c')\epsilon$ if $\epsilon < \frac{1}{2}(a+2c-3c')$; $\frac{1}{9}{(a-c+\epsilon )}^2-\frac{1}{16}{(a-2c-\epsilon +c')}^{2}$ if $\frac{1}{2}(a+2c-3c') \le \epsilon < a+2c-3c'$;¹⁷$\frac{4}{9}(a-c)\epsilon$ if $a+2c-3c' \le \epsilon < a-2c+c'$; and $\frac{1}{9}(a-c+\epsilon )^2$ in the case of drastic invention. It follows that the equilibrium profits of the firms are:

$$\begin{aligned} \begin{array}{llllll} {\pi ^{F}_{1}}^{*} & =& \dfrac{1}{16}{(a-2[c-\epsilon ]+c')}^{2}+F_{2}^{*} & \text{ for } \epsilon<\dfrac{1}{2}(a+2c-3c'),\\ \\ & =& \dfrac{1}{9}{(a-c+\epsilon )}^{2}+F_{2}^{*} & \text{ otherwise },\\ \\ {\pi ^{F}_{2}}^{*} & =& \dfrac{1}{16}{(a-2[c-\epsilon ]+c')}^{2}-F_{2}^{*} & \text{ for } \epsilon<\dfrac{1}{2}(a+2c-3c'),\\ \\ & =& \dfrac{1}{9}{(a-c+\epsilon )}^{2}-F_{2}^{*} & \text{ otherwise },\\ \\ {\pi ^{F}_{3}}^{*} & =& \dfrac{1}{16}{(a+2[c-\epsilon ]-3c')}^{2} & \text{ for } \epsilon <\dfrac{1}{2}(a+2c-3c'),\\ \\ & =& 0 & \text{ otherwise }, \end{array} \end{aligned}$$

(13)

where the Eq. 12 defines $F_{2}^{*}$. I can now compare the profit of Firm 1 with and without technology transfer and determine the equilibrium of this case. Suppose that using the existing technology is better than imitation for Firm 2.¹⁸ Then technology transfer will always occur in equilibrium if $\epsilon < \frac{2}{3}(a-c)$, because only in this interval of $\epsilon$ does licensing maximize the patent holder’s profit and satisfy the strong competitor’s participation constraints. Now suppose that imitation is at least as beneficial to Firm 2 as using the existing technology. Then comparative statics show that technology transfer will always occur in equilibrium, regardless of the type of invention.

License to Firm 3: In this case, the unit cost is $c_{1}=c-\epsilon$ for Firm 1, $c_{2}=c$ or $c_{2}=c-\epsilon$, and $c_{3}=c'-\epsilon$ for Firm 3. Suppose that using the existing technology is better than imitation for Firm 2 so $c_{2}=c$. In addition, let $F_{3}^{ND}$, where $F_{3}^{ND}>0$, and $\pi ^{F}_{i}(ND)$ for $i=\{1,2,3\}$ be the fixed fee and the profit of Firm i in this case, respectively. Considering the fixed fee and substituting the unit costs in Eq. 9, it can be shown that: $\pi ^{F}_{1}(ND)=\frac{1}{16}{(a-2[c-\epsilon ]+c')}^{2}+F_{3}^{ND}$, $\pi ^{F}_{2}(ND)=\frac{1}{16}{(a-2[c+\epsilon ]+c')}^{2}$, and $\pi ^{F}_{3}(ND)=\frac{1}{16}{(a+2[c+\epsilon ]-3c')}^{2}-F_{3}^{ND}$. Obviously, Firm 2 will produce output greater than zero if $\epsilon < \frac{1}{2}(a-2c+c')$ and the industry will become a duopoly otherwise. That is, the firms’ profits become: $\pi ^{F}_{1}(ND)=\frac{1}{9}{(a-2c+\epsilon +c')}^{2}+F_{3}^{ND}$, $\pi ^{F}_{2}(ND)=0$, and $\pi ^{F}_{3}(ND)=\frac{1}{9}{(a+c+\epsilon -2c')}^{2}-F_{3}^{ND}$. Firm 3 will accept a licensing offer if it will gain a profit at least as much as that from using the existing technology. Given that $\pi ^{F}_{3}(ND) \ge {\pi ^{ND}_{3}}^{*}$, I find that the optimal fixed fee that Firm 1 can charge Firm 3 when $c'<\frac{1}{7}(a+6c)$ is:

$$\begin{aligned} {F_{3}^{ND}}^{*}= & \left\{ \begin{array}{ll} \frac{3}{16}(2a+4c+\epsilon -6c')\epsilon & \text{ for } \epsilon< \frac{1}{2}(a-2c+c') \text{ and } c'<\frac{1}{7}(a+6c),\\ \\ \frac{1}{9}{(a+c+\epsilon -2c')}^{2}-\frac{1}{16}{(a+2c-\epsilon -3c')}^2 & \text{ for } \frac{1}{2}(a-2c+c') \le \epsilon< a+2c-3c'\\ & \text{ and } c'<\frac{1}{7}(a+6c),\\ \\ \frac{1}{9}{(a+c+\epsilon -2c')}^2 & \text{ for } \epsilon \ge a+2c-3c' \text{ and } c'<\frac{1}{7}(a+6c),\\ \end{array} \right. \end{aligned}$$

(14)

and if $\frac{1}{7}(a+6c) \le c' < \frac{1}{3}(a+2c)$, then the optimal fixed fee is:

$$\begin{aligned} {F_{3}^{ND}}^{*}= & \left\{ \begin{array}{ll} \frac{3}{16}(2a+4c+\epsilon -6c')\epsilon & \text{ for } \epsilon< a+2c-3c' \text{ and } c'\ge \frac{1}{7}(a+6c),\\ \\ \frac{1}{16}{(a+2[c+\epsilon ]-3c')}^{2} & \text{ for } a+2c-3c' \le \epsilon < \frac{1}{2}(a-2c+c') \text{ and } c'\ge \frac{1}{7}(a+6c),\\ \\ \frac{1}{9}{(a+c+\epsilon -2c')}^2 & \text{ for } \epsilon \ge \frac{1}{2}(a-2c+c') \text{ and } c'\ge \frac{1}{7}(a+6c).\\ \end{array} \right. \end{aligned}$$

(15)

According to the Eq. 14, if $\epsilon <\frac{1}{2}(a-2c+c')$, then the optimal fixed fee that the patent holder can charge Firm 3 is $\frac{3}{16}(2a+4c+\epsilon -6c')\epsilon$; if $\frac{1}{2}(a-2c+c')\le \epsilon < a+2c-3c'$, the optimal fee is $\frac{1}{9}{(a+c+\epsilon -2c')}^{2}-\frac{1}{16}{(a+2c-3c'-\epsilon )}^2$; and if $\epsilon \ge a+2c-3c'$, the optimal fee is $\frac{1}{9}(a+c+\epsilon -2c')^2$. Similarly, Eq. 15 shows that if $\epsilon <a+2c-3c'$, the optimal fixed fee that Firm 1 can charge is $\frac{3}{16}(2a+4c+\epsilon -6c')\epsilon$; if $a+2c-3c' \le \epsilon < \frac{1}{2}(a-2c+c')$, the optimal fee is $\frac{1}{16}{(a+2c+2\epsilon -3c')}^{2}$; and if $\epsilon \ge \frac{1}{2}(a-2c+c')$, then the optimal fee is $\frac{1}{9}{(a+c+\epsilon -2c')}^2$. I can now derive the firms’ profits in equilibrium as follows:

$$\begin{aligned} \begin{array}{llllll} {\pi ^{F}_{1}}^{*}(ND) & =& \dfrac{1}{16}{(a-2[c-\epsilon ]+c')}^{2}+{F_{3}^{ND}}^{*} & \text{ for } \epsilon<\dfrac{1}{2}(a-2c+c'),\\ \\ & =& \dfrac{1}{9}{(a-2c+\epsilon +c')}^{2}+{F_{3}^{ND}}^{*} & \text{ otherwise },\\ \\ {\pi ^{F}_{2}}^{*}(ND) & =& \dfrac{1}{16}{(a-2[c+\epsilon ]+c')}^{2} & \text{ for } \epsilon<\dfrac{1}{2}(a-2c+c'),\\ \\ & =& 0 & \text{ otherwise },\\ \\ {\pi ^{F}_{3}}^{*}(ND) & =& \dfrac{1}{16}{(a+2[c+\epsilon ]-3c')}^{2}-{F_{3}^{ND}}^{*} & \text{ for } \epsilon <\dfrac{1}{2}(a-2c+c'),\\ \\ & =& \dfrac{1}{9}{(a+c+\epsilon -2c')}^{2}-{F_{3}^{ND}}^{*} & \text{ otherwise }, \end{array} \end{aligned}$$

(16)

where the Eqs. 14 and 15 define ${F_{3}^{ND}}^{*}$ if $c'< \frac{1}{7}(a+6c)$ or $c'\ge \frac{1}{7}(a+6c)$, respectively.

Now suppose that imitation is at least as good as not imitating for Firm 2. The unit cost in this case is $c_{1}=c_{2}=c-\epsilon$ for Firms 1 and 2, and $c_{3}=c'-\epsilon$ for Firm 3. Let $F_{3}^{D}$, where $F_{3}^{D}>0$, and $\pi ^{F}_{i}(D)$ for $i=\{1,2,3\}$ be the fee for Firm 3 and the profit of Firm i in this case, respectively. The profits of the firms are then: $\pi ^{F}_{1}(D) = \frac{1}{16}{(a-2c+\epsilon +c')}^{2}+F_{3}^{D}$, $\pi ^{F}_{2}(D) = \frac{1}{16}{(a-2c+\epsilon +c')}^{2}-C$, and $\pi ^{F}_{3}(D) = \frac{1}{16}{(a+2c+\epsilon -3c')}^{2}-F_{3}^{D}$. Considering that Firm 3 will accept a licensing offer only if it is better off with it than without it, i.e. $\pi _{3}^{F}(D) \ge {\pi _{3}^{D}}^{*}$, I obtain the optimal fixed fee as follows:

$$\begin{aligned} {F_{3}^{D}}^{*}= & \left\{ \begin{array}{ll} \frac{3}{16}(2a+4c-\epsilon -6c')\epsilon & \text{ for } \epsilon < \frac{1}{2}(a+2c-3c'),\\ \\ \frac{1}{16}{(a+2c+\epsilon -3c')}^2 & \text{ otherwise }. \end{array} \right. \end{aligned}$$

(17)

Equation 17 shows that the optimal fixed fee that the patent holder can charge Firm 3 is $\frac{3}{16}(2a+4c-\epsilon -6c')\epsilon$ if $\epsilon < \frac{1}{2}(a+2c-3c')$ and $\frac{1}{16}{(a+2c+\epsilon -3c')}^2$ otherwise. Therefore, the profits of the firms in equilibrium are:

$$\begin{aligned} \begin{array}{llllll} {\pi ^{F}_{1}}^{*}(D) & =& \dfrac{1}{16}{(a-2c+\epsilon +c')}^{2}+{F_{3}^{D}}^{*},\\ \\ {\pi ^{F}_{2}}^{*}(D) & =& \dfrac{1}{16}{(a-2c+\epsilon +c')}^{2}-C,\\ \\ {\pi ^{F}_{3}}^{*}(D) & =& \dfrac{1}{16}{(a+2c+\epsilon -3c')}^{2}-{F_{3}^{D}}^{*}, \end{array} \end{aligned}$$

(18)

where the Eq. 17 defines ${F_{3}^{D}}^{*}$.

I can now determine when an exclusive license to the weak competitor is optimal for the patent holder. Suppose that for Firm 2, using the existing technology is better than imitation. Then the comparative statics show that if $c'<\frac{1}{7}(a+6c)$, technology transfer will occur if $\epsilon <\frac{2}{3}(a-c)$, while if $c'\ge \frac{1}{7}(a+6c)$, technology transfer will occur only if $\epsilon <2(a+4c-5c')$. Now suppose that imitation is at least as good as not imitating for Firm 2. If $c'< \frac{1}{11}(a+10c)$, technology transfer will always occur in equilibrium, while if $c'\ge \frac{1}{11}(a+10c)$, technology transfer will occur if $\epsilon \le \frac{2}{3}(a+4c-5c')$ or $\epsilon \ge 15c'-14c-a$.

Licenses to Firms 2 and 3: The last case to consider is when the patent holder offers a license to each competitor. The unit cost in this case is $c_{1}=c_{2}=c-\epsilon$ for Firms 1 and 2, and $c_{3}=c'-\epsilon$ for Firm 3. Let $\mathbb {F}_{i}$ for $i=\{2, 3\}$, where $\mathbb {F}_{i}>0$, be the fixed fee for Firm i if the patent holder offers a license to each competitor. In addition, let $\pi _{i}^{\mathbb {F}}$ for $i=\{1, 2, 3\}$ be the profit of Firm i in this case. Then, considering the fixed fees and substituting the unit costs in Eq. 9, the profits of the firms are as follows: $\pi ^{\mathbb {F}}_{1}=\frac{1}{16}{(a-2c+\epsilon +c')}^{2}+\mathbb {F}_{2}+\mathbb {F}_{3}$, $\pi ^{\mathbb {F}}_{2}=\frac{1}{16}{(a-2c+\epsilon +c')}^{2}-\mathbb {F}_{2}$, and $\pi ^{\mathbb {F}}_{3}=\frac{1}{16}{(a+2c+\epsilon -3c')}^{2}-\mathbb {F}_{3}$. Firms 2 and 3 will accept the licenses if each is at least better off with the technology transfer than without it. Specifically, a technology transfer will occur if $\pi ^{\mathbb {F}}_{2} \ge \text {max}$ $({\pi ^{ND}_{2}}^{*}, {\pi ^{D}_{2}}^{*})$ and $\pi ^{\mathbb {F}}_{3} \ge {\pi ^{ND}_{3}}^{*}$.¹⁹ Thus, the optimal fixed fee for Firm 2 is:

$$\begin{aligned} \mathbb {F}_{2}^{*}= & \left\{ \begin{array}{ll} \text{ min } \text{( }\frac{1}{4}[a-2c+c']\epsilon \text{, } A^{\mathbb {F}}\text{) } & \text{ for } \epsilon< \frac{1}{2}(a+2c-3c'),\\ \\ \text{ min } \text{( }\frac{1}{4}[a-2c+c']\epsilon \text{, } B^{\mathbb {F}}\text{) } & \text{ for } \frac{1}{2}(a+2c-3c')\le \epsilon< a+2c-3c',\\ \\ \text{ min } \text{( }\frac{1}{16}{[a-2c+\epsilon +c']}^{2}-\frac{1}{9}{[a-c-\epsilon ]}^{2}\text{, } B^{\mathbb {F}}\text{) } & \text{ for } a+2c-3c'\le \epsilon < a-2c+c',\\ \\ \text{ min } \text{( }\frac{1}{16}{[a-2c+\epsilon +c']}^{2}\text{, } B^{\mathbb {F}}\text{) } & \text{ otherwise }, \end{array} \right. \end{aligned}$$

(19)

where $A^{\mathbb {F}} \equiv C-\frac{1}{16}(2a-4c+2c'+3\epsilon )\epsilon$ and $B^{\mathbb {F}} \equiv \frac{1}{16}{(a-2c+\epsilon +c')}^{2}-\frac{1}{9}{(a-c+\epsilon )}^{2}+C$.²⁰ Similarly, the maximum fixed fee for Firm 3 is:

$$\begin{aligned} \mathbb {F}_{3}^{*}= & \left\{ \begin{array}{ll} \dfrac{1}{4}(a+2c-3c')\epsilon & \text{ for } \epsilon < a+2c-3c',\\ \\ \dfrac{1}{16}{(a+2c+\epsilon -3c')}^{2} & \text{ otherwise }. \end{array} \right. \end{aligned}$$

(20)

According to the Eq. 19, if $\epsilon < \frac{1}{2}(a+2c-3c')$, the optimal fee that Firm 1 can charge Firm 2 is the minimum of $\frac{1}{4}(a-2c+c')\epsilon$ and $A^{\mathbb {F}}$; if $\frac{1}{2}(a+2c-3c') \le \epsilon < a+2c-3c'$, the optimal fee is the minimum of $\frac{1}{2}(a+2c-3c')$ and $B^{\mathbb {F}}$; if $a+2c-3c' \le \epsilon < a-2c+c'$, the optimal fixed fee is the minimum of $\frac{1}{16}{(a-2c+\epsilon +c')}^{2}-\frac{1}{9}{(a-c-\epsilon )}^{2}$ and $B^{\mathbb {F}}$; and in the case of a drastic invention, Firm 1 will charge the minimum of $\frac{1}{16}{(a-2c+\epsilon +c')}^{2}$ and $B^{\mathbb {F}}$. Furthermore, Eq. 20 shows that the optimal fee that the patent holder can charge Firm 3 is $\frac{1}{4}(a+2c-3c')\epsilon$ if $\epsilon < a+2c-3c'$ and $\frac{1}{16}{(a+2c+\epsilon -3c')}^{2}$ otherwise. The profits of the firms in equilibrium are therefore equal to:

$$\begin{aligned} \begin{array}{llllll} {\pi ^{\mathbb {F}}_{1}}^{*} & =& \dfrac{1}{16}{(a-2c+\epsilon +c')}^{2}+\mathbb {F}_{2}^{*}+\mathbb {F}_{3}^{*},\\ \\ {\pi ^{\mathbb {F}}_{2}}^{*} & =& \dfrac{1}{16}{(a-2c+\epsilon +c')}^{2}-\mathbb {F}_{2}^{*},\\ \\ {\pi ^{\mathbb {F}}_{3}}^{*} & =& \dfrac{1}{16}{(a+2c+\epsilon -3c')}^{2}-\mathbb {F}_{3}^{*}, \end{array} \end{aligned}$$

(21)

where the Eqs. 19 and 20 define $\mathbb {F}_{2}^{*}$ and $\mathbb {F}_{3}^{*}$, respectively. I can now determine the equilibrium of this last case. If using the existing technology is better than imitation for Firm 2, the patent holder will transfer the patented technology if $\epsilon < \frac{1}{2}(a+2c-3c')$ and will not do so otherwise. However, if imitation is at least as beneficial to Firm 2 as not imitating, it is always optimal for the patent holder to offer licenses to both competitors, regardless of the type of invention.

Finally, I can characterize the equilibrium of the game by comparing the patent holder’s fixed-fee licensing decisions described above. I establish the following proposition:

Proposition 4

If using the existing technology is better than imitation for Firm 2, then technology transfer through a fixed-fee license will occur only if $\epsilon <\frac{2}{3}(a-c)$; in this case, offering an exclusive license to the strong competitor is at least as beneficial from the patent owner’s perspective as offering an exclusive license to the weak competitor or offering licenses to both competitors. If imitation is at least as beneficial to Firm 2 as using the existing technology, then licensing to both competitors is at least as beneficial to the patent holder as the other two options. Technology transfer exclusively to the weak competitor will not occur in equilibrium.

Technology Transfer Through Royalty Licensing

Similar to the fixed-fee licensing game, I will now consider licensing through a unit royalty. The patent holder can offer an exclusive license to Firm 2, an exclusive license to Firm 3, or licenses to Firms 2 and 3. I will consider each licensing option in turn.

License to Firm 2: Let $r_{2}$, where $r_{2} > 0$, be the unit royalty that the patent holder charges Firm 2 in return for an exclusive license, and $\pi _{i}^{R}$ for $i=\{1,2,3\}$ be the profit of Firm i. Since the unit cost is $c_{1}=c-\epsilon$ for Firm 1, $c_{2}=c-\epsilon +r_{2}$ for Firm 2, and $c_{3}=c'$ for Firm 3, I derive the firms’ profits as follows: $\pi ^{R}_{1}=\frac{1}{16}{(a-2[c-\epsilon ]+r_{2}+c')}^{2}+\frac{1}{4}(a-2[c-\epsilon ]-3r_{2}+c')r_{2}$, $\pi ^{R}_{2}=\frac{1}{16}{(a-2[c-\epsilon ]-3r_{2}+c')}^{2}$, and $\pi ^{R}_{3}=\frac{1}{16}{(a+2[c-\epsilon ]+r_{2}-3c')}^{2}$. It can be shown that Firm 3 will choose not to produce if $\epsilon \ge \frac{1}{2}(a+2c+r_{2}-3c')$; in this case, the industry becomes a duopoly. I showed that in a duopoly setting, technology transfer through a unit royalty will occur only if the invention is non-drastic, i.e. $\epsilon < a-c$. I have also shown that royalty licensing is at least as good as fixed-fee licensing in this case. Consequently, I will now consider only the case where $\epsilon < \frac{1}{2}(a+2c+r_{2}^{*}-3c')$. Firm 2 will accept a license offer if $\pi ^{R}_{2} \ge \text {max}$ $({\pi ^{ND}_{2}}^{*}, {\pi ^{D}_{2}}^{*})$ and reject it otherwise. Solving Firm 2’s maximization problem yields the optimal unit royalty as follows:

$$\begin{aligned} r_{2}^{*}= & \left\{ \begin{array}{ll} \text{ min } \text{( }\epsilon \text{, } A^{R}\text{) } & \text{ for } \epsilon< \frac{1}{2}(a+2c-3c'),\\ \\ \text{ min } \text{( }\epsilon \text{, } B^{R}\text{) } & \text{ for } \frac{1}{2}(a+2c-3c') \le \epsilon < \frac{3}{5}(a-2c+c'),\\ \\ \text{ min } \text{( }\frac{3}{11}[a-2(c-\epsilon )+c']\text{, } B^{R}\text{) } & \text{ otherwise }, \end{array} \right. \end{aligned}$$

(22)

where $A^{R} \equiv \frac{1}{3}(a-2[c-\epsilon ]+c')-\frac{1}{3}\sqrt{{(a-c+\epsilon )}^{2}-16C}$ and $B^{R} \equiv \frac{1}{3}(a-2[c-\epsilon ]+c')-\frac{4}{3}\sqrt{{(a-c+\epsilon )}^{2}-9C}$.²¹

Proof

See the Appendix.

According to the Eq. 22, if $\epsilon <\frac{1}{2}(a+2c-3c')$, the optimal unit royalty is the minimum of the size of the innovation and $A^{R}$; if $\frac{1}{2}(a+2c-3c') \le \epsilon < \frac{3}{5}(a-2c+c')$, the optimal unit royalty is the minimum of the size of the innovation and $B^{R}$; and if $\epsilon \ge \frac{3}{5}(a-2c+c')$, then Firm 1 will charge the minimum of $\frac{3}{11}(a-2[c-\epsilon ]+c')$ and $B^{R}$. The profits of the firms in equilibrium can be shown to be:

$$\begin{aligned} \begin{array}{llll} {\pi ^{R}_{1}}^{*} & =& \dfrac{1}{16}{(a-2[c-\epsilon ]+r_{2}^{*}+c')}^{2}+\dfrac{1}{4}(a-2[c-\epsilon ]-3r_{2}^{*}+c')r_{2}^{*},\\ \\ {\pi ^{R}_{2}}^{*} & =& \dfrac{1}{16}{(a-2[c-\epsilon ]-3r_{2}^{*}+c')}^{2},\\ \\ {\pi ^{R}_{3}}^{*} & =& \dfrac{1}{16}{(a+2[c-\epsilon ]+r_{2}^{*}-3c')}^{2}, \end{array} \end{aligned}$$

(23)

where the Eq. 22 defines $r_{2}^{*}$. Comparing the profits of the patent holder with and without technology transfer shows that licensing may be optimal for the patent holder. Suppose that for Firm 2, using the existing technology is better than imitation. Then technology transfer will always occur, since I consider $\epsilon < \frac{1}{2}(a+2c+r_{2}^{*}-3c')$. However, if $\epsilon \ge \frac{1}{2}(a+2c+r_{2}^{*}-3c')$, the industry becomes a duopoly and thus technology transfer will not occur unless the invention is drastic. Finally, if imitation is at least as beneficial to Firm 2 as not imitating, technology transfer will always occur in equilibrium, regardless of the type of invention.

License to Firm 3: In this case, the patent holder offers a royalty license to the weak competitor. First, suppose that using the existing technology is better than imitation for Firm 2. Let $r_{3}^{ND}$, where $r_{3}^{ND}>0$, and $\pi _{i}^{R}(ND)$ for $i=\{1,2,3\}$ be the unit royalty and profit of Firm i. Since the unit cost is $c_{1}=c-\epsilon$ for Firm 1, $c_{2}=c$ for Firm 2, and $c_{3}=c'-\epsilon +r_{3}^{ND}$ for Firm 3, I obtain the firms’ profits as follows: $\pi ^{R}_{1}(ND)= \frac{1}{16}{(a-2[c-\epsilon ]+r_{3}^{ND}+c')}^{2}+\frac{1}{4}(a+2[c+\epsilon ]-3[r_{3}^{ND}+c'])r_{3}^{ND}$, $\pi ^{R}_{2}(ND)=\frac{1}{16}{(a-2[c+\epsilon ]+r_{3}^{ND}+c')}^{2}$, and $\pi ^{R}_{3}(ND)=\frac{1}{16}{(a+2[c+\epsilon ]-3[r_{3}^{ND}+c'])}^{2}$. In this case, it is optimal for Firm 2 not to produce if $\epsilon \ge \frac{1}{2}(a-2c+r_{3}^{ND}+c')$, in which case the industry becomes a duopoly. This is similar to the royalty licensing case in the duopoly game, except that the unit cost of the less efficient incumbent is now c’ instead of c, where $c'>c$. Therefore, the profit of a duopolistic patent holder and the trigger value corresponding to a drastic invention will both be larger. Otherwise, the results of the duopoly game hold, so I only need to consider the case where $\epsilon < \frac{1}{2}(a-2c+r_{3}^{ND}+c')$. Firm 3 will accept a license offer if $\pi ^{R}_{3}(ND) \ge {\pi ^{ND}_{3}}^{*}$. Solving the patent holder’s maximization problem, I find that if $c'<\frac{1}{5}(a+4c)$, the optimal unit royalty is:

$$\begin{aligned} {r_{3}^{ND}}^{*}= & \left\{ \begin{array}{ll} \epsilon & \text{ for } \epsilon< \frac{1}{5}(3a+2c-5c') \text{ and } c'< \frac{1}{5}(a+4c)\\ \\ \frac{1}{11}(3a+2c+6\epsilon -5c') & \text{ for } \epsilon \ge \frac{1}{5}(3a+2c-5c') \text{ and } c' < \frac{1}{5}(a+4c). \end{array} \right. \end{aligned}$$

(24)

Proof

See the Appendix.

Furthermore, if $\frac{1}{5}(a+4c) \le c'< \frac{1}{3}(a+2c)$, then the optimal unit royalty is:

$$\begin{aligned} {r_{3}^{ND}}^{*}= & \left\{ \begin{array}{ll} \epsilon & \text{ for } \epsilon< a+2c-3c' \text{ and } \epsilon \ge \frac{1}{5}(a+4c),\\ \\ \frac{1}{3}(a+2[c+\epsilon ]-3c') & \text{ for } a+2c-3c' \le \epsilon < \frac{1}{2}(9c'-8c-a) \text{ and } \epsilon \ge \frac{1}{5}(a+4c),\\ \\ \frac{1}{11}(3a+2c+6\epsilon -5c') & \text{ for } \epsilon \ge \frac{1}{2}(9c'-8c-a) \text{ and } \epsilon \ge \frac{1}{5}(a+4c). \end{array} \right. \end{aligned}$$

(25)

Proof

See the Appendix.

Equation 24, which corresponds to the case when $c'<\frac{1}{5}(a+4c)$, shows that the optimal unit royalty is equal to the size of the innovation if $\epsilon < \frac{1}{5}(3a+2c-5c')$ and $\frac{1}{11}(3a+2c+6\epsilon -5c')$ otherwise. Furthermore, according to the Eq. 25, which corresponds to the case when $c'\ge \frac{1}{5}(a+4c)$, if $\epsilon < a+2c-3c'$, the optimal unit royalty is equal to the size of the innovation; if $a+2c-3c' \le \epsilon < \frac{1}{2}(9c'-8c-a)$, the optimal unit royalty is $\frac{1}{3}(a+2[c+\epsilon ]-3c')$; and if $\epsilon \ge \frac{1}{2}(9c'-8c-a)$, then the optimal unit royalty is $\frac{1}{11}(3a+2c+6\epsilon -5c')$. The profits of the firms in equilibrium are:

$$\begin{aligned} \begin{array}{llll} {\pi ^{R}_{1}}^{*}(ND) & =& \dfrac{1}{16}{(a-2[c-\epsilon ]+{r_{3}^{ND}}^{*}+c')}^{2}\\ & \quad +& \dfrac{1}{4}(a+2[c+\epsilon ]-{3[r_{3}^{ND}}^{*}+c']){r_{3}^{ND}}^{*},\\ \\ {\pi ^{R}_{2}}^{*}(ND) & =& \dfrac{1}{16}{(a-2[c+\epsilon ]+{r_{3}^{ND}}^{*}+c')}^{2},\\ \\ {\pi ^{R}_{3}}^{*}(ND) & =& \dfrac{1}{16}{(a+2[c+\epsilon ]-3[{r_{3}^{ND}}^{*}+c'])}^{2}, \end{array} \end{aligned}$$

(26)

where the Eqs. 24 or 25 defines ${r_{3}^{ND}}^{*}$ if $c'<\frac{1}{5}(a+4c)$ or $c'\ge \frac{1}{5}(a+4c)$, respectively.

Now, suppose that imitation is at least as beneficial as not imitating for Firm 2. In addition, let $r_{3}^{D}$, where $r_{3}^{D}>0$, and $\pi _{i}^{R}(D)$ for $i=\{1,2,3\}$ be the unit royalty and the profit of Firm i in this case, respectively. Then the unit cost is $c_{1}=c_{2}=c-\epsilon$ for Firms 1 and 2, and $c_{3}=c'-\epsilon +r_{3}^{D}$ for Firm 3. This yields the firms’ profits, as follows: $\pi ^{R}_{1}(D)=\frac{1}{16}{(a-2c+\epsilon +r_{3}^{D}+c')}^{2}+\frac{1}{4}(a+2c+\epsilon -3[r_{3}^{D}+c'])r_{3}^{D}$, $\pi ^{R}_{2}(D)=\frac{1}{16}{(a-2c+\epsilon +r_{3}^{D}+c')}^{2}-C$, and $\pi ^{R}_{3}(D)=\frac{1}{16}{(a+2c+\epsilon -3[r_{3}^{D}+c'])}^{2}$. It can be shown that, in this case, Firm 3 will accept a license offer if $\pi ^{R}_{3}(D) \ge {\pi ^{D}_{3}}^{*}$. Solving the patent holder’s maximization problem, I find that if $c'<\frac{1}{7}(a+6c)$, the optimal unit royalty is:

$$\begin{aligned} {r_{3}^{D}}^{*}= & \left\{ \begin{array}{ll} \epsilon & \text{ for } \epsilon< \frac{1}{8}(3a+2c-5c') \text{ and } c'< \frac{1}{7}(a+6c),\\ \\ \frac{1}{11}(3[a+\epsilon ]+2c-5c') & \text{ for } \epsilon \ge \frac{1}{8}(3a+2c-5c') \text{ and } c' < \frac{1}{7}(a+6c).\\ \\ \end{array} \right. \end{aligned}$$

(27)

Proof

See the Appendix.

In addition, if $\frac{1}{7}(a+6c) \le c' < \frac{1}{3}(a+2c)$, then the optimal unit royalty is:

$$\begin{aligned} {r_{3}^{D}}^{*}= & \left\{ \begin{array}{ll} \epsilon & \text{ for } \epsilon< \frac{1}{2}(a+2c-3c') \text{ and } c' \ge \frac{1}{7}(a+6c),\\ \\ \frac{1}{3}(a+2c+\epsilon -3c') & \text{ for } \frac{1}{2}(a+2c-3c') \le \epsilon < 9c'-8c-a \text{ and } c' \ge \frac{1}{7}(a+6c),\\ \\ \frac{1}{11}(3[a+\epsilon ]+2c-5c') & \text{ for } \epsilon \ge 9c'-8c-a \text{ and } c' \ge \frac{1}{7}(a+6c). \end{array} \right. \end{aligned}$$

(28)

Proof

See the Appendix.

Equation 27, which corresponds to the case when $c'<\frac{1}{7}(a+6c)$, shows that the optimal unit royalty is equal to the size of the innovation when $\epsilon < \frac{1}{8}(3a+2c-5c')$, and equal to $\frac{1}{11}(3[a+\epsilon ]+2c-5c')$ otherwise. Similarly, according to the Eq. (28), which corresponds to the case when $c'\ge \frac{1}{7}(a+6c)$, if $\epsilon < \frac{1}{2}(a+2c-3c')$, the optimal unit royalty is equal to the size of the innovation. In addition, the optimal unit royalty is $\frac{1}{3}(a+2c+\epsilon -3c')$ if $\frac{1}{2}(a+2c-3c') \le \epsilon < 9c'-8c-a$ and $\frac{1}{11}(3[a+\epsilon ]+2c-5c')$ otherwise.²² Consequently, the equilibrium profits of the firms in this case are:

$$\begin{aligned} \begin{array}{llll} {\pi ^{R}_{1}}^{*}(D) & =& \dfrac{1}{16}{(a-2c+\epsilon +{r_{3}^{D}}^{*}+c')}^{2}+\dfrac{1}{4}(a+2c+\epsilon -3[{r_{3}^{D}}^{*}+c']){r_{3}^{D}}^{*},\\ \\ {\pi ^{R}_{2}}^{*}(D) & =& \dfrac{1}{16}{(a-2c+\epsilon +{r_{3}^{D}}^{*}+c')}^{2}-C,\\ \\ {\pi ^{R}_{3}}^{*}(D) & =& \dfrac{1}{16}{(a+2c+\epsilon -3[{r_{3}^{D}}^{*}+c'])}^{2}, \end{array} \end{aligned}$$

(29)

where the Eq. 27 defines ${r_{3}^{D}}^{*}$ if $c'<\frac{1}{7}(a+6c)$ and the Eq. 28 defines ${r_{3}^{D}}^{*}$ if $c' \ge \frac{1}{7}(a+6c)$.

Comparing the patent holder’s profit with and without technology transfer, I get the following results. If using the existing technology is better than imitation for Firm 2, technology transfer will always occur in equilibrium, since $\epsilon < \frac{1}{2}(a-2c+r_{3}^{ND}+c')$. Not surprisingly, when $\epsilon \ge \frac{1}{2}(a-2c+{r_{3}^{ND}}^{*}+c')$, the industry becomes a duopoly, in which case technology transfer does not occur unless the invention is drastic. Finally, if imitation is at least as beneficial to Firm 2 as the decision not to imitate, then licensing is at least as beneficial to the patent holder as the decision not to license.

Licenses to Firms 2 and 3: The final case to consider is when the patent holder offers a license to each competitor. I define $\mathfrak {r}_{i}$ for $i=\{2,3\}$ and $\pi _{i}^{\mathbb {R}}$ for $i=\{1,2,3\}$ as the unit royalty and the profit of Firm i, respectively. Then the cost per unit is $c_{1}=c-\epsilon$ for the patent holder, $c_{2}=c-\epsilon +\mathfrak {r}_{2}$ for Firm 2, and $c_{3}=c'-\epsilon +\mathfrak {r}_{3}$ for Firm 3. Thus, I can derive the firms’ profit functions as follows: $\pi ^{\mathbb {R}}_{1}= \frac{1}{16}{(a-2c+\epsilon +\mathfrak {r}_{2}+\mathfrak {r}_{3}+c')}^{2}+\frac{1}{4}(a-2c+\epsilon -3\mathfrak {r}_{2}+\mathfrak {r}_{3}+c')\mathfrak {r}_{2} +\frac{1}{4}(a+2c+\epsilon +\mathfrak {r}_{2}-3[\mathfrak {r}_{3}+c'])\mathfrak {r}_{3}$, $\pi ^{\mathbb {R}}_{2}=\frac{1}{16}{(a-2c+\epsilon -3\mathfrak {r}_{2}+\mathfrak {r}_{3}+c')}^{2}$, and $\pi ^{\mathbb {R}}_{3}=\frac{1}{16}{(a+2c+\epsilon +\mathfrak {r}_{2}-3[\mathfrak {r}_{3}+c'])}^{2}$. Obviously, in this case technology transfer occurs when $\pi ^{\mathbb {R}}_{2} \ge \text {max}$ $({\pi ^{ND}_{2}}^{*}, {\pi ^{D}_{2}}^{*})$ and $\pi ^{\mathbb {R}}_{3} \ge {\pi ^{ND}_{3}}^{*}$.²³ Solving the patent holder’s maximization problem, I obtain the optimal unit royalty corresponding to Firm 2 as follows:

$$\begin{aligned} \mathfrak {r}_{2}^{*}= & \left\{ \begin{array}{ll} \text{ min } \text{( }\epsilon \text{, } A^{\mathbb {R}}\text{) } & \text{ for } \epsilon< \frac{1}{2}(a+2c-3c'),\\ \\ \text{ min } \text{( }\epsilon \text{, } B^{\mathbb {R}}\text{) } & \text{ for } \frac{1}{2}(a+2c-3c')\le \epsilon<\frac{1}{6}(6a-7c+c'),\\ \\ \text{ min } \text{( }\frac{1}{12}[6a-7c+6\epsilon +c']\text{, } B^{\mathbb {R}}\text{) } & \text{ for } \frac{1}{6}(6a-7c+c') \le \epsilon < a-2c+c',\\ \\ \text{ min } \text{( }\frac{1}{2}[a-c+\epsilon ]\text{, } B^{\mathbb {R}}\text{) } & \text{ for } \epsilon \ge a-2c+c', \end{array} \right. \end{aligned}$$

(30)

where $A^{\mathbb {R}} \equiv \frac{3}{8}(a-2[c-\epsilon ]+c')-\frac{3}{8}\sqrt{(a-2[c-\epsilon ]+c')^{2}-16C}$ and $B^{\mathbb {R}} \equiv \frac{1}{2}(a-c+\epsilon )-\frac{1}{2}\sqrt{(a-c+\epsilon )^{2}-9C}$.²⁴

Proof

See the Appendix.

Furthermore, the optimal royalty rate for Firm 3 can be shown to be:

$$\begin{aligned} \mathfrak {r}_{3}^{*}= & \left\{ \begin{array}{ll} \epsilon & \text{ for } \epsilon< \frac{1}{6}(6a-7c+c'),\\ \\ \frac{1}{12}(6a-c+6\epsilon -5c') & \text{ for } \frac{1}{6}(6a-7c+c') \le \epsilon < a-2c+c',\\ \\ \frac{1}{3}(a+c+\epsilon -2c') & \text{ for } \epsilon \ge a-2c+c'. \end{array} \right. \end{aligned}$$

(31)

Proof

See the Appendix.

Equation 30 shows that if $\epsilon <\frac{1}{2}(a+2c-3c')$, the optimal unit royalty is the minimum of the size of the innovation and $A^{\mathbb {R}}$; if $\frac{1}{2}(a+2c-3c')\le \epsilon <\frac{1}{6}(6a-7c+c')$, the optimal unit royalty is the minimum of the size of the innovation and $B^{\mathbb {R}}$; if $\frac{1}{6}(6a-7c+c') \le \epsilon < a-2c+c'$, the optimal unit royalty is the minimum of $\frac{1}{12}(6a-7c+6\epsilon +c')$ and $B^{\mathbb {R}}$; and in the case of a drastic invention, the optimal unit royalty is the minimum of $\frac{1}{2}(a-c+\epsilon )$ and $B^{\mathbb {R}}$. Similarly, Eq. 31 shows that if $\epsilon < \frac{1}{6}(6a-7c+c')$, the patent holder will charge Firm 3 a unit royalty equal to the size of the innovation. Furthermore, the optimal unit royalty rate is $\frac{1}{12}(6a-c+6\epsilon -5c')$ if $\frac{1}{6}(6a-7c+c') \le \epsilon < a-2c+c'$ and $\frac{1}{3}(a+c+\epsilon -2c')$ if the invention is drastic. The equilibrium profits of the firms are then:

$$\begin{aligned} \begin{array}{llll} {\pi ^{\mathbb {R}}_{1}}^{*} & =& \dfrac{1}{16}{(a-2c+\epsilon +\mathfrak {r}_{2}^{*}+\mathfrak {r}_{3}^{*}+c')}^{2}\\ & \quad +& \dfrac{1}{4}(a-2c+\epsilon -3\mathfrak {r}_{2}^{*}+\mathfrak {r}_{3}^{*}+c')\mathfrak {r}_{2}^{*}+\dfrac{1}{4}(a+2c+\epsilon +\mathfrak {r}_{2}^{*}-3[\mathfrak {r}_{3}^{*}+c'])\mathfrak {r}_{3}^{*},\\ \\ {\pi ^{\mathbb {R}}_{2}}^{*} & =& \dfrac{1}{16}{(a-2c+\epsilon -3\mathfrak {r}_{2}^{*}+\mathfrak {r}_{3}^{*}+c')}^{2},\\ \\ {\pi ^{\mathbb {R}}_{3}}^{*} & =& \dfrac{1}{16}{(a+2c+\epsilon +\mathfrak {r}_{2}^{*}-3[\mathfrak {r}_{3}^{*}+c'])}^{2}, \end{array} \end{aligned}$$

(32)

where the Eqs. 30 and 31 define $\mathfrak {r}_{2}^{*}$ and $\mathfrak {r}_{3}^{*}$, respectively.

I can now determine the equilibrium of this case. If using the existing technology is better than imitation for Firm 2, technology transfer will always occur if $\epsilon <a+2c-3c'$. In addition, if $\epsilon \ge a+2c-3c'$, the industry will become a duopoly and thus technology transfer will not occur only if the invention is drastic. Finally, if imitation is at least as beneficial to Firm 2 as the decision not to imitate, then offering a license to each competitor is at least as beneficial as the decision not to license.

Finally, I determine the equilibrium of the royalty licensing game and establish the following:

Proposition 5

If using the existing technology is better than imitation for Firm 2, then technology transfer through a per-unit royalty license will occur in equilibrium as follows: if $\epsilon <\frac{1}{2}(a+2c+r_{2}^{*}-3c')$, technology transfer to both players is at least as beneficial from the patent holder’s perspective as offering an exclusive license to any competitor; if $\frac{1}{2}(a+2c+r_{2}^{*}-3c') \le \epsilon < \frac{1}{2}(a-2c+r_{3}^{ND}+c')$, technology transfer exclusively to Firm 2 is at least as beneficial to the patent holder as the other options; and if $\frac{1}{2}(a-2c+r_{3}^{ND}+c') \le \epsilon < a-c$, technology transfer exclusively to Firm 3 is at least as beneficial to the patent holder as the other options. Technology transfer will not occur only in the case of a drastic invention. However, if, imitation is at least as beneficial to Firm 2 as using the existing technology, then licensing to both competitors is at least as beneficial to the patent holder as the other two options.

The Equilibrium

Finally, I can compare technology transfer through a fixed fee with that through a unit royalty, and determine the equilibrium of the oligopoly game. I establish the following proposition:

Proposition 6

If using the existing technology is better than imitation for Firm 2, then technology transfer through a per-unit royalty license is at least as beneficial as technology transfer through a fixed-fee license from the patent holder’s perspective. In particular, if $\epsilon <\frac{1}{2}(a+2c+r_{2}^{*}-3c')$, then royalty licensing to both competitors is at least as beneficial as all other equilibrium options; if $\frac{1}{2}(a+2c+r_{2}^{*}-3c')\le \epsilon < \frac{1}{2}(a-2c+{r_{3}^{ND}}^{*}+c')$, royalty licensing exclusively to the strong competitor is at least as beneficial as any other equilibrium option; and if $\frac{1}{2}(a-2c+{r_{3}^{ND}}^{*}+c') \le \epsilon <a-c$, royalty licensing exclusively to the weak competitor is at least as beneficial as any other equilibrium option. No technology transfer will occur in equilibrium if $\epsilon \ge a-c$. However, if imitation is at least as beneficial to Firm 2 as using the existing technology, then offering licenses to both competitors is at least as beneficial to the patent holder as the other two options.

One explanation for the overall superiority of royalties may be the cost advantage to the patent holder of licensing through a unit royalty. However, the analysis suggests that the cost advantage of royalties does not exceed the efficiency gain in the case of a drastic invention and when imitation is not an equilibrium option. In other words, the difference between the monopoly profit and the sum of the duopoly profits is greater than the cost advantage of royalties when imitation is not optimal.

Moreover, comparing the equilibrium quantity outputs shows that fixed-fee licensing is at least as attractive to consumers as royalty licensing. The analysis also suggests that patent holders can use technology transfer through licensing to prevent imitation, which is consistent with findings in the existing literature. Finally, these results do not support the proposition that licensing is selective and thus that patent holders could use it to harm competition, which seems to be the case when considering the effects of licensing on competition after patent expiration. For example, Rockett (1990) suggests that patent holders may use licensing strategically to select competition in order to maintain their dominant position in the market even after patent expiration. One reason to explain this result is that imitation may reduce the competitive advantage of the patent holder because of royalty licensing.

Conclusion

I analyze how imitation affects a patent holder’s licensing behavior, market structure, and consumer surplus. Specifically, I consider an incumbent patent holder who can license a cost-reducing technology to one or two competing firms in the same industry while facing a potential threat of imitation. Technology transfer increases the overall production efficiency of the industry; however, it can also increase the concentration level of the industry and lead to the formation of monopolies. The efficiency gain from technology transfer may make consumers indifferent to licensing or imitation if it does not transfer surplus from them to the firms. Identifying such a surplus threshold in an oligopolistic setting has received less attention in the existing literature. I contribute to this literature by providing a theoretical framework that explains why firms in concentrated industries enter into technology licensing agreements.

I find that royalty licensing is preferable to fixed-fee licensing from the patent holder’s perspective, while the opposite is true from the consumer’s perspective. In particular, licensing through a unit royalty, whether exclusive or to all competitors, occurs in equilibrium in several cases. The cost advantage of royalty licensing is offset by efficiency gains only in the case of a drastic invention and when imitation is not an optimal choice for the potential imitator. This result is consistent with Wang (1998), who shows that royalty licensing is superior due to the cost advantage arising from the patent holder’s royalties. In addition, I find that a patent holder can use technology licensing to prevent imitation, but cannot use it strategically to select competition during the patent term. From a consumer perspective, however, per-unit royalty licensing is at most as beneficial as fixed-fee licensing. This suggests a need for greater caution in intellectual property licensing.

Patents and licenses are increasingly important in the economy, and a better understanding of their strategic use will benefit all stakeholders. By identifying the factors that facilitate technology licensing and influence competition and consumer surplus, policy makers can revise regulations and policies to encourage beneficial licensing practices while preventing anti-competitive behavior that could harm consumer surplus. Understanding how different regulatory environments affect strategic licensing decisions could help policy makers design more effective intellectual property laws and competition policies. This is particularly important in complex industries such as ICT, semiconductors, and electronics, where technology transfer can have a significant impact on market structure. Practitioners can use insights into strategic licensing to optimize their strategies and business policies, such as entering into licensing agreements that maximize their competitive advantage while promoting collaborative innovation. This approach can promote competition, foster innovation, and ensure that the benefits of technological improvements are shared among firms and consumers.

Nevertheless, there are theoretical issues that I do not address in this analysis. For example, I do not consider random imitation or a lengthy imitation process. Future research could examine the impact of random imitation on licensing strategies and market outcomes, potentially challenging the conclusions drawn in this paper regarding strategic patenting and optimal licensing. In addition, I leave the inclusion of an arbitrary number of firms for future research. Examining the effects of including an arbitrary number of firms could provide deeper insights into how market structure influences the strategic behavior of patent holders and licensees. Another important avenue for future research is the replicability of these results in specific industry contexts, particularly in industries with different competitive dynamics and innovation patterns. This could involve empirical analysis to validate the theoretical propositions presented here. Finally, future studies could also relax the main assumptions made in this analysis by incorporating more realistic scenarios, thereby enhancing the applicability of this theoretical framework to a general context.²⁵

Appendix

Proof of Equation 7

First, I solve the maximization problem of Firm 1, subject to the participation constraints of Firm 2:

$$\begin{aligned} \begin{array}{ll} \underset{r\ge 0}{\text {max}} & \quad \pi ^{R}_{1},\\ \text { s.t.} & \pi _{2}^{R} \ge \pi _{2}^{ND},\\ & \pi _{2}^{R} \ge \pi _{2}^{D}. \end{array} \end{aligned}$$

(33)

Second, I form the Lagrange equation and define the critical points. The first-order conditions of the Lagrange equation are:

$$\begin{aligned} \begin{array}{llll} r & =& \dfrac{1}{2}(a-c+\epsilon ),\\ \\ r & \le & \epsilon & \text{ for } \epsilon < a-c,\\ \\ r & \le & \dfrac{1}{2}(a-c+\epsilon ) & \text{ for } \epsilon \ge a-c,\\ \\ r & \le & \dfrac{1}{2}(a-c+\epsilon )-\dfrac{1}{2}\sqrt{{(a-c+\epsilon )}^{2}-9C} & \forall \epsilon . \end{array} \end{aligned}$$

(34)

Finally, I compute the value of the objective function at each critical point and derive the solution. In all of the proofs in the Appendix, I also take into account the non-negativity constraints and the Lagrange multipliers, and consider that the optimal royalty cannot be greater than the size of the innovation in any case. This procedure is equivalent to using the Kuhn and Tucker Theorem to solve an inequality-constrained optimization problem. Equation 7 follows from simple calculations.

Proof of Equation 22

I can apply the same basic analysis to this case. First, I solve Firm 1’s maximization problem subject to Firm 2’s participation constraints:

$$\begin{aligned} \begin{array}{ll} \underset{r_{2}\ge 0}{\text {max}} & \quad \pi ^{R}_{1},\\ \text { s.t} & \pi _{2}^{R} \ge \pi _{2}^{ND},\\ & \pi _{2}^{R} \ge \pi _{2}^{D}. \end{array} \end{aligned}$$

(35)

Second, I form the Lagrange equation and define the critical points. The first-order conditions of the Lagrange equation are:

$$\begin{aligned} \begin{array}{llll} r_{2} & =& \dfrac{3}{11}(a-2[c-\epsilon ]+c'),\\ \\ r_{2} & \le & \epsilon & \text{ for } \epsilon< a+2c-3c',\\ \\ 0 & \le & \dfrac{1}{16}(a-2[c-\epsilon ]-3r_{2}+c')^{2}-\dfrac{1}{9}(a-c-\epsilon )^{2} & \text{ for } a\!+\!2c-3c' \le \epsilon \!<\! a-2c\!+\!c',\\ \\ r_{2} & \le & \dfrac{1}{3}(a-2[c-\epsilon ]+c')-\dfrac{1}{2}\sqrt{{(a-c+\epsilon )}^{2}-9C} & \text{ for } \epsilon \ge a-2c+c',\\ \\ 0 & \le & \dfrac{1}{16}(a-2[c-\epsilon ]-3r_{2}+c')^{2}-\dfrac{1}{16}(a-2c+c'+2\epsilon )^{2}-C & \text{ for } \epsilon < \dfrac{1}{2}(a+2c-3c'),\\ \\ 0 & \le & \dfrac{1}{16}(a-2[c-\epsilon ]-3r_{2}+c')^{2}-\dfrac{1}{9}(a-c+\epsilon )^{2}-C & \text{ for } \epsilon \ge \dfrac{1}{2}(a+2c-3c'). \end{array} \end{aligned}$$

(36)

Finally, I compute the value of the objective function at each critical point and derive the solution. By focusing specifically on the different intervals of the size of the innovation and using simple algebra, the results shown in Eq. 22 follow.

Proof of Equations 24 and 25

Similar to the optimization problems above, I solve Firm 1’s maximization problem subject to Firm 3’s participation constraint:

$$\begin{aligned} \begin{array}{ll} \underset{r_{3}^{ND}\ge 0}{\text {max}} & \quad \pi ^{R}_{1}(ND),\\ \text { s.t} & \pi _{3}^{R}(ND) \ge \pi _{3}^{ND}. \end{array} \end{aligned}$$

(37)

Second, I form the Lagrange equation and define the critical points. The first-order conditions of the Lagrange equation are:

$$\begin{aligned} \begin{array}{llll} r_{3}^{ND} & =& \dfrac{3}{11}(3a+2c+6\epsilon -5c'),\\ \\ r_{3}^{ND} & \le & \epsilon & \text{ for } \epsilon < a+2c-3c',\\ \\ r_{3}^{ND} & \le & \dfrac{1}{3}(a+2[c+\epsilon ]-3c') & \text{ for } \epsilon \ge a-2c+c'. \end{array} \end{aligned}$$

(38)

Finally, I compute the value of the objective function at each critical point and derive the solution. By focusing specifically on the different intervals of the size of the innovation and using simple algebra, the results in Eqs. 24 and 25 follow. Note that the relationship between $r_{3}^{ND}$ and $\epsilon$ changes for values of $c'$ less or greater than $\frac{1}{5}(a+4c)$.

Proof of Equations 27 and 28

I first solve Firm 1’s maximization problem subject to Firm 3’s participation constraint:

$$\begin{aligned} \begin{array}{ll} \underset{r_{3}^{D}\ge 0}{\text {max}} & \quad \pi ^{R}_{1}(D),\\ \text { s.t} & \pi _{3}^{R}(D) \ge \pi _{3}^{D}. \end{array} \end{aligned}$$

(39)

Second, I form the Lagrange equation and define the critical points. The first-order conditions of the Lagrange equation in this case are:

$$\begin{aligned} \begin{array}{llll} r_{3}^{D} & =& \dfrac{1}{11}(3[a+\epsilon ]+2c-5c'),\\ \\ r_{3}^{D} & \le & \epsilon & \text{ for } \epsilon < \dfrac{1}{2}(a+2c-3c'),\\ \\ r_{3}^{D} & \le & \dfrac{1}{3}(a+2c+\epsilon -3c') & \text{ for } \epsilon \ge \dfrac{1}{2}(a+2c-3c'). \end{array} \end{aligned}$$

(40)

Finally, I compute the value of the objective function at each critical point and derive the solution. By focusing specifically on the different intervals of the size of the innovation and using simple algebra, the results in Eqs. 27 and 28 follow. Note that the relationship between $r_{3}^{D}$ and $\epsilon$ changes for values of $c'$ less or greater than $\frac{1}{7}(a+6c)$.

Proof of Equations 30 and 31

I can apply the same basic analysis as above to this last case. First, I solve the maximization problem of Firm 1, subject to the participation constraints of Firms 2 and 3:

$$\begin{aligned} \begin{array}{ll} \underset{\mathfrak {r}_{2}, \mathfrak {r}_{3}\ge 0}{\text {max}} & \quad \pi ^{\mathbb {R}}_{1},\\ \text { s.t} & \pi ^{\mathbb {R}}_{2} \ge \pi _{2}^{ND},\\ & \pi ^{\mathbb {R}}_{2} \ge \pi _{2}^{D},\\ & \pi ^{\mathbb {R}}_{3} \ge \pi _{3}^{ND}. \end{array} \end{aligned}$$

(41)

Second, I form the Lagrange equation and define the critical points. The first-order conditions of the Lagrange equation are:

$$\begin{aligned} \begin{array}{llll} \mathfrak {r}_{2} & =& \frac{1}{11}(3a-6c+3\epsilon +5\mathfrak {r}_{3}+3c')=0,\\ \\ \mathfrak {r}_{3} & =& \frac{1}{11}(3a+2c+3\epsilon +5\mathfrak {r}_{2}-5c')=0,\\ \\ \mathfrak {r}_{2} & \le & \frac{1}{3}(2\epsilon +\mathfrak {r}_{3}) & \text{ for } \epsilon \le a+2c-3c',\\ \\ 0 & \le & \frac{1}{16}(a-2c+\epsilon -3\mathfrak {r}_{2}+\mathfrak {r}_{3}+c')-\frac{1}{9}(a-c-\epsilon )^{2} & \text{ for } a+2c-3c' \le \epsilon< a-2c+c',\\ \\ \mathfrak {r}_{2} & \le & \frac{1}{3}(a-2c+\epsilon +\mathfrak {r}_{3}+c') & \text{ for } \epsilon \ge a-2c+c',\\ \\ 0 & \le & \frac{1}{16}(a-2c+\epsilon -3\mathfrak {r}_{2}+\mathfrak {r}_{3}+c')^{2}-\frac{1}{16}(a-2c+c'+2\epsilon )^{2}+C & \text{ for } \epsilon \le \frac{1}{2}(a+2c-3c'),\\ \\ 0 & \le & \frac{1}{16}(a-2c+\epsilon -3\mathfrak {r}_{2}+\mathfrak {r}_{3}+c')^{2}-\frac{1}{9}(a-c+\epsilon )^{2}+C & \text{ for } \epsilon \ge \frac{1}{2}(a+2c-3c'),\\ \\ \mathfrak {r}_{3} & \le & \frac{1}{3}(2\epsilon +\mathfrak {r}_{2}) & \text{ for } \epsilon < a+2c-3c',\\ \\ \mathfrak {r}_{3} & \le & \frac{1}{3}(a+2c+\epsilon +\mathfrak {r}_{2}-3c') & \text{ for } \epsilon \ge a+2c-3c'.\\ \\ \end{array} \end{aligned}$$

(42)

Finally, I compute the value of the objective function at each critical point and derive the solution. Obviously, this case is relatively more complicated than the other cases. By solving the first two conditions simultaneously, I obtain the two critical points as follows:

$$\begin{aligned} \begin{array}{llll} \mathfrak {r}_{2} & =& \dfrac{1}{12}(6a-7c+6\epsilon +c'),\\ \\ \mathfrak {r}_{3} & =& \dfrac{1}{12}(6a-c+6\epsilon -5c'). \end{array} \end{aligned}$$

(43)

In addition, considering the fifth and last conditions, I get the following critical points:

$$\begin{aligned} \begin{array}{llll} \mathfrak {r}_{2} & =& \dfrac{1}{2}(a-c+\epsilon ),\\ \\ \mathfrak {r}_{3} & =& \dfrac{1}{3}(a+c+\epsilon -2c'). \end{array} \end{aligned}$$

(44)

I also get two more critical points from the sixth and seventh conditions:

$$\begin{aligned} \begin{array}{llll} \mathfrak {r}_{2} & =& \dfrac{3}{8}(a-2[c-\epsilon ]+c')-\dfrac{3}{8}\sqrt{(a-2[c-\epsilon ]+c')^{2}-16C},\\ \\ \mathfrak {r}_{2} & =& \dfrac{1}{2}(a-c+\epsilon )-\dfrac{1}{2}\sqrt{(a-c+\epsilon )^{2}-9C}. \end{array} \end{aligned}$$

(45)

Clearly, the size of the innovation is also a critical point. Again, by focusing specifically on the different intervals of the size of the innovation and using simple algebra, the results in Eqs. 30 and 31 follow.