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Consumers’ choices, infringements and market competition

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Abstract

The study fits the individuals’ characteristics of consumption into an analysis of the demands for both genuine goods and counterfeit goods. The consumers’ substitutability of genuine goods for counterfeits and attitudes toward a dispersed consumption of counterfeit varieties are the dimensions that affect the niche markets for counterfeits. We show that it is not necessary to increase the competition among counterfeits to reduce the demand for individual counterfeits if at the margin the variety of counterfeits enhances the value of consuming the good. The enforcements against counterfeiting deter the number of counterfeit firms, but encourage the output of individual counterfeits if the market includes a significant number of counterfeiters. The optimal private enforcement against counterfeiting is also fully discussed in the model.

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Notes

  1. In China, people refer to products like this as “shanzhai”, which in Chinese means a bandit stronghold in the mountain. These products infringe the property rights of branded firms in relation to both software and hardware. For instance, several shanzhai mobile phones, e.g., Hiphone, Ophone, or Uphone instead of iphone or NCKIA rather than NOKIA, as well as PC/NB, Netbook, GPS, monitor and TFT-LCD TV, etc., give rise to trademark infringements with confusing similarity. The phones also duplicate some of the functions that are patented by brand firms such as Apple, Nokia and Samsung.

  2. For example, as the 3C hardware industries become more mature, shanzhai 3C products manufactured in China have become more popular in the markets in China, India, Pakistan and Russia in the last 4 years. The mayor of Shenzhen city estimates that there were more than 0.2 billion shanzhai mobile phones produced for nearly 1 million families in 2008 and nearly ninety thousand types of mobile phones in 2009 in Guangdong province alone. We can expect that shanzhai 3C products could constitute one of the business innovation models in the developing countries that must be able to significantly affect the development of the global IT industry.

  3. OECD (2007) announced that the international trade in counterfeit and pirated products could have reached $200 billion in 2005. The Business Software Alliance (BSA) and International Data Corporation (IDC) estimated the total value of software piracy to be $53 billion in 2008 while customers paid $88 billion for the use of copyrighted software in the same year.

  4. Chaudhry (2006) argued that one of the main reasons why China has become the primary source of counterfeit goods is the lack of intellectual property rights (IPR) enforcement. If China is willing to enforce intellectual property rights, the infringements of the right should decrease. For example, the Chinese government fiercely protected its Olympic logo, and punished the selling of fake Olympic items much more severely than of other counterfeits. China has thus successfully protected its property rights in relation to its Olympic logo.

  5. For example, more than $16 billion worth of counterfeit goods are sold each year inside China. The US trade representative based in China reckons that the amount that American industries lose to counterfeit goods ranges from $200 to $250 billion (The Economist 2003/5/17). The IT software firms could lose nearly $100 billion per year according to KPMG and the Alliance for Gray Market and Counterfeit Abatement (Bednarz 2006). The International Chamber of Commerce in Geneva estimated that the global value of counterfeit products may exceed $650 billion per year (Kurtenbach 2006). OECD (2007) also announced that the international trade in counterfeit and pirated products could have been as high as $200 billion in 2005. It is not hard to imagine how huge the total value of products whose property rights have been violated must be.

  6. For example, Higgins and Rubin (1986) provide data indicating that 27% of the buyers of the counterfeit goods already owned genuine Rolexes.

  7. Counterfeit goods have become increasingly rampant over the years: from small scale copying of luxury goods in the 1980s, to large-scale production of low-tech products in the late 1980s and 1990s, and to the much larger scale production of high-tech electrical products (Chaudhry 2006). In 2004, the European Union experienced rapid growth of counterfeits with data on seizures revealing a dramatic percentage increase in counterfeits in the sectors for electrical equipment (+707%) and computer equipment (+899%) (EU Taxation and Customs Union 2005). Counterfeit goods account for 12% of products being sold in the toys and sports marketplace in the UK (Jones and Cline, 2002).

  8. Counterfeit products can be classified into four distinct types: (1) True counterfeit products that look as much like the original as possible and use the same brand name. (2) Look-alikes that duplicate the original and bear a different name, but not a private label of a branded industrial product. (3) Reproductions that are not exact copies. (4) Unconvincing imitations.

  9. Ordover (1991) and Maskus and McDaniel (1999) document the evidence that the rapid postwar industrialization in East Asian countries such as Japan and South Korea was accomplished under relatively weak IPR system and that a hasty imposition of a strong IPR regime could slow down the industrial development of today’s developing countries. Maskus(2000) also mentions these issues. Chin and Grossman (1990), Deardorff (1992), Helpman (1993), Lai (1998), Helpman and Lai (2005) and Branstetter et al. (2007) theoretically address the question of whether a country with limited capacity to innovate will benefit from extending IPRs to foreign inventors.

  10. Nordhaus (1969) argued that the optimal policy render the marginal dynamic benefit equal to the marginal static efficiency loss. Strengthening intellectual property right provides greater incentives for innovations and thus the benefit that come from having more and better products.

    However, McCalman (2001) and Chaudhuri et al. (2006) stress that a strong IPR also incurs static welfare losses.

  11. Given that the individual counterfeit producer is symmetric, the demand for counterfeit varieties is \( Q = (\alpha_{C} \beta_{B} - \alpha_{B} \gamma_{B} )/\delta - (\beta_{B} /\delta ) \cdot p + (\gamma_{B} /\delta ) \cdot p_{B} , \)where \( \delta \equiv \beta_{B} (\beta_{C} /n + \gamma_{C} ) - (\gamma_{B} )^{2} . \) Since δ > 0, it needs \( (\alpha_{C} \beta_{B} - \alpha_{B} \gamma_{B} ) > 0 \) to guarantee that the demand for the counterfeit varieties is not negative when p and p B are equal to zero.

  12. As for the general setting of the cost function, the expenditure on detecting the infringement of the intellectual property rights is convex in the level of the probability of each counterfeiter i being caught.

  13. In the long run, the equilibrium number of the counterfeit varieties is derived from the zero profit condition for the production of individual counterfeit that makes the output and price of the individual counterfeit be the constants q le and p le. The output of the genuine good affects the market size of the counterfeit varieties, and hence the firm with the genuine good takes this into consideration to maximize its profit. So, the first order condition of profit-maximization for the production of the genuine good includes the effect of the output of the genuine good on the number of the counterfeit varieties. However, in the short run, the output of the genuine good does not affect the number of the counterfeit varies, but does affect the output and price of individual counterfeit. So, equating the short run outputs q se and the long run outputs q le of individual counterfeit and solving for n should not arrive at the long run equilibrium number of the counterfeit varieties \( n_{R}^{\text{le}} \). This can be seen from the equalization \( q^{\text{se}} = q^{\text{le}} \) that is specified as

    \( \left( {{\frac{{2\beta_{C} (2\alpha_{C} \beta_{B} - \gamma_{B} \alpha_{B} ) + n\{ \gamma_{C} (\alpha_{C} \beta_{B} - \gamma_{B} \alpha_{B} ) + \alpha_{C} (\gamma_{C} \beta_{B} - \gamma_{B} \gamma_{B} )\} }}{{(2\beta_{C} + n\gamma_{C} )[4\beta_{C} \beta_{B} + 2n(\beta_{B} \gamma_{C} - \gamma_{B} \gamma_{B} )]}}}} \right) = \sqrt {{{\rho (E)F} \mathord{\left/ {\vphantom {{\rho (E)F} {(1 - \rho (E))\beta_{C} }}} \right. \kern-\nulldelimiterspace} {(1 - \rho (E))\beta_{C} }}} \). The solution for n in this equation is not equal to the long run equilibrium number of the counterfeit varieties \( n_{R}^{\text{le}} \).

  14. \( p_{B}^{e} = \left( {{\frac{{\alpha_{B} }}{2}} + {\frac{{\gamma_{B} (2\sqrt {{{\beta_{c} \rho (E)F} \mathord{\left/ {\vphantom {{\beta_{c} \rho (E)F} {(1 - \rho (E))}}} \right. \kern-\nulldelimiterspace} {(1 - \rho (E))}}} - \alpha_{C} )}}{{2\gamma_{C} }}}} \right)\;\left( {1 + {\frac{{\gamma_{B} \alpha_{C}^{\prime } ( \cdot ){{{\text{d}}n_{R} } \mathord{\left/ {\vphantom {{{\text{d}}n_{R} } {{\text{d}}q_{B} }}} \right. \kern-\nulldelimiterspace} {{\text{d}}q_{B} }}}}{{2\left( {2(\beta_{B} \gamma_{C} - \gamma_{B} \gamma_{B} ) + \gamma_{B} \alpha_{C}^{\prime } ( \cdot ){{{\text{d}}n_{R} } \mathord{\left/ {\vphantom {{{\text{d}}n_{R} } {{\text{d}}q_{B} }}} \right. \kern-\nulldelimiterspace} {{\text{d}}q_{B} }}} \right)}}}} \right) \)

  15. \( \rho^{''} (E) = {\frac{\psi (1 - \psi )}{{\left( {1 - \psi (1 - G(E)} \right)^{2} }}}G^{''} (E) - 2{\frac{{\psi^{2} (1 - \psi )}}{{\left( {1 - \psi (1 - G(E)} \right)^{3} }}}(G^{'} (E))^{2} \)

  16. \( R_{G}^{nc\prime \prime } (E) = \left( {{\frac{{2\gamma_{B} \sqrt {{{\beta_{c} \psi \cdot F} \mathord{\left/ {\vphantom {{\beta_{c} \psi \cdot F} {(1 - \psi )}}} \right. \kern-\nulldelimiterspace} {(1 - \psi )}}} \mu^{2} \left( {1 - G(E)} \right)}}{{4\gamma_{C} \left( {\beta_{B} \gamma_{C} - \gamma_{B} \gamma_{B} } \right)\sqrt {G(E)} }}}} \right)\; \times \left( {{\frac{{ - \left( {\alpha_{c} \gamma_{B} } \right)\left( {{\frac{1 + G(E)}{2}}} \right) - 2\gamma_{B} \sqrt {{{\beta_{c} \psi \cdot F} \mathord{\left/ {\vphantom {{\beta_{c} \psi \cdot F} {(1 - \psi )}}} \right. \kern-\nulldelimiterspace} {(1 - \psi )}}} G(E)^{3/2} }}{G(E)}}} \right) \)

  17. The marginal revenue and compensation with respect to private enforcement are

    \( R_{G}^{{C^{\prime } }} (E) = {\frac{{\gamma_{B} \left( {\alpha_{B} \gamma_{C} - \gamma_{B} \left( {\alpha_{C} - p^{ge} } \right)} \right)}}{{4\gamma_{C} \left( {\beta_{B} \gamma_{C} - \gamma_{B} \gamma_{B} } \right)}}}2\;{\frac{{\partial p^{ge} }}{\partial E}}, \) \( CPS^{\prime } (E) = {\frac{{2\left( {2\beta_{B} \gamma_{C} - \gamma_{B} \gamma_{B} } \right)\alpha_{C} - 2\alpha_{B} \gamma_{B} \gamma_{C} - \left( {4\beta_{B} \gamma_{C} - 3\gamma_{B} \gamma_{B} } \right)2p^{\text{ge}} }}{{\left( {4(\beta_{B} \gamma_{C} - \gamma_{B} \gamma_{B} )} \right)\gamma_{C} }}}\;{\frac{{\partial p^{ge} }}{\partial E}}. \)

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Authors and Affiliations

  1. Department of Economics, National Chi-Nan University, Nan-Tau, Taiwan

    Fav Tsoin Lai

  2. Department of Management Administration, National Taiwan University of Science and Technology, Taipei, Taiwan

    Shun-Chiao Chang

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  1. Fav Tsoin Lai
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  2. Shun-Chiao Chang
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Correspondence to Fav Tsoin Lai or Shun-Chiao Chang.

Appendix

Appendix

1.1 Illustration of remark 2

Similarly, if \( \alpha_{C}^{\prime } ( \cdot ) > 0, \) the equilibrium outputs are

$$ q_{B}^{le} = {\frac{{\alpha_{B} \gamma_{C} - \gamma_{B} \left( {\alpha_{C} - 2\sqrt {{{\beta_{C} \rho (E)F} \mathord{\left/ {\vphantom {{\beta_{C} \rho (E)F} {(1 - \rho (E))}}} \right. \kern-\nulldelimiterspace} {(1 - \rho (E))}}} } \right)}}{{\left( {2(\beta_{B} \gamma_{C} - \gamma_{B} \gamma_{B} ) + \gamma_{B} \alpha_{C}^{\prime } ( \cdot ){{dn_{R} } \mathord{\left/ {\vphantom {{dn_{R} } {dq_{B} }}} \right. \kern-\nulldelimiterspace} {dq_{B} }}} \right)}}} $$
$$ \begin{gathered} Q_{R} (q_{B}^{le} ) = {\frac{{\alpha_{C} }}{{2\gamma_{C} }}} + {\frac{{\alpha_{C} (\beta_{B} + \frac{1}{2}\gamma_{B} \alpha_{C}^{'} ( \cdot ){{dn_{R} } \mathord{\left/ {\vphantom {{dn_{R} } {dq_{B} }}} \right. \kern-\nulldelimiterspace} {dq_{B} }}) - \gamma_{B} \alpha_{B} }}{{\left( {2(\beta_{B} \gamma_{C} - \gamma_{B} \gamma_{B} ) + \gamma_{B} \alpha_{C}^{\prime } ( \cdot ){{dn_{R} } \mathord{\left/ {\vphantom {{dn_{R} } {dq_{B} }}} \right. \kern-\nulldelimiterspace} {dq_{B} }}} \right)}}} \\ \, - {\frac{{(2\beta_{B} \gamma_{C} - \gamma_{B} \gamma_{B} + \gamma_{B} \alpha_{C}^{\prime } ( \cdot ){{dn_{R} } \mathord{\left/ {\vphantom {{dn_{R} } {dq_{B} }}} \right. \kern-\nulldelimiterspace} {dq_{B} }}) \cdot 2\sqrt {{{\beta_{c} \rho (E)F} \mathord{\left/ {\vphantom {{\beta_{c} \rho (E)F} {(1 - \rho (E))}}} \right. \kern-\nulldelimiterspace} {(1 - \rho (E))}}} }}{{\gamma_{C} \left( {2(\beta_{B} \gamma_{C} - \gamma_{B} \gamma_{B} ) + \gamma_{B} \alpha_{C}^{\prime } ( \cdot ){{dn_{R} } \mathord{\left/ {\vphantom {{dn_{R} } {dq_{B} }}} \right. \kern-\nulldelimiterspace} {dq_{B} }}} \right)}}} \\ \end{gathered} $$

the equilibrium number of the counterfeit varieties is

$$ n_{R}^{le} = {\frac{{\left( {2\beta_{B} \gamma_{C} - \gamma_{B} \gamma_{B} + \gamma_{B} \alpha_{C}^{\prime } ( \cdot ){{dn_{R} } \mathord{\left/ {\vphantom {{dn_{R} } {dq_{B} }}} \right. \kern-\nulldelimiterspace} {dq_{B} }}} \right)\left( {\alpha_{c} - 2\sqrt {{{\beta_{c} \rho (E)F} \mathord{\left/ {\vphantom {{\beta_{c} \rho (E)F} {(1 - \rho (E))}}} \right. \kern-\nulldelimiterspace} {(1 - \rho (E))}}} } \right) - \alpha_{B} \gamma_{B} \gamma_{C} }}{{\left( {2(\beta_{B} \gamma_{C} - \gamma_{B} \gamma_{B} ) + \gamma_{B} \alpha_{C}^{\prime } ( \cdot ){{dn_{R} } \mathord{\left/ {\vphantom {{dn_{R} } {dq_{B} }}} \right. \kern-\nulldelimiterspace} {dq_{B} }}} \right)\gamma_{C} \sqrt {{{\rho (E)F} \mathord{\left/ {\vphantom {{\rho (E)F} {(1 - \rho (E))\beta_{C} }}} \right. \kern-\nulldelimiterspace} {(1 - \rho (E))\beta_{C} }}} }}} $$

and the equilibrium price of the genuine good is

$$ p_{B}^{le} = \left( {{\frac{{\alpha_{B} }}{2}} + {\frac{{\gamma_{B} (2\sqrt {{{\beta_{c} \rho (E)F} \mathord{\left/ {\vphantom {{\beta_{c} \rho (E)F} {(1 - \rho (E))}}} \right. \kern-\nulldelimiterspace} {(1 - \rho (E))}}} - \alpha_{C} )}}{{2\gamma_{C} }}}} \right)\;\left( {1 + {\frac{{\gamma_{B} \alpha_{C}^{\prime } ( \cdot ){{dn_{R} } \mathord{\left/ {\vphantom {{dn_{R} } {dq_{B} }}} \right. \kern-\nulldelimiterspace} {dq_{B} }}}}{{2\left( {2(\beta_{B} \gamma_{C} - \gamma_{B} \gamma_{B} ) + \gamma_{B} \alpha_{C}^{\prime } ( \cdot ){{dn_{R} } \mathord{\left/ {\vphantom {{dn_{R} } {dq_{B} }}} \right. \kern-\nulldelimiterspace} {dq_{B} }}} \right)}}}} \right) $$

where \( {{dn_{R} } \mathord{\left/ {\vphantom {{dn_{R} } {dq_{B} }}} \right. \kern-\nulldelimiterspace} {dq_{B} }} = - {{\gamma_{B} } \mathord{\left/ {\vphantom {{\gamma_{B} } {(\gamma_{C} \sqrt {{{\rho (E)F} \mathord{\left/ {\vphantom {{\rho (E)F} {(1 - \rho (E))\beta_{C} }}} \right. \kern-\nulldelimiterspace} {(1 - \rho (E))\beta_{C} }}} - \alpha_{C}^{\prime } ( \cdot ))}}} \right. \kern-\nulldelimiterspace} {(\gamma_{C} \sqrt {{{\rho (E)F} \mathord{\left/ {\vphantom {{\rho (E)F} {(1 - \rho (E))\beta_{C} }}} \right. \kern-\nulldelimiterspace} {(1 - \rho (E))\beta_{C} }}} - \alpha_{C}^{\prime } ( \cdot ))}}. \) If \( \alpha_{C}^{\prime } ( \cdot ) \) is close to zero in equilibrium, the equilibrium total output of counterfeits is approaching to

$$ Q_{R}^{le} = {\frac{{\left( {2\beta_{B} \gamma_{C} - \gamma_{B} \gamma_{B} } \right)\left( {\alpha_{C} - 2p^{e} } \right) - \alpha_{B} \gamma_{B} \gamma_{C} }}{{\gamma_{C} \cdot \left( {2(\beta_{B} \gamma_{C} - \gamma_{B} \gamma_{B} )} \right)}}}, $$

and the equilibrium price is very close to

$$ p_{B}^{le} = \left( {{\frac{{\alpha_{B} }}{2}} + {\frac{{\gamma_{B} (2\sqrt {{{\beta_{c} \rho (E)F} \mathord{\left/ {\vphantom {{\beta_{c} \rho (E)F} {(1 - \rho (E))}}} \right. \kern-\nulldelimiterspace} {(1 - \rho (E))}}} - \alpha_{C} )}}{{2\gamma_{C} }}}} \right). $$

1.2 Illustration of Table 1

$$ \begin{gathered} R_{G}^{nc\prime } (E) = \hfill \\ {\frac{{\gamma_{B} \left( {\alpha_{B} \gamma_{C} - \gamma_{B} \left( {\alpha_{c} - 2\sqrt {{{\beta_{c} \psi \cdot F} \mathord{\left/ {\vphantom {{\beta_{c} \psi \cdot F} {(1 - \psi )}}} \right. \kern-\nulldelimiterspace} {(1 - \psi )}}} \sqrt {G(E)} } \right)} \right)2\sqrt {{{\beta_{c} \psi \cdot F} \mathord{\left/ {\vphantom {{\beta_{c} \psi \cdot F} {(1 - \psi )}}} \right. \kern-\nulldelimiterspace} {(1 - \psi )}}} \left( {\mu (1 - G(E))} \right)}}{{4\gamma_{C} \left( {\beta_{B} \gamma_{C} - \gamma_{B} \gamma_{B} } \right)\sqrt {G(E)} }}} \hfill \\ \end{gathered} $$

To simplify the notations, we define the following value functions:

$$ \begin{gathered} \Upomega \left( {\sqrt {G(E)} } \right) \equiv \gamma_{B} \left( {\alpha_{B} \gamma_{C} - \gamma_{B} \left( {\alpha_{c} - 2\sqrt {{{\beta_{c} \psi \cdot F} \mathord{\left/ {\vphantom {{\beta_{c} \psi \cdot F} {(1 - \psi )}}} \right. \kern-\nulldelimiterspace} {(1 - \psi )}}} \sqrt {G(E)} } \right)} \right) \\ \times 2\sqrt {{{\beta_{c} \psi \cdot F} \mathord{\left/ {\vphantom {{\beta_{c} \psi \cdot F} {(1 - \psi )}}} \right. \kern-\nulldelimiterspace} {(1 - \psi )}}} \left( {\mu (1 - G(E))} \right) \\ \end{gathered} $$
$$ \Uptheta (\sqrt {G(E)} ) \equiv 4\gamma_{C} \left( {\beta_{B} \gamma_{C} - \gamma_{B} \gamma_{B} } \right)\sqrt {G(E)} . $$

From the first-order condition, we can derive the static comparisons.

$$ {\frac{{\partial E^{nc} }}{\partial F}} = - {\frac{{\Upomega \left( {\sqrt {G(E)} } \right) + 4(\gamma_{B} )^{2} \left( {{{\beta_{C} \psi \cdot F} \mathord{\left/ {\vphantom {{\beta_{C} \psi \cdot F} {(1 - \psi )}}} \right. \kern-\nulldelimiterspace} {(1 - \psi )}}} \right)\sqrt {G(E)} \left( {\mu (1 - G(E))} \right)}}{{2R_{G}^{nc\prime \prime } (E)\Uptheta (\sqrt {G(E)} )F}}} > 0 $$
$$ {\frac{{\partial E^{nc} }}{{\partial \beta_{C} }}} = - {\frac{{\Upomega \left( {\sqrt {G(E)} } \right) + 4(\gamma_{B} )^{2} \left( {{{\beta_{C} \psi \cdot F} \mathord{\left/ {\vphantom {{\beta_{C} \psi \cdot F} {(1 - \psi )}}} \right. \kern-\nulldelimiterspace} {(1 - \psi )}}} \right)\sqrt {G(E)} \left( {\mu (1 - G(E))} \right)}}{{2R_{G}^{nc\prime \prime } (E)\Uptheta (\sqrt {G(E)} )\beta_{C} }}} > 0 $$
$$ {\frac{{\partial E^{nc} }}{{\partial \alpha_{B} }}} = - {\frac{{\gamma_{B} \gamma_{C} 2\sqrt {{{\beta_{c} \psi \cdot F} \mathord{\left/ {\vphantom {{\beta_{c} \psi \cdot F} {(1 - \psi )}}} \right. \kern-\nulldelimiterspace} {(1 - \psi )}}} \left( {\mu (1 - G(E))} \right)}}{{R_{G}^{nc\prime \prime } (E)\Uptheta (\sqrt {G(E)} }}} > 0. $$
$$ {\frac{{\partial E^{nc} }}{{\partial \alpha_{C} }}} = {\frac{{\left( {\gamma_{B} } \right)^{2} 2\sqrt {{{\beta_{C} \psi \cdot F} \mathord{\left/ {\vphantom {{\beta_{C} \psi \cdot F} {(1 - \psi )}}} \right. \kern-\nulldelimiterspace} {(1 - \psi )}}} \left( {\mu (1 - G(E))} \right)}}{{R_{G}^{nc\prime \prime } (E)\Uptheta (\sqrt {G(E)} )}}} < 0 $$
$$ {\frac{{\partial E^{nc} }}{{\partial \beta_{B} }}} = {\frac{{R_{G}^{nc\prime } (E)4\gamma_{C}^{2} \sqrt {G(E)} }}{{R_{G}^{nc\prime \prime } (E)\Uptheta \left( {\sqrt {G(E)} } \right)}}} < 0 $$
$$ \begin{gathered} {\frac{{\partial E^{nc} }}{{\partial \gamma_{B} }}} = {\frac{{ - \Upomega \left( {\sqrt {G(E)} } \right)\left( {{{2\Uptheta \left( {\sqrt {G(E)} } \right)} \mathord{\left/ {\vphantom {{2\Uptheta \left( {\sqrt {G(E)} } \right)} {\gamma_{B} }}} \right. \kern-\nulldelimiterspace} {\gamma_{B} }} + 8\gamma_{B} \gamma_{C} \sqrt {G(E)} } \right)}}{{R_{G}^{\prime \prime } (E)\left( {\Uptheta \left( {\sqrt {G(E)} } \right)} \right)^{2} }}} \\ {\frac{{ + \alpha_{B} \gamma_{C} 2\sqrt {{{\beta_{c} \psi \cdot F} \mathord{\left/ {\vphantom {{\beta_{c} \psi \cdot F} {(1 - \psi )}}} \right. \kern-\nulldelimiterspace} {(1 - \psi )}}} \left( {\mu (1 - G(E))} \right)}}{{R_{G}^{\prime \prime } (E)\left( {\Uptheta \left( {\sqrt {G(E)} } \right)} \right)^{2} }}} \\ \end{gathered} $$
$$ \begin{gathered} {\frac{{\partial E^{nc} }}{{\partial \gamma_{C} }}} = {\frac{{4\gamma_{C} \beta_{B} \Upomega \left( {\sqrt {G(E)} } \right)\sqrt {G(E)} }}{{R_{G}^{\prime \prime } (E)\left( {\Uptheta \left( {\sqrt {G(E)} } \right)} \right)^{2} }}} \hfill \\ - {\frac{{\Uptheta \left( {\sqrt {G(E)} } \right)({{\gamma_{B} )^{2} } \mathord{\left/ {\vphantom {{\gamma_{B} )^{2} } {\gamma_{C} }}} \right. \kern-\nulldelimiterspace} {\gamma_{C} }}\left( {\alpha_{c} - 2\sqrt {{{\beta_{c} \psi \cdot F} \mathord{\left/ {\vphantom {{\beta_{c} \psi \cdot F} {(1 - \psi )}}} \right. \kern-\nulldelimiterspace} {(1 - \psi )}}} \sqrt {G(E)} } \right)2\sqrt {{{\beta_{c} \psi \cdot F} \mathord{\left/ {\vphantom {{\beta_{c} \psi \cdot F} {(1 - \psi )}}} \right. \kern-\nulldelimiterspace} {(1 - \psi )}}} \left( {\mu (1 - G(E))} \right)}}{{R_{G}^{\prime \prime } (E)\left( {\Uptheta \left( {\sqrt {G(E)} } \right)} \right)^{2} }}} \hfill \\ \end{gathered} $$

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Lai, F.T., Chang, SC. Consumers’ choices, infringements and market competition. Eur J Law Econ 34, 77–103 (2012). https://doi.org/10.1007/s10657-011-9225-z

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