Technological leapfrogging and country strategic patent policy

Abstract

In this paper, the term “Country Strategic Patent Policy” refers to the case in which the examination of foreign firms’ patent applications may be deliberately manipulated by national patent offices to protect domestic firms, as a means to leapfrogging their foreign counterparts in technological strategic sectors. However, it is empirically questionable to distinguish the impact of discriminatory patent policy from the effect of the liabilities of foreignness. Therefore, international intervention to eliminate discriminatory treatment has been controversial. In this paper, we try to solve this conundrum by proposing a game-theory model to simulate the effect of strategic patent policy. The simulation results suggest that strategic patent policy measures are more likely to impede foreign patents that are (1) associated with R&D-intensive industries, (2) related to sectors where local firms’ absorptive capability is weak, and (3) registered in other countries. These hypotheses are then tested empirically by using patent data bases of six major economies in the world. The empirical analysis provides evidence of the possible existence of strategic patent policy against foreign companies in Japan and China, especially in high-technology and medium-high-technology industries.

Résumé

Dans cet article, l'expression "Politique Stratégique Nationale des Brevets" fait référence au cas où les offices nationaux des brevets manipuleraient délibérément l’examen des demandes de brevets des entreprises étrangères afin de protéger les entreprises nationales et ainsi leur permettre de devancer leurs homologues étrangers dans des secteurs technologiques stratégiques. Néanmoins, il est empiriquement discutable de distinguer l'impact de la politique discriminatoire des brevets de l'effet des handicaps liés au statut étranger de l’entreprise. Reste par conséquent controversée l'intervention internationale visant à éliminer le traitement discriminatoire. Cet article vise à résoudre cette énigme en proposant un modèle de théorie des jeux pour simuler l'effet de la politique stratégique des brevets. Les résultats de la simulation suggèrent que les mesures de la politique stratégique des brevets sont plus susceptibles d'entraver les brevets étrangers lorsqu’ils sont (1) associés à des industries à forte intensité de R&D, (2) liés à des secteurs où la capacité d'absorption des entreprises locales est faible, et (3) enregistrés dans d'autres pays. Ces hypothèses sont ensuite testées empiriquement en utilisant les bases de données de brevets de six grandes économies mondiales. L'analyse empirique apporte des preuves de l'existence possible d'une politique stratégique des brevets à l'encontre des entreprises étrangères au Japon et en Chine, en particulier dans les industries de haute technologie et de moyenne-haute technologie.

Resumen

En este artículo, el término “Política Estratégica de Patentes del País" se refiere al caso en el que el examen de las solicitudes de patentes de las empresas extranjeras puede ser manipulado deliberadamente por las oficinas nacionales de patentes para proteger a las empresas nacionales para aventajarlas con relación a las contrapartes extranjeras en sectores tecnológicos estratégicos. Sin embargo, es empíricamente cuestionable distinguir el impacto de la política de patentes discriminatoria del efecto de las desventajas de extranjería. Por lo tanto, la intervención internacional para eliminar el trato discriminatorio ha sido controversial. En este artículo, intentamos resolver este acertijo proponiendo un modelo de teoría de juegos para simular el efecto de la política estratégica de patentes. Los resultados de la simulación sugieren que es más probable que las medidas de política estratégica de patentes impidan las patentes extranjeras que están (1) asociadas a industrias intensivas en I+D, (2) relacionadas con sectores en los que la capacidad de absorción de las empresas locales es débil, y (3) registradas en otros países. Estas hipótesis se comprueban empíricamente utilizando bases de datos de patentes de seis grandes economías del mundo. El análisis empírico aporta evidencias de la posible existencia de una política estratégica de patentes contra las empresas extranjeras en Japón y China, especialmente en las industrias de alta y media-alta tecnología.

Resumo

Neste artigo, o termo “Política de Patentes Estratégica do País” refere-se ao caso em que o exame de pedidos de patentes de empresas estrangeiras pode ser deliberadamente manipulado pelos escritórios nacionais de patentes para proteger empresas nacionais como um meio de ultrapassar suas contrapartes estrangeiras em setores tecnológicos estratégicos. No entanto, é empiricamente questionável distinguir o impacto da política de patentes discriminatória do efeito da desvantagem de ser estrangeiro. Portanto, a intervenção internacional para eliminar o tratamento discriminatório tem sido controversa. Neste artigo, tentamos resolver esse dilema propondo um modelo baseado na teoria dos jogos para simular o efeito da política de patentes estratégica. Os resultados da simulação sugerem que medidas de política de patentes estratégica são mais propensas a impedir patentes estrangeiras que são (1) associadas a indústrias intensivas em P&D, (2) relacionadas a setores em que a capacidade de absorção de empresas locais é fraca e (3) registradas em outros países. Essas hipóteses são então testadas empiricamente usando bases de dados de patentes das seis maiores economias do mundo. A análise empírica fornece evidências da possível existência de política de patentes estratégica contra empresas estrangeiras no Japão e China, especialmente em indústrias de alta e média-alta tecnologia.

摘要

在本文中, “国家战略性专利政策”一词指的是国家专利局为了保护国内公司可能故意操纵对外国公司专利申请的审查, 从而助力国内公司在技术战略领域反超外国同行企业。然而, 将歧视性专利政策的影响与外来者劣势的影响区分开来在实证上是有难度的。因此, 消除歧视性待遇的国际干预一直存在争议。在本文中, 我们试图通过提出一个博弈论模型来揭示战略专利政策的影响以解决这个“识别”难题。对数理模型的仿真求解结果表明, 相比一般性的歧视政策，战略性专利政策措施更有可能阻碍如下国外专利的申请: (1) 与研发密集型产业相关的专利, (2) 与本地公司吸收能力较弱的行业相关的专利, 以及 (3) 在其他国家注册次数较多的专利。然后, 这些假设通过用世界六个主要经济体的专利数据库被实证检验。实证分析揭示日本和中国的外国公司可能面临较强的战略性专利政策, 特别是在高科技和中高科技行业。

Appendices

APPENDIX A: MODELING THE IMPACTS OF STRATEGIC PATENT POLICY

In this appendix, we demonstrate how a strategic patent policy can affect the profit of a domestic company and that of a foreign company, and then show how the use of this policy can further stimulate a domestic company to choose a technological leapfrogging strategy.

The Structure of the Model

As explained in the main text, the model has three players, namely a national patent office, a foreign MNE (Firm 1), and a domestic firm (Firm 2). Firm 1 is the technological leader, and Firm 2 is the follower. The game is played in sequential order. In stage 1, the national patent office decides whether to impose a strategic patent policy. If it does, Firm 1 is confronted with discrimination, which is reflected by two indicators: patent grant probability (P) and grant lag (T). In stage 2, Firm 1 decides whether to introduce its new product or invention in the country. If it does, it will bring a higher-quality (s₁) product into the existing market. In stage 3, Firm 2 decides whether to imitate Firm 1’s product and chooses its own quality (s₂) if it enters the market.

Suppose that Firm 2 is the only firm that takes the risk of imitating Firm 1’s new product. Hence, during the examination period (T years), a dual oligopolistic market structure appears, and the firms earn oligopolistic profits ($\Pi_{1}^{{\text{O}}}$ and $\Pi_{2}^{{\text{O}}}$, respectively). However, after the patent application is rejected or after the patent protection expires, additional firms will enter the market because the risk of patent infringement no longer exists. This will push the profit close to zero, as in a perfectly competitive market.

To start, we derive the mathematical expressions of the profit functions. We assume that production is costless, and Firm 2 invests in R&D (C₂) to absorb Firm 1’s invention (in this case $0 < s_{2} \le s_{1}$) or to develop a higher-quality product (in this case $s_{2} \ge s_{1} > 0$). If the patent application is rejected, then Firm 2 will keep its profit. If the patent is granted, Firm 1 can claim royalty damages and hence collect compensation, usually in the form of fines (F). If we denote the probability that the patent will be granted as P, then Firm 2’s expected profit can be written as:

$$E(\Pi_{2} ) = [(1 - P)\Pi_{2}^{{\text{O}}} + P(\Pi_{2}^{{\text{O}}} - F)] - C_{2} .$$

(8)

Correspondingly, Firm 1’s profit also depends on whether its patent is granted. Because of competition from Firm 2, Firm 1 only obtains oligopolistic profit ($\Pi_{1}^{{\text{O}}}$) during the examination period. If the patent application is rejected, even the oligopolistic profit will disappear in perfect competition. However, if the patent is granted, Firm 1 can enjoy monopoly profits ($\Pi_{1}^{{\text{M}}}$). Hence, Firm 1’s expected profit can be written as:

$$E(\Pi_{1} ) = [(1 - P)\Pi_{1}^{{\text{O}}} + P(\Pi_{1}^{{\text{O}}} + \Pi_{1}^{{\text{M}}} + F)] - C_{1} ,$$

(9)

where C₁ is the R&D cost function for Firm 1. Different countries have different rules on determining the amount of compensation (F). According to Coury (2003), in practice it is common to calculate the fine on the basis of the infringers’ profits. Therefore, we construct the compensation function as:

$$F = \mu \beta {\text{Max}}\left( {0,\Pi_{2}^{O} - C_{2} } \right),$$

(10)

where β is the degree of punishment for patent infringement, and μ is the degree of infringement. Because a high degree of imitation may increase the probability of litigation, we use the quality differential to measure the degree of infringement, that is, $\mu = 1 - \left| {s_{1} - s_{2} } \right|$.

To play the game, we need to rewrite the profits E(Π₁) and E(Π₂) as functions of s₁ and s₂. To achieve this goal, the following two sections will derive costs (C₁ and C₂) and profits ($\Pi_{1}^{{\text{O}}} ,\Pi_{2}^{{\text{O}}}$ and $\Pi_{1}^{{\text{M}}}$) as functions of s₁ and s₂.

Deriving Cost Functions

We derive the R&D cost function from a knowledge production function. New knowledge is created by new R&D investment based on established knowledge, and the level of established knowledge is implicated in the quality (s) of the product (Griliches, 1979; Klette & Kortum, 2004; Lederman, 2010). Thus, the knowledge production function can be written as:

$$s_{t} = K(s_{t - 1} ,r_{t} ),$$

(11)

where s_t reflects the new knowledge needed to obtain a higher-quality product. The new knowledge is created by R&D investment (r_t) in period t together with the existing quality of the product (s_t−1).

For simplicity, we assume an additive form for Eq. (11) and obtain:

$$s_{t} = \gamma s_{t - 1} + \hat{K}(r_{t} ),$$

(12)

where $\gamma \in [0,1]$ represents the absorptive capacity of the existing knowledge stock. Function $\hat{K}(r_{t} )$ measures how much new knowledge can be generated by R&D investment.

Essentially, the marginal product of R&D investment is constrained by the stock of existing knowledge. For example, even infinite R&D investment could not lead to development of iPhone in the 1950s. Thus, the marginal product of R&D investment should converge to zero as investment increases to infinity. The zero marginal product of a specific input in a production function has long been acknowledged. For instance, Färe (1974) and Suliman (1997) studied the zero marginal product of labor in production functions. Essentially, given a certain stock of knowledge, it means only a bounded quality improvement ($\overline{s}$) is feasible. Accordingly, the knowledge production function is constrained by the following conditions:

$$\mathop {\lim }\limits_{{r_{t} \to \infty }} s_{t} = \gamma s_{t - 1} + \mathop {\lim }\limits_{{r_{t} \to \infty }} \hat{K}(r_{t} ) = \gamma s_{t - 1} + \overline{s},$$

(13)

and

$$\mathop {\lim }\limits_{{r_{t} \to 0}} s_{t} = \gamma s_{t - 1} + \mathop {\lim }\limits_{{r_{t} \to 0}} \hat{K}(r_{t} ) = \gamma s_{t - 1} .$$

(14)

The commonly employed quadratic form of cost functions cannot satisfy these conditions. A typical quadratic cost function can be represented as $r = (1/2)s^{2}$ (Kovac & Zigic, 2014). This cost function implies that as $r_{t} \to \infty ,s_{t} \to \infty$, which violates Eq. (13).

The simplest form of a cost function that satisfies Eqs. (13) and (14) is

$$r_{t} = C(s_{t} ) = \frac{{s_{t} - \gamma s_{t - 1} }}{{\gamma s_{t - 1} + \overline{s} - s_{t} }}.$$

(15)

This is the baseline cost function to be used for Firms 1 and 2. Firm 1 is an MNE, hence its cost of R&D can be divided by the number of markets or countries (n) in which it operates. Thus, the R&D cost function of Firm 1 can be written as:

$$C_{1} (s_{1,t} ) = \frac{1}{n}\frac{{s_{1,t} - \gamma s_{1,t - 1} }}{{\gamma s_{1,t - 1} + \overline{s} - s_{1,t} }},$$

(16)

where $s_{1,t - 1}$ reflects the stock of knowledge at Firm 1.

Without loss of generality, we normalize the baseline knowledge stock ($s_{1,t - 1}$) to 0 and standardize the upper bound of R&D productivity ($\overline{s}$) to 1. This gives Firm 1’s cost function as:

$$C_{1} (s_{1} ) = \frac{1}{n}\frac{{s_{1} }}{{1 - s_{1} }},\quad (0 < s_{1} < 1).$$

(17)

Firm 2 operates only in the domestic market; hence its cost cannot be shared. However, its R&D activities can benefit from Firm 1’s product in the market (with quality s₁). Thus, Firm 2’s cost function can be written as:

$$C_{2} (s_{2} |s_{1} ) = \frac{{s_{2} - \gamma s_{1} }}{{\gamma s_{1} + 1 - s_{2} }},\quad (\gamma s_{1} \le s_{2} < 1 + \gamma s_{1} ).$$

(18)

Finally, we add parameter α to capture the R&D expenditure as a reasonable ratio over earned revenue. The share of a firm’s revenue spent on R&D investment varies substantially across sectors. Thus, the final forms of the cost functions take into account the industrial heterogeneity:

$$C_{1} (s_{1} ) = \frac{1}{n}\frac{{\alpha s_{1} }}{{1 - s_{1} }},\quad (0 < s_{1} < 1),$$

(19)

$$C_{2} (s_{2} |s_{1} ) = \frac{{\alpha (s_{2} - \gamma s_{1} )}}{{1 + \gamma s_{1} - s_{2} }},\quad (\gamma s_{1} \le s_{2} < 1 + \gamma s_{1} ).$$

(20)

Deriving Profit Functions

The profits ($\Pi_{1}^{{\text{O}}} ,\Pi_{2}^{{\text{O}}}$) of both Firms 1 and 2 depend on the examination duration (T years) of Firm 1’s patent application. Thus, they are the summed values of the annual oligopolistic profits $\pi_{1}^{{\text{O}}}$ and $\pi_{2}^{{\text{O}}}$ for T years, respectively. Similarly, profit $\Pi_{1}^{{\text{M}}}$ is the sum of the annual monopoly profit $\pi_{1}^{{\text{M}}}$. Because patents are usually protected for 20 years, we also sum $\pi_{1}^{{\text{M}}}$ for 20 years. Denoting the annual discount rate as ρ, we can write:

$$\Pi_{2}^{{\text{O}}} = \int\limits_{0}^{T} {e^{ - \rho \, t} \pi_{2}^{{\text{O}}} {\text{d}}t} = \frac{{1 - e^{ - \rho T} }}{\rho }\pi_{2}^{{\text{O}}} ,$$

(21)

$$\Pi_{1}^{{\text{O}}} = \int\limits_{0}^{T} {e^{ - \rho \, t} \pi_{1}^{{\text{O}}} {\text{d}}t} = \frac{{1 - e^{ - \rho T} }}{\rho }\pi_{1}^{{\text{O}}} ,$$

(22)

and

$$\Pi_{1}^{{\text{M}}} = \int\limits_{T}^{T + 20} {e^{ - \rho \, t} \pi_{1}^{{\text{M}}} {\text{d}}t} = \frac{{e^{ - \rho T} (1 - e^{ - 20\rho } )}}{\rho }\pi_{1}^{{\text{M}}} .$$

(23)

Firms 1 and 2 compete with each other and receive oligopolistic profits $\pi_{1}^{{\text{O}}} = p_{1} \times q_{1} \left( {p_{1} ,p_{2} } \right)$ and $\pi_{2}^{{\text{O}}} = p_{2} \times q_{2} (p_{1} ,p_{2} )$, where p and q are price and quantity, respectively.

Thus, we need to derive the demand functions q₁(p₁, p₂) and q₂(p₁, p₂) from consumers' utility function. We assume that products are vertically differentiated. Each consumer is characterized by a parameter θ and has the following utility function (Motta, 1993):

$$U_{\theta } = \left\{ {\begin{array}{*{20}l} {\theta s - p,} \hfill & {\quad {\text{if}}\,{\mkern 1mu} {\text{the}}\,{\mkern 1mu} {\text{consumer}}\,{\mkern 1mu} {\text{buys}}\,{\mkern 1mu} {\text{good}}\,{\mkern 1mu} {\text{with}}\,{\mkern 1mu} {\text{quality}}\,{\mkern 1mu} s\,{\mkern 1mu} {\text{and}}\,{\mkern 1mu} {\text{price}}\,{\mkern 1mu} p,} \hfill \\ {0,} \hfill & {\quad {\text{if}}\, {\mkern 1mu} {\text{the}}\,{\mkern 1mu} {\text{consumer}}\,{\mkern 1mu} {\text{does}}\,{\mkern 1mu} {\text{not}}\,{\mkern 1mu} {\text{buy}}.} \hfill \\ \end{array} } \right.$$

(24)

θ indicates the consumer’s appreciation of quality. It can also be interpreted as the marginal rate of substitution between income and quality, so a high θ corresponds to a low marginal utility of income (Shaked & Sutton, 1987; Tirole, 1988: 96). For the sake of simplicity, we assume that θ is uniformly distributed over the interval [0, 1], and each consumer buys only one unit of the product (Li & Song, 2009).

Based on the quality and price offered by Firms 1 and 2, consumers can choose either to buy the product from Firm 1 or Firm 2 or not to buy it. Their choices determine the two firms' demand functions. In the case of s₁ > s₂, the marginal consumer who is indifferent between buying either quality s₁ or quality s₂ is determined by $\theta s_{1} - p_{1} = \theta s_{2} - p_{2}$. Hence, $\theta_{12} = (p_{1} - p_{2} )/(s_{1} - s_{2} )$. As a result, the demand function for a good of quality s₁ can be derived as:

$$q_{1} = \int_{{\theta_{12} }}^{1} {\text{d}} \theta = 1 - \frac{{p_{1} - p_{2} }}{{s_{1} - s_{2} }}.$$

(25)

Similarly, the marginal consumer who is indifferent between buying a good of quality s₂ and not buying anything at all is determined by $\theta s_{2} - p_{2} = 0$, Hence, $\theta_{20} = p_{2} /s_{2}$. Thus, the demand function for a good of quality s₂ is:

$$q_{2} = \int_{{\theta_{20} }}^{{\theta_{12} }} {\text{d}} \theta = \frac{{p_{1} - p_{2} }}{{s_{1} - s_{2} }} - \frac{{p_{2} }}{{s_{2} }}.$$

(26)

Based on the same method, in which Firm 2 produces a higher-quality good (s₂>s₁), the demand functions are:

$$q_{2} = 1 - \frac{{p_{2} - p_{1} }}{{s_{2} - s_{1} }}$$

(27)

and

$$q_{1} = \frac{{p_{2} - p_{1} }}{{s_{2} - s_{1} }} - \frac{{p_{1} }}{{s_{1} }}.$$

(28)

Firm 1 will obtain the monopoly profit $\pi_{1}^{{\text{M}}}$ after its patent is granted. In this case, the marginal consumer who is indifferent between buying the good of quality s₁ and not buying any good at all is determined by:

$$\theta s_{1} - p_{1} = 0$$

(29)

Hence $\theta_{10} = p_{1} /s_{1}$. Now the demand function is:

$$q_{1} = \int_{{\theta_{10} }}^{1} {\text{d}} \theta = 1 - \frac{{p_{1} }}{{s_{1} }}.$$

(30)

Winning the Game

In the previous section, two versions of the demand functions $q_{1} (p_{1} ,p_{2} )$ and $q_{2} (p_{1} ,p_{2} )$ are derived. If Firm 2 produces a good with inferior quality (s₂<s₁), then the demand functions are $q_{1} = 1 - \frac{{p_{1} - p_{2} }}{{s_{1} - s_{2} }}$ and $q_{2} = \frac{{p_{1} - p_{2} }}{{s_{1} - s_{2} }} - \frac{{p_{2} }}{{s_{2} }}$. In this case, the Bertrand duopoly solutions are:

$$p_{1}^{*} = \frac{{2s_{1}^{2} - 2s_{1} s_{2} }}{{4s_{1} - s_{2} }}$$

(31)

and

$$p_{2}^{*} = \frac{{s_{1} s_{2} - s_{2}^{2} }}{{4s_{1} - s_{2} }}.$$

(32)

The corresponding profits are:

$$\pi_{1}^{{\text{O}}} = \frac{{4(s_{1} - s_{2} )s_{1}^{2} }}{{(4s_{1} - s_{2} )^{2} }}$$

(33)

and

$$\pi_{2}^{{\text{O}}} = \frac{{s_{1} s_{2} (s_{1} - s_{2} )}}{{(4s_{1} - s_{2} )^{2} }}.$$

(34)

If Firm 2 produces a good of superior quality (s₂>s₁), the demand functions change to $\hat{q}_{1} = \frac{{p_{1} - p_{2} }}{{s_{1} - s_{2} }} - \frac{{p_{2} }}{{s_{2} }}$ and $\hat{q}_{2} = 1 - \frac{{p_{1} - p_{2} }}{{s_{1} - s_{2} }}$. In this case, the annual profits change to:

$$\hat{\pi }_{1}^{{\text{O}}} = \frac{{s_{1} s_{2} (s_{1} - s_{2} )}}{{(4s_{1} - s_{2} )^{2} }}$$

(35)

and:

$$\hat{\pi }_{2}^{{\text{O}}} = \frac{{4(s_{1} - s_{2} )s_{1}^{2} }}{{(4s_{1} - s_{2} )^{2} }}$$

(36)

If Firm 1’s patent is granted, the demand for Firm 1’s product is $q_{1} = 1 - p_{1} /s_{1}$. In this case, Firm 1 obtains a monopolistic profit:

$$\pi_{1}^{{\text{M}}} = s_{1} /4.$$

(37)

By integrating these results with the expected profit functions in Equations (8) and (9), we obtain Eqs. (38) and (39), in the case of s₂<s₁:

$$\begin{aligned} E(\Pi_{2} ) & = [(1 - P)\Pi_{2}^{O} + P(\Pi_{2}^{O} - F)] - C_{2} = \Pi_{2}^{O} - P \times F - C_{2} = \frac{{1 - e^{ - \rho T} }}{\rho }\frac{{s_{1} s_{2} (s_{1} - s_{2} )}}{{(4s_{1} - s_{2} )^{2} }} \\ & \quad - P \times (1 - s_{1} + s_{2} )\beta {\text{Max}}\left( {0,\quad \frac{{1 - e^{ - \rho T} }}{\rho }\frac{{s_{1} s_{2} (s_{1} - s_{2} )}}{{(4s_{1} - s_{2} )^{2} }} - \frac{{\alpha (s_{2} - \gamma s_{1} )}}{{1 + \gamma s_{1} - s_{2} }}} \right) - \frac{{\alpha (s_{2} - \gamma s_{1} )}}{{1 + \gamma s_{1} - s_{2} }}, \\ \end{aligned}$$

(38)

and:

$$\begin{aligned} E(\Pi_{1} ) & = [(1 - P)\Pi_{1}^{{\text{O}}} + P(\Pi_{1}^{{\text{O}}} + \Pi_{1}^{{{\text{M}}20}} + F)] - C_{1} = \Pi_{1}^{{\text{O}}} + P(\Pi_{1}^{{\text{M}}} + F) - C_{1} \\ & = \frac{{1 - e^{ - \rho T} }}{\rho }\frac{{4(s_{1} - s_{2} )s_{1}^{2} }}{{(4s_{1} - s_{2} )^{2} }} + P\frac{{e^{ - \rho T} (1 - e^{ - 20\rho } )}}{\rho }\frac{{s_{1} }}{4} \\ & \quad + P \times (1 - s_{1} + s_{2} )\beta {\text{Max}}\left( {0,\frac{{1 - e^{ - \rho T} }}{\rho }\frac{{s_{1} s_{2} (s_{1} - s_{2} )}}{{(4s_{1} - s_{2} )^{2} }} - \frac{{\alpha (s_{2} - \gamma s_{1} )}}{{1 + \gamma s_{1} - s_{2} }}} \right) - \frac{1}{n}\frac{{\alpha s_{1} }}{{1 - s_{1} }}. \\ \end{aligned}$$

(39)

In the case of s₂>s₁, the expected profits are:

$$\begin{aligned} E(\hat{\Pi }_{2} ) & = \hat{\Pi }_{2}^{{\text{O}}} - P \times F - C_{2} = \frac{{1 - e^{ - \rho T} }}{\rho }\frac{{4(s_{2} - s_{1} )s_{2}^{2} }}{{(4s_{2} - s_{1} )^{2} }} \\ & \quad - P \times (1 - s_{2} + s_{1} )\beta {\text{Max}}\left[ {0,\frac{{1 - e^{ - \rho T} }}{\rho }\frac{{4(s_{2} - s_{1} )s_{2}^{2} }}{{(4s_{2} - s_{1} )^{2} }} - \frac{{\alpha (s_{2} - \gamma s_{1} )}}{{1 + \gamma s_{1} - s_{2} }}} \right] - \frac{{\alpha (s_{2} - \gamma s_{1} )}}{{1 + \gamma s_{1} - s_{2} }} \\ \end{aligned}$$

(40)

and:

$$\begin{aligned} E(\hat{\Pi }_{1} ) & = \hat{\Pi }_{1}^{{\text{O}}} + P(\Pi_{1}^{{\text{M}}} + F) - C_{1} = \frac{{1 - e^{ - \rho T} }}{\rho }\frac{{s_{1} s_{2} (s_{2} - s_{1} )}}{{(4s_{2} - s_{1} )^{2} }} + P\frac{{e^{ - \rho T} (1 - e^{ - 20\rho } )}}{\rho }\frac{{s_{1} }}{4} \\ & \quad + P \times (1 - s_{2} + s_{1} )\beta {\text{Max}}\left( {0,\frac{{1 - e^{ - \rho T} }}{\rho }\frac{{s_{1} s_{2} (s_{1} - s_{2} )}}{{(4s_{1} - s_{2} )^{2} }} - \frac{{\alpha (s_{2} - \gamma s_{1} )}}{{1 + \gamma s_{1} - s_{2} }}} \right) - \frac{1}{n}\frac{{\alpha s_{1} }}{{1 - s_{1} }}. \\ \end{aligned}$$

(41)

In stage 3, Firm 2 chooses its quality to maximize its profit. That is:

$${\text{Max}}\left[ {E(\Pi_{2}^{{}} )} \right],\quad {\text{subject}}\,{\text{to}}\,0 < \gamma s_{1} \le s_{2} < s_{1} < 1$$

(42)

or:

$${\text{Max}}\left[ {E(\hat{\Pi }_{2}^{{}} )} \right],\quad {\text{subject}}\,{\text{to}}\,0 < s_{1} < s_{2} < 1 + \gamma s_{1} .$$

(43)

Solutions to these maximization problems depend on both s₁ and s₂. In stage 2, Firm 1 can calculate Firm 2’s best response curve, and thus decides its best quality level ($s_{1}^{*}$). Then, in stage 3, Firm 2 compares ${\text{Max}}\left[ {E(\Pi_{2}^{{}} )} \right]$ with ${\text{Max}}\left[ {E(\hat{\Pi }_{2}^{{}} )} \right]$ to decide whether to leapfrog Firm 1.According to Zermelo’s theorem, a Nash equilibrium exists in a finite game with perfect information (Schwalbe & Walker, 2001). Furthermore, we can prove that, in this game, s₂ = s₁ is not an equilibrium solution: Proof When s₂ equals s₁, Eqs. (38) and (40) become identical:

$$E(\Pi_{2} ) = E(\hat{\Pi }_{2} ) = 0 - P \times \beta {\text{Max}}\left( {0,\quad - \frac{{\alpha (s_{2} - \gamma s_{1} )}}{{1 + \gamma s_{1} - s_{2} }}} \right) - \frac{{\alpha (s_{2} - \gamma s_{1} )}}{{1 + \gamma s_{1} - s_{2} }} \le 0$$

(44)

Thus, if Firm 2 chooses the same quality as s₁, it cannot obtain positive profit. Therefore, the case in which s₂ = s₁ will not occur as an equilibrium solution.

Appendix B: DETERMINING THE BASELINE VALUES OF α AND γ FOR NUMERICAL SIMULATIONS

In the numeric simulation exercises (Section 3), we need to choose the baseline values for α and γ. Chan, Lakonishok, and Sougiannis (2001) found that R&D expenditures of the US companies typically equal 30–80 percent of firm earnings. Higón and Manjón Antolín (2012) estimated that the ratio of R&D investment in revenue was 0.4019 for foreign multinationals in the UK. Hence, we assume that the baseline ratio of R&D investment in revenue is 0.4 and obtain:

$$\frac{1}{n}\frac{{\alpha s_{1} }}{{1 - s_{1} }} = 0.4 \times \pi_{1} ,\quad {\text{which}}\,{\text{derives}}\,\alpha = 0.4 \times \pi_{1} \times \left( {n\frac{{1 - s_{1} }}{{s_{1} }}} \right)$$

(45)

The maximum profit that Firm 1 can obtain is $\pi_{1}^{{\text{M}}} = s_{1} /4$ (Eq. 37). Thus, we get:

$$\alpha \le 0.4 \times \frac{{s_{1} }}{4} \times \left( {n\frac{{1 - s_{1} }}{{s_{1} }}} \right) = 0.1 \, n \times (1 - s_{1} ),\quad {\text{where}}\,s_{1} \in (0,1).$$

(46)

In Section 3, the baseline value of n is set at 2. In this case, α is bound within the interval (0, 0.2], but 0.2 becomes possible only if Firm 1 is a monopoly. In reality, what Firm 1 can obtain will be far less than its monopoly profit. Hence, a reasonable baseline value for α is 0.02.

The choice of the baseline value for γ needs more discussions. To start, we can compare Firm 2’s expected profit in two particular cases, namely, $s_{2} = \gamma s_{1}^{*} < s_{1}^{*}$ and $s_{2} = s_{1}^{*}$. The first case ($s_{2} = \gamma s_{1}^{*}$) occurs when Firm 2 does not invest in its own research, hence, C₂ = 0. Thus,

$$\left. {E(\Pi_{2} )} \right|_{{s_{2} = \gamma s_{1}^{*} }} = \Pi_{2}^{{\text{O}}} - P \times F = \Pi_{2}^{{\text{O}}} - P \times (1 - s_{1} + s_{2} )\beta \, \Pi_{2}^{{\text{O}}}$$

(47)

Therefore, if β ≤ 1, Firm 2 can obtain a positive profit. However, as discussed in Appendix A, if $s_{2} = s_{1}^{*}$, it will have no positive profit. Thus, as s₂ increases from $\gamma s_{1}^{*}$ to $s_{1}^{*}$, if Firm 2’s expected profit decreases monotonically, then $s_{2} = \gamma s_{1}^{*}$ will be the only Nash equilibrium solution if s₂<s₁. This implies that Firm 2 will never invest in R&D. In practice, however, it is not rare for technological laggards to invest in innovation ($\gamma s_{1}^{*} < s_{2} < s_{1}^{*}$). To mimic reality, we need to include $s_{2} \in (\gamma s_{1}^{*} ,s_{1}^{*} )$ as the Nash equilibrium solution.

A sufficient condition for $s_{2} \in (\gamma s_{1}^{*} ,s_{1}^{*} )$ as an equilibrium solution is that the first-order derivative of Firm 2’s profit with respect to s₂ is strictly greater than zero at the point $s_{2} = \gamma s_{1}^{*}$. If this condition is satisfied, Firm 2 will obtain larger profit when $s_{2} > \gamma s_{1}^{*}$ (quality is higher) than when $s_{2} = \gamma s_{1}^{*}$. The first-order derivative is:

$$\left. {\frac{{{\text{d}}E(\Pi_{2} )}}{{{\text{d}}s_{2} }}} \right|_{{s_{2} = \gamma s_{1}^{*} }} = \left[ {\frac{{1 - e^{ - \rho T} }}{\rho }\frac{4 - 7\gamma }{{(4 - \gamma )^{3} }} - \alpha } \right] \times \left[ {1 - P\beta (1 - s_{1}^{*} + \gamma s_{1}^{*} )} \right] + \left[ {\frac{{1 - e^{ - \rho T} }}{\rho }\frac{{P\beta s_{1}^{*3} (\gamma^{2} - \gamma )}}{{(4s_{1}^{*} - \gamma s_{1}^{*} )^{2} }}} \right].$$

(48)

It is not clear whether $\left[ {\frac{{1 - e^{ - \rho T} }}{\rho }\frac{4 - 7\gamma }{{(4 - \gamma )^{3} }} - \alpha } \right]$ is positive or negative, but it is easy to determine that $\left[ {1 - P\beta (1 - s_{1}^{*} + \gamma s_{1}^{*} )} \right]$ is positive when β ≤ 1 and $\left[ {\frac{{1 - e^{ - \rho T} }}{\rho }\frac{{P\beta s_{1}^{*3} (\gamma^{2} - \gamma )}}{{(4s_{1}^{*} - \gamma s_{1}^{*} )^{2} }}} \right]$ is negative. Therefore, if β ≤ 1, then the necessary (but not sufficient) condition for $\left. {\frac{{{\text{d}}E(\Pi_{2} )}}{{{\text{d}}s_{2} }}} \right|_{{s_{2} = \gamma s_{1}^{*} }} > 0$ is $4 - 7\gamma > 0$. In other words, $s_{2} \in (\gamma s_{1}^{*} ,s_{1}^{*} )$ can be an equilibrium solution, given β ≤ 1 and $\gamma < \frac{4}{7} \approx 0.57$. Thus, if γ is allowed to move through the interval [0, 0.6], then the numerical simulation can have Nash equilibrium solutions. Thus, γ = 0.3 is chosen as the baseline value. As the discussion of Figure 1 shows, in this case, equilibrium can be obtained when $s_{2}^{*} > \gamma s_{1}^{*}$.