## Abstract

This paper examines a foreign technology holder’s licensing choices between royalty and fixed-fee scheme. We emphasize that foreign licensor chooses the quality of licensed technology when the licensee country does not implement perfect intellectual property protection for licensor’s technology. We study quality choice as the foreign licensor’s selection for a particular grade of technical skills. We show that fixed fee emerges as the equilibrium licensing scheme when both the transfer of his technology is relatively efficient and the licensee is sufficiently cost competitive in the domestic market, and that royalty licensing prevails otherwise. We further show it need not hold the general belief that welfare in the licensor country unambiguously rise with a stronger patenting system in the licensee country when, in particular, such patenting system in place is sufficiently lax.

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## Notes

In this paper, the terms of “patenting” and “intellectual property protection” are used interchangeably.

This requires USTR to identify as the “priority watch list” foreign countries that deny adequate and effective protection of

*IPR*s, or fair and equitable market access for U.S. persons that rely on IP protection. Indeed, The USTR has requested and received submissions from U.S. industries suggesting that several nations be included on priority, priority watch, and watch lists. These submissions include many of the nations, which opposed the TRIPS negotiations and putting it into force, such as India and Brazil.In an interesting contribution, Colombo and Filippini (2016) study different licensing schemes and compare two-part tariff to both schemes of ad valorem royalty contracts and revenue-royalty contracts, where the price, rather than the quantity, is the basis of the licensing contract.

The authors thank an anonymous referee for suggesting a clarification on the linkage between quality choice and cost-reducing technology.

Put differently, a weak domestic patenting implies the licensee’s output ex post licensing is not observable and, thus, royalty licensing is not feasible under

*WP*.In the output stage, the cost of transferring technology is sunk and, thus, has no effect on the equilibrium fixed licensing fee.

Elsewhere, we explicit investigate the case of \(9\gamma -10<0\) and show that the licensor’s profit is convex in its choice of technology quality [see the discussion in Mukherjee and Tsai (2015, p. 68)].

For any \(\alpha \in (0,1]\) and \(\gamma >10/9\), it is easy to verify that \(s^{F}=\frac{2-10\alpha c}{9\gamma -10}<\alpha c\) holds for any \(\alpha c<\frac{1}{5}\) and \(\gamma >\frac{2}{9\alpha c}\), i.e., \(\frac{2}{9\gamma c}<\alpha <\frac{1}{5c}\) ; and that \(s^{F}=\alpha c\) holds for any \(\frac{10}{9}<\gamma <\frac{2}{9\alpha c}\) and \(\alpha c<\frac{1}{5}\), implying that \(\alpha c<\min \big \{\frac{2}{9\gamma },\frac{1}{5}\big \}=\frac{1}{5}\) and \(\gamma \le \frac{2}{9\alpha c}\).

The authors thank an anonymous referee for suggesting a graphical presentation with numerical example for better presentation of the results.

The licensee (firm 2) has no incentive to use the licensed technology if the ex post licensing contract is such that \(c-(s-r)>c\). It follows that the constraint \(r\le s\) must hold.

The equilibrium royalty of \(r^{*}=(5-4c+4s)/10\) is obtained from solving Eq. (7). For any \(s\in [0,c]\) and \(c<1/2\), it is easy to verify that \(r^{*}=(5-4c+4s)/10<s\) holds only if \(s>1/2\), which violates \(s\in [0,c]\) and \(c<1/2\). Hence, the royalty rate in equilibrium must bind and is, therefore, given by \(r^{*}=s\).

This is evident, from Eq. (10), that \(\partial \pi _1^R /\partial s \gg 0\) for any \(s\in [0,c]\) and \(c<1/2\) if \(\gamma \) approximates zero.

Colombo and Filippini (2016) distinguish ad valorem royalty from revenue royalty and argue that “revenue royalty” allows the patentee for extracting a quota of the licensee’s revenues.

The authors thank an anonymous referee for raising this issue.

This is evident from the unambiguously negative impact on licensor’s output of a lax domestic

*IPRs*system, i.e., \(\frac{\partial ^{2}y_1^F }{\partial \alpha ^{2}}=\frac{2}{9}\big (c-\frac{\partial s^{F}}{\partial \alpha }\big )^{2}>0\), and the ambiguous effect on the gains from licensing, \(\frac{\partial ^{2}F^{*}}{\partial \alpha ^{2}}=\frac{2}{9}\left[ {(-2)\left\{ {\big (-2c+\frac{-10c}{9\gamma -10}\big )\frac{9\gamma c}{9\gamma -10}-(-2c^{2})} \right\} } \right] \frac{>}{<}0\), which obviously rests upon the interplays between efficiency of technology transfer and post-licensing marginal cost of the licensee.The authors thank an anonymous referee for raising this issue.

See Levhari and Peles (1973) for a justification on this characterization.

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## Acknowledgements

Yingyi Tsai grateful acknowledges partial financial support from Ministry of Science and Technology (MOST), Taiwan under Research Grant No.: MOST 103-2923-H-390-001-MY2. The authors thank two anonymous referees for valuable comments and A. Hsu excellent research assistance in revising this paper. The usual disclaimer applies.

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## Appendices

### Appendix 1

Notice, under a strong patenting system in the licensee country, that fixed-fee licensing emerges as the equilibrium scheme if and only if \(\pi _1^F >\pi _1^R \). Using Eqs. (12a)–(12c) and (13a)–(13c), it is easy to verify, for any of the following conditions: (i) \(\gamma >\max \left\{ {\frac{2}{9c},\;\frac{1-2c}{3c}} \right\} =\frac{2}{9c}\) and \(c\in (1/6,\;1/5)\), (ii) \(\gamma >\max \left\{ {\frac{2}{9c},\;\frac{1-2c}{3c}} \right\} =\frac{1-2c}{3c}\) and \(c\in (0,\;1/6)\) [cf. Eqs. (12a) and (13a)], (iii) \(\gamma <\min \left\{ {\frac{2}{9c},\;\frac{1-2c}{3c}} \right\} =\frac{2}{9c}\) and \(c\in (0,\;1/6)\), and (iv) \(\gamma <\min \left\{ {\frac{2}{9c},\;\frac{1-2c}{3c}} \right\} =\frac{1-2c}{3c}\) and \(c\in (1/6,\;1/5)\) [cf. Eqs. (12b) and (13b)], that \(\pi _1^F>\pi _1^R \) if and only if \(Z>0\), where

To prove the result contained in Proposition 3, we sketch our proof by contradiction and proceed in three steps.

### Proof

First, given the *SPE* outcome of \(s^{F}\) and \(s^{R}\), we rewrite *Z* as

and we note that *Z* consists of three terms \(\Delta , \Omega \) and \(\Psi \), where \(\Omega \) is strictly positive and \(\Psi \) is nonnegative if \(s^{F}-s^{R}\ge 0\). Hence, Z is strictly positive provided that \(\Delta \) is non-negative. It is evident, for any \(s^{F}-s^{R}>0\), that \(\Delta \) is strictly positive if and only if

Rewriting Eq. (17), we have \(\frac{s^{R}}{4(1-c)s^{F}}+\frac{4(1-c)(s^{F}-s^{R})}{4(1-c)s^{F}}>\frac{4s^{F}(1-c-s^{F})}{4(1-c)s^{F}}\), i.e., \(4{(s^{F})^{2}}-[4(1-c)-1]s^{R}>0\). Given \(s^{R}\), it is evident that \(\Delta \) is strictly convex in \(s^{F}\). Hence, \(\Delta \) rises with an increase in \(s^{F}\) for any \(s^{F}\in [s^{{F}^{*}},\;c)\), where \(s^{{F}^{{*}}}\equiv \frac{\sqrt{(3-4c)s^{R}}}{2}\). Note the sign of \(s^{F}-s^{R}\) is decisive in identifying that of *Z*, we investigate further in the next two steps.

Second, we note that \(s^{F}-s^{R}<0\) occurs if \(s^{F}\equiv \frac{2(1-5c)}{9\gamma -10}<c\) for any \(\frac{10}{9}<\frac{2}{9c}<\gamma \) and \(c\in (1/6,\;1/5)\) and \(s^{R}=c\) for any \(\gamma \le \frac{1-2c}{3c}\) and \(c\in (1/6,\;1/5)\). The boundary conditions of \(\frac{2}{9c}<\gamma \) and \(\gamma \le \frac{1-2c}{3c}\), however, violates \(c\in (1/6,\;1/5)\). Hence, \(s^{F}-s^{R}<0\) does not hold. Next, we study whether or not \(s^{F}-s^{R}\ge 0\) can hold. It is evident \(s^{F}=c\) for any \(c\in (0,\;1/5)\) and \(\frac{10}{9}<\gamma \le \frac{2}{9c}\), and \(s^{R}=\frac{1-2c}{3\gamma }<c\) for any \(\gamma >\frac{1-2c}{3c}\) and \(c\in (0,\;1/6)\). Again, the boundary conditions of \(\frac{2}{9c}<\gamma \) and \(\gamma >\frac{1-2c}{3c}\) fails to meet the condition of \(c\in (0,\;1/6)\).

Third, we then examine whether or not \(s^{F}-s^{R}\ge 0\) given that \(s^{F}=c\) for any \(c\in (0,\;1/5)\) and \(\frac{10}{9}<\gamma \le \frac{2}{9c}\), and \(s^{R}=c\) for any \(\gamma \le \frac{1-2c}{3c}\) and \(c\in (1/6,\;1/5)\). It is easy to verify that \(s^{F}-s^{R}\ge 0\) if and only if \(\frac{10}{9}<\gamma<\frac{1-2c}{3c}<\frac{2}{9c}\) and \(c\in (1/6,\;3/16)\).

Hence, we have established that \(\pi _1^F >\pi _1^R \) if and only if \(\frac{10}{9}<\gamma<\frac{1-2c}{3c}<\frac{2}{9c}\) and \(c\in (1/6,\;3/16)\).

### Appendix 2

Welfare in the licensor’s country is now simplified to \(W_N^F =\pi _1^F\). Using Eqs. (5) and (6), we differentiate \(\pi _1^F\) with respect to \(\alpha \) and obtain that

where

Using the results contained in Propositions 1 and 2, we have

Hence, we have established that

where \(\omega =A\alpha +B, A=[(9\gamma +4)(9\gamma -10)-50(9\gamma -1)]c\) and \(B=[(9\gamma -2)(9\gamma -10)+10(9\gamma -1)]\).

It follows that \(\frac{d\pi _1^F }{d\alpha }\gg 0\) for any \(\frac{10}{9\gamma }<\alpha \le \frac{2}{9\gamma c}\) since \(\alpha <\frac{4}{(8+9\gamma )c}\) trivially satisfies the boundary condition, and that, for any \(\frac{2}{9\gamma c}<\alpha<\frac{1}{5c}, \frac{d\pi _1^F}{d\alpha }<0\) if \(\alpha \le \tilde{\alpha }\) and otherwise if \(\alpha >\tilde{\alpha }\), where \(\tilde{\alpha }=\frac{(9\gamma -2)(9\gamma -10)+10(9\gamma -1)}{[(9\gamma +4)(9\gamma -10)-50(9\gamma -1)]c}\) and \(\tilde{\alpha }\in \big (\frac{1}{5c},\;1\big )\).

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Tsai, Y., Mukherjee, A. Domestic patenting systems and foreign licensing choices.
*J Econ* **121**, 173–191 (2017). https://doi.org/10.1007/s00712-017-0523-y

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DOI: https://doi.org/10.1007/s00712-017-0523-y