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Revenue royalties

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Abstract

Under the assumption of decreasing returns to scale, we compare several licensing mechanisms—per-unit royalty, an ad valorem royalty, and a revenue-royalty, and combinations with fixed fees—for an insider patentee. In the case of a non-drastic innovation, the patentee maximizes its profits by offering, respectively, an ad valorem royalty, a revenue-royalty and a two-part per-unit royalty, if the cost function is scarcely or highly convex, moderately-low convex, and moderately-high convex. In the case of a drastic innovation, the patentee always offers an ad valorem royalty contract.

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Notes

  1. The existing literature has somewhat adopted a confused terminology when describing these licensing methods. Indeed, the term “ad valorem licensing” has been used to describe indifferently both the case where the patentee extracts a quota of the licensee’s profits and the case where the patentee extracts a quota of the licensee’s revenues. Sometimes, a royalty imposed on the licensee’s profits has also been defined “profit-sharing” licensing (Niu 2013, 2014). Note that when the production costs after the innovation are zero, ad valorem royalty contracts and revenue-royalty contracts cannot be distinguished. The scarce existing literature on price-based licensing contracts (San Martín and Saracho 2010, 2015; Colombo and Filippini 2015) assumes that after the innovation the production costs cancel out (for an exception, see Heywood et al. (2014)). At the best of our knowledge, no theoretical analysis has been provided yet where ad valorem royalties and revenue-royalties are clearly distinguished and compared.

  2. Ad valorem licensing by an insider innovator has been also considered by Mukhopadhyay et al. (1999), that compare ad valorem licensing with fixed fee licensing and find that ad valorem licensing is preferred to fixed fee. Niu (2013) compares per-unit and ad valorem licensing and finds that the two mechanisms are equivalent if the licensing contracts are restricted by the government in order to prevent anti-competitive effects of licensing. When the innovator is an outsider, revenue-royalties have been analysed by Bousquet et al. (1998), Hernández-Murillo and Llobet (2006) and Llobet and Padilla (2014).

  3. As observed by Anand and Khanna (2000) and Amir et al. (2014), empirical evidence in patent licensing is quite scarce, the reason probably being that the firms are reluctant to disclose the terms of their patent licensing agreements.

  4. The Economist, Nov. 1st 2014.

  5. The existence of royalties based on the licensee’s profits is documented also in the past [see Burhop and Lubbers (2012), that document both types of licensing contracts in the Imperial Germany].

  6. http://www.nrel.gov/technologytransfer/pdfs/license_agreement_template-exclusive.pdf.

  7. http://research.ucdavis.edu/pgc/d/tt/Exclusive%20Patent%20License%20%28Template%29%20%28pdf%29.

  8. “The licensee commits to pay for each unit of product a royalty equal to [...]% of the price charged to customers, minus rebates and taxes imposed on sales, or the licensee commits to pay [...] euros for each unit of product it sells” (http://www.confindustria.venezia.it/nordestimpresa%5CGuideSchede.nsf/weblink/296c6?OpenDocument).

  9. The distinction between drastic and non-drastic innovation is popular in patent licensing literature (see Wang (1998), for an example). Non-drastic innovation means that, even in the absence of a license, the patentee cannot act as a monopolist. The innovation is non-drastic when the difference between the marginal costs of a firm that owns the innovation and the marginal costs of a firm that does not own the innovation is sufficiently low.

  10. See Sen and Stamatopoulos (2009, 2013) and Mukherjee (2014).

  11. Therefore, the innovation reduces the linear component of the total costs function, but it does not cancel out the costs. It follows that ad valorem royalties and revenue-royalties remain clearly distinguished from each other.

  12. We do not allow for negative fees. Indeed, as shown by Katz and Shapiro (1985), in this case there would be the possibility for the patentee to “bribe firm 2 to exit from the market ... and would likely to hold to be illegal by antitrust authorities”. We briefly discuss the case of a negative fixed fee at the end of this section (see later footnote 25).

  13. The following equilibrium quantities are valid when \(d\ne 1\). The case \(d=1\) is discussed in Sect. 3.4.

  14. Note that when \(k=0\) Proposition 1 corresponds to Proposition 4.1 in Faulí-Oller and Sandonìs (2002).

  15. The argument runs as follows. If the innovation is drastic, the profits of Firm 2 are zero without license. Therefore, solving the incentive compatibility constraint of Firm 2 yields \(d=1\). That is, the patentee extracts all the licensee’s profits, which obtains zero profits whatever is the quantity it sets. Therefore, if the patentee sets the monopoly quantity \(q_1 =\frac{1}{4+k}\), one of the best-response of the licensee is \(q_2 =\frac{1}{4+k}\) (and the best-response of the patentee to \(q_2 =\frac{1}{4+k}\) is \(q_1 =\frac{1}{4+k})\). As a consequence, the patentee gets the profits of a multiplant monopolist.

  16. Indeed, we have a fourth-degree equation in \(h{^{*}} \) which is not easy to solve analytically.

  17. As for the case of ad valorem royalties, a higher c, by reducing the disagreement profits of the licensee, allows the patentee to set a higher revenue royalty.

  18. We thank an anonymous referee for this explanation.

  19. We use the Mathematica software. The file with the numerical simulations is available from the authors.

  20. Note that when \(k=0\), we are back to the situation described in San Martìn and Saracho (2010): in this case, the ad valorem royalty and the revenue-royalty licensing coincide, and they are preferred to the two-part per-unit royalty contract.

  21. The thresholds \(\underline{k}(c)\) and \(\bar{k}(c)\) refer to the second preferred option for the patentee. Therefore, the complete comparison between the equilibrium profits under the different licensing contracts can be summarized as follows. When \(k\le k_1 (c)\), we have: \(\Pi _1^V {^{*}} \ge \Pi _1^R {^{*}} \ge \Pi _1^U {^{*}} \ge \Pi _1^{NL} {{^{*}}} \); when \(k\in [k_1 (c),\underline{k}(c)]\), we have: \(\Pi _1^R {^{*}} \ge \Pi _1^V {^{*}} \ge \Pi _1^U {^{*}} \ge \Pi _1^{NL} {{^{*}}} \); when \(k\in [\underline{k}(c),k_2 (c)]\) we have \(\Pi _1^R {^{*}} \ge \Pi _1^U {^{*}} \ge \Pi _1^V {^{*}} \ge \Pi _1^{NL} {{^{*}}} \); when \(k\in [k_2 (c),\bar{k}(c)]\) we have \(\Pi _1^U {^{*}} \ge \Pi _1^R {^{*}} \ge \Pi _1^V {^{*}} \ge \Pi _1^{NL} {{^{*}}} \); when \(k\in [\bar{k}(c),k_3 (c)]\), we have \(\Pi _1^U {^{*}} \ge \Pi _1^V {^{*}} \ge \Pi _1^R {^{*}} \ge \Pi _1^{NL} {{^{*}}} \), and when \(k\ge k_3 (c)\), we have \(\Pi _1^V {^{*}} \ge \Pi _1^U {^{*}} \ge \Pi _1^R {^{*}} \ge \Pi _1^{NL} {{^{*}}} \).

  22. Note that when k is very high, the difference between the profits under the different licensing scheme becomes negligible.

  23. Note that the higher is k, the lower is \(q_1^v \), as \({\partial b_1^V }/{\partial k}<0\).

  24. We also numerically check the impact of the innovation size, c, over the parameter space supporting the various cases. We find that \(k_1 \), \(k_2 \) and \(k_3 \) decrease with c. Therefore, the higher is c, the more likely is that the ad valorem royalty contract is the preferred licensing scheme for the patentee.

  25. In the case of negative fixed fees, the optimal per-unit royalty is always given by \(r{^{*}} =r^{\wedge } \). Therefore, the profits of the patentee under the per-unit royalty are given by (7). By numerically comparing the patentee’s profits under the different licensing schemes, it may be observed that for low levels of k, the preferred option of the patentee is now the two-part per-unit-royalties instead of the ad valorem royalties (details are available upon request). For the other values of k, we have the same results as in the case of non-negative fixed fees. That is, if k is intermediate-low, the patentee chooses the revenue-royalty contract, if k is intermediate-high, the patentee chooses the two-part per-unit contract, and if k is high, the patentee chooses the ad valorem royalty contract. When the fee is not constrained to be non-negative, the patentee has a higher degree of freedom when choosing the optimal two-part per-unit contract. It follows that the parameter space where the two-part per-unit royalty licensing scheme is the preferred option of the patentee must now be larger.

  26. It should be noted that \(\tilde{k}_1 \) may also be higher than \(\tilde{k}_2 \), thus implying that the two-part per-unit licensing in the sub-case \(r^{\wedge } \ge c\) is never the best option in terms of consumer surplus.

  27. This is contrast with the common wisdom that profits sharing agreements are detrimental for consumers, and therefore they raise antitrust concerns (Shapiro and Willig 1990). Indeed, we show that when the returns to scale are sufficiently decreasing, ad valorem licensing yields a higher consumer surplus than the other licensing mechanisms.

  28. Indeed, when \(r\ge c\) the incentive-compatibility constraint of the licensee is not satisfied.

  29. The complete comparison between the equilibrium profits is the following. When \(k\le \tilde{k}_1 (c)\), we have: \(\Pi _1^V {^{*}} \ge \Pi _1^R {^{*}} \ge \Pi _1^{PU} {^{*}} \ge \Pi _1^{NL} {{^{*}}} \); when \(k\in [\tilde{k}_1 (c),\underline{\tilde{k}}(c)]\), we have: \(\Pi _1^R {^{*}} \ge \Pi _1^V {^{*}} \ge \Pi _1^{PU} {^{*}} \ge \Pi _1^{NL} {{^{*}}} \); when \(k\in [\underline{\tilde{k}}(c),\bar{\tilde{k}}(c)]\) we have \(\Pi _1^R {^{*}} \ge \Pi _1^{PU} {^{*}} \ge \Pi _1^V {^{*}} \ge \Pi _1^{NL} {{^{*}}} \); when \(k\in [\bar{\tilde{k}}(c),\tilde{k}_2 (c)]\) we have \(\Pi _1^R {^{*}} \ge \Pi _1^V {^{*}} \ge \Pi _1^{PU} {^{*}} \ge \Pi _1^{NL} {{^{*}}} \); and when \(k\ge \tilde{k}_2 (c)\), we have \(\Pi _1^V {^{*}} \ge \Pi _1^R {^{*}} \ge \Pi _1^U {^{*}} \ge \Pi _1^{NL} {{^{*}}} \).

  30. It is simply to show that a licensing contract based on fixed fee only is never the preferred option, as the profits in this case are equal to \(\Pi _1^F {^{*}}=\frac{(2+k)[1+4c-4c^2+k^2(1+2c-c^2)+2k(1+3c-2c^2)]}{2(k^2+4k+3)^2}\).

  31. The expression of \(\tilde{d}{^{*}} \) is available on request.

  32. See “Appendix 1”.

  33. It can be shown that this three-part contract reduces to a two-part contract, as the optimal per-unit royalty is equal to zero.

  34. Non drastic innovation in this case requires that \(\gamma \le \frac{2+k}{2(k^2+6k+9)}\).

  35. We do not report the derivation of the optimal contracts under this specification of the cost function, as they can be derived following the same procedure adopted in the main text. The complete analysis is available from the authors upon request.

  36. Note that the patentee’s profits in the case of no-licensing and in the case of revenue-royalty licensing do not depend on the innovation size, \(\gamma \) (Tables 5, 6).

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Acknowledgments

We are grateful to Vincenzo Denicolò, Yair Tauman, Debapriya Sen, Salvatore Piccolo, Giorgios Stamatopoulos, Arijit Mukherjee, Farok Contractor for comments received on a previous version of this paper. Ana Saracho, Marta San Martìn, John Heywood and Jos Jansen are greatly acknowledged for suggestions about the current version. We also thank the participants to ASSET Meeting 2014, Aix-en-Provence, and MaCCI Conference 2015, Mannehim, for valuable comments, and in particular David Perez-Castrillo, Florian Schuett and Clara Graziano. We thank the Editor Giacomo Corneo and two anonymous reviewers for useful comments. Usual disclaimers apply.

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  1. Università Cattolica del Sacro Cuore, Milano, Largo A. Gemelli 1, Milano, I-20123, Italy

    Stefano Colombo & Luigi Filippini

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  1. Stefano Colombo
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Correspondence to Luigi Filippini.

Appendices

Appendix 1

In Tables 1, 2, 3 and 4, k is selected following the switching points of the licensing contract choice. The complete numerical simulations are performed through Mathematica and are available from the authors. Moreover, c has been chosen in such a way to avoid a shift from drastic innovation to non-drastic innovation. Indeed, if \(c>0.5\), non drastic innovation would emerge only if k is sufficiently high.

Table 1 Numerical example with a two-part per-unit royalty contract
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Table 2 Numerical example with a pure per-unit royalty contract
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Table 3 Numerical example with two-part ad valorem and revenue royalty contracts
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Table 4 Numerical example with three-part contracts
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Appendix 2

In this appendix we discuss an alternative cost function and we show that even in this case the degree of convexity has an impact on the optimal choice of the patentee. In particular, we consider a situation where the cost function, in the absence of the innovation, is the following: \(\hat{C}_i (c,k,q_i )=\gamma +k\frac{q_i ^2}{2}\), while in the case of innovation it is the following: \(\hat{C}_i (0,k,q_i )=k\frac{q_i ^2}{2}\), with \(i=1,2\). Therefore, under this specification, the innovation reduces the fixed component of the total costs, rather than the variable component. We consider only non-drastic innovation, and we compare the two-part and the pure per-unit royalty contract, the revenue-royalty contract, the ad-valorem contract and the no-licensing case.Footnote 34 We can state the following propositions.Footnote 35 \(^{, }\) Footnote 36

Proposition 9

Suppose that the innovation is non-drastic. If k is low, the patentee chooses the revenue-royalty contract, while if k is high, the patentee chooses the two-part per-unit royalty contract.

Table 5 Numerical example with a two-part per-unit contract and alternative cost function
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Table 6 Numerical example with a pure per unit contract and alternative cost function
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On the other hand, if the fixed fee is not implementable, we have:

Proposition 10

Suppose that the innovation is non-drastic and that the fixed fee is not implementable. If k is low, the patentee chooses the revenue-royalty contract, whereas if k is high, the patentee chooses the ad valorem royalty contract.

Tables 5 and 6 provide numerical examples of Propositions 9 and 10, respectively.

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Colombo, S., Filippini, L. Revenue royalties. J Econ 118, 47–76 (2016). https://doi.org/10.1007/s00712-015-0459-z

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