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International licensing under an endogenous tariff in vertically-related markets

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Abstract

We construct a trading framework involving vertically-related markets to examine the foreign licensor’s optimal licensing contract, the optimal tariff, and the welfare difference between licensing and no technology transfer, in which a foreign vertically-integrated firm has a cost-reducing technology for the downstream product competing against host upstream and downstream firms in host markets. We obtain the following interesting results. First, international licensing lowers the welfare of the host country under non-drastic innovation, while the reverse occurs under drastic innovation. Second, the foreign licensor will choose royalty licensing with an optimal royalty rate higher than the innovation size if the innovation size is small, while selecting mixed licensing otherwise. Third, the optimal tariff rises, is followed by a vertical jump, and then falls (the optimal royalty rate increases, is followed by a vertical drop, and then continues to increase), when the innovation size becomes larger under non-drastic innovation.

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Notes

  1. See Vishwasrao (2007, p. 741).

  2. See Gentile (2020), and Smith and Anti (2022).

  3. Refer to Kabiraj and Marjit (2003, p. 114).

  4. In addition, Sinha (2010) and Chen et al. (2016) also discuss the international licensing issue. Sinha (2010) constructs a game where the foreign licensor’s licensing contract is chosen prior to the determination of entry mode, i.e., exports or FDI, given an exogenous tariff so that international licensing can strategically affect the foreign firm’s subsequent entry mode. Chen et al. (2016) find that given that the tariff is high, it is optimal for the foreign firm to adopt an inferior technology for its production when it licenses its most advanced technology to the host firm. Such licensing may improve the welfare of the two countries.

  5. The reader may refer to http://iknow.stpi.narl.org.tw/Post/Read.aspx?PostID=13731 (in Chinese). We find from this website that Haier America Trading, L.L.C. filed a lawsuit in 2017 against the patent union organized by Samsung Electronics Co. Ltd., five other corporations, and the Trustees of Columbia University for charging an improperly high royalty to the patent composition, ATSC tuner, on Digital TV. The royalty rate was US$5 for each unit, which was much higher than the US$0.67 charged to European and US$1.0 to Japanese licensees. This led to unfair competition in the digital TV market of the U.S.

  6. Refer to the websites: https://equalocean.com/news/2020101714964and https://aikahao.xcar.com.cn/item/421135.html.

  7. Refer to the website: https://www.aisin.com/en/investors/stock/.

  8. Refer to the website: https://en.wikipedia.org/wiki/Shengrui_Transmission.

  9. The foreign vertically-integrated firm may try to foreclose the host downstream firm by strategically buying the upstream product from the host upstream firm such that there are two potential buyers in the upstream market.

  10. Lin et al. (2022) and Tsao et al. (2023) also obtain the result that licensing may lower the welfare level. However, the factors leading the welfare to decline in their papers are different from ours. Lin et al. (2022) set up a successive oligopoly model with two complementary inputs. The reason why they can derive this result is that the downstream firm may worsen the production efficiency by setting an input price higher than the marginal cost under a Nash bargaining agreement with a two-part tariff through strategically licensing its core input technology to an external licensee to extract a part of the profit from the complementary input supplier. Tsao et al. (2023) construct a successive monopoly model, and show that the two-tier technology licensing in vertically related markets may lower the welfare level due to a change in the input price.

  11. There is a strand of the literature studying the vertically-integrated firm’s behavior of strategic selling in the upstream market, which contains the vertically-integrated firm strategically reducing its sales in the upstream market or foreclosing the upstream market in order to enhance its competitiveness in the downstream market. Related studies include Greenhut and Ohta (1979), Salop and Scheman (1983, 1987), Salinger (1988), Schrader and Martin (1998), and Higgens (1999). In addition, Spencer and Jones (1992), and Wang et al. (2011) analyze the trade policies of the host country under vertically-related markets.

  12. In general, the substitutability between the products of the licensor and licensee plays a key role in determining the licensing means. It has been shown in the related literature that the more (less) substitutability there is, the more likely it is that the licensor will choose royalty (fixed-fee) licensing. The related literature includes Kamien and Tauman (1986, 2002), Wang (1998, 2002), and Wang and Yang (1999, 2004).

  13. This assumption is reasonable. Hwang et al. (2017, pp. 1597–1598) indicate that it can be frequently observed in the real world that the downstream tariff is much higher than the upstream tariff. The main results in the paper remain valid if the upstream tariff is sufficiently low relative to the downstream tariff.

  14. Indeed, given the existence of an independent upstream firm in the host country, the foreign vertically-integrated firm may foreclose the host downstream firm by strategically purchasing the upstream product. Intuitively, the greater the number of the independent upstream firms or the higher the marginal cost of the vertically-integrated firm, the less likely it is that the vertically-integrated firm will engage in such a market foreclosure. Please see Schrader and Martin (1998) and Higgens (1999) for the discussions on the market foreclosure. In this paper, we exclude the possibility of the market foreclosure by Assumption 1. In Assumption 1, we assume that the tariff is sufficiently high to ensure that \({q}_{D}^{N}>0.\) Assumption 1 is the non-drastic innovation condition in licensing, while it is the condition for the market foreclosure to be unable to occur from the viewpoint of the independent downstream firm.

  15. By Assumptions 1 and 2, we can calculate \(\overline{t}-\underline{t}=\frac{32}{33}(1-{c}_{U}-{c}_{D})>0.\)

  16. Two-part tariff licensing is a more general means of licensing. Under a two-part tariff, the optimal licensing contract can be royalty, mixed, or fixed-fee licensing.

  17. The second-order condition for \({x}_{U}^{L}\) equals (-3), for \({x}_{I}^{L}\) it is (-5/2), and the stability condition equals 6.

  18. Please refer to Wang (1998) and Kamien and Tauman (2002).

  19. The antitrust law requires that both the royalty rate and fixed fee be non-negative.

  20. The second-order condition equals (-53/96).

  21. We obtain from Assumption 2 that \(t<\overline{t}\equiv \frac{7}{11}\left(1-{c}_{U}\right)+\frac{4}{11}{c}_{D}=\left(1-{c}_{U}\right)-\frac{4}{11}\left(1-{c}_{U}-{c}_{D}\right).\) As \(\left(1-{c}_{U}-{c}_{D}\right)> 0\), we can derive the relationship that \(\overline{t}<\left(1-{c}_{U}\right).\) Thus, \({r}^{*}\left(\overline{t }\right)=\frac{1}{2}\left(1-{c}_{U}-\frac{49}{53}\overline{t }\right)>0\).

  22. The take-it-or-leave-it licensing contract is an extreme case of a Nash bargaining licensing contract, in which the bargaining power of the foreign licensor is \(\beta =1\) and the fixed fee equals \(F={\pi }_{D}^{OL}-{\pi }_{D}^{N}\). The optimal royalty rate is not affected by the bargaining power. The bargaining power only affects the fixed fee. This is the characteristic of the Nash bargaining game, in which the bargaining power does not influence the achievement of the cooperation, but only affects the distribution of the cooperative profits between the two firms. Thus, the results derived in the paper remain unchanged in the Nash bargaining licensing contract.

  23. By solving \(\widetilde{t}\) and \(\underline{t}\), we obtain the upper bound of \({c}_{D}\) at point D for the royalty licensing as \({c}_{D}={c}_{0}\equiv \frac{1}{4}(1-{c}_{U}).\)

  24. By solving \(\widetilde{t}\) and \(\overline{t}\), we obtain the lower bound of \({c}_{D}\) at point B for the mixed licensing as \({c}_{D}={c}_{1}\equiv \frac{15}{158}(1-{c}_{U}).\) It is obvious that \({c}_{1}<{c}_{0}\).

  25. When the two upstream firms engage in a Bertrand competition, the firms will undercut each other until the equilibrium input price is equal to the marginal cost of the upstream product, \({c}_{U},\) with the upstream products being homogeneous in the paper. In this case, there exists no distortion in the upstream market such that the marginal cost of the foreign vertically-integrated firm is \({c}_{U}+t\) while that of the host downstream firm is \({c}_{U}+{c}_{D}.\) Since the foreign licensor is unable to strategically increase its upstream exports to lower the input price to exploit the host upstream firm such that the strategic royalty revenue effect vanishes, the results will be similar to those derived in Kabiraj and Kabiraj (2017), i.e., the host government will impose a high tariff to induce the foreign licensor to choose fixed-fee licensing.

  26. Please refer to Wang (1998) and Kamien and Tauman (2002).

  27. Filippini (2005) introduces a Stackelberg model, in which the licensor serves as a leader while the licensee is a follower. Since the licensor is the leader, it can strategically decrease its output in the prior stage so that the licensee can produce a larger output in the latter stage than that under no licensing to mitigate the competition in the market and to increase the industry profit. Then the licensor can extract the licensee’s entire extra profit by charging a royalty rate higher than the innovation size, as the industry profit rises post licensing.

  28. Note that the host downstream firm’s profit remains unchanged after licensing, i.e., \(\pi_{D}^{L} = \pi_{D}^{N}\), and the consumer surplus equals \(\frac{1}{2}\left( {Q^{L} } \right)^{2}\).

  29. \(\frac{d{q}_{I}^{L}}{dt}=\frac{\partial {q}_{I}^{L}}{\partial t}+\left(\frac{\partial {q}_{I}^{L}}{\partial r}\right)\left(\frac{\partial r}{\partial t}\right)=-\frac{79}{106}<0.\)

  30. \(\frac{d\left({\pi }_{U}^{L}+{\pi }_{D}^{L}\right)}{dt}=\frac{\partial \left({\pi }_{U}^{L}+{\pi }_{D}^{N}\right)}{\partial t}+\left(\frac{\partial \left({\pi }_{U}^{L}+{\pi }_{D}^{N}\right)}{\partial r}\right)\left(\frac{\partial r}{\partial t}\right)=\frac{24}{53}{x}_{U}^{L}+\frac{3}{4}{q}_{D}^{N}>0.\)

  31. \(\frac{d{Q}^{L}}{dt}=\frac{\partial {Q}^{L}}{\partial t}+\left(\frac{\partial {Q}^{L}}{\partial r}\right)\left(\frac{\partial r}{\partial t}\right)=\left(-\frac{5}{16}\right)+\left(-\frac{1}{8}\right)\left(-\frac{49}{106}\right)=-\frac{27}{106}<0.\)

  32. It should be noted that \({c}_{1}<{c}_{2}<{c}_{0}.\)

  33. It should be noted that \({c}_{3}>{c}_{0}>{c}_{2}.\)

  34. It should be noted that \({c}_{2}<{c}_{0}<{c}_{4}<{c}_{3}\).

  35. It should be noted that \({c}_{2}<{c}_{5}<{c}_{3}\).

  36. The calculation procedures are available from the authors upon request.

  37. The calculation procedures are available from the authors upon request.

  38. The calculation procedures are available from the authors upon request.

  39. The second-order condition for \({x}_{U}^{N}\) equals (-3), for \({x}_{I}^{N}\) is (-5/2), and the stability condition equals 6.

  40. The second-order condition equals (-761/768).

  41. We can solve for an optimal tariff \({t}^{*Lr}\) confined by 0 \(<{t}^{*Lr}<\overline{t}\) for \({c}_{D}\le {c}_{1},\) and confined by 0 \(<{t}^{*Lr}<\widetilde{t}\) for \({c}_{1}<{c}_{D}<{c}_{6}\). However, the optimal tariff is higher than the restriction for royalty licensing, i.e., \({t}^{*Lr}\ge \widetilde{t},\) for \({c}_{6}\le {c}_{D}\le {c}_{0}.\) Thus, the optimal tariff has to become \(\widetilde{t}\) in this interval. Please note that \({c}_{1}<{c}_{2}<{c}_{6}<{c}_{0}\).

  42. The second-order condition is (-96727/89888).

  43. It should be noted that \({c}_{1}<{c}_{7}<{c}_{2}<{c}_{6}<{c}_{3}\).

References

  • Chang RY, Hwang H, Peng CH (2013) Technology licensing, R&D and welfare. Econ Lett 118(2):396–399

    Article  Google Scholar 

  • Chen HL, Hwang H, Mukherjee A, Shih PC (2016) Tariff, technology licensing and adoption. Int Rev Econ Financ 43:234–240

    Article  Google Scholar 

  • Filippini L (2005) Licensing contract in a stackelberg model. Manch Sch 73:582–598

    Article  Google Scholar 

  • Gentile E (2020) Intellectual property rights and foreign technology licensing in developing countries: an empirical investigation. Econ Dev Cult Change 68(2):655–698

    Article  Google Scholar 

  • Greenhut ML, Ohta H (1979) Vertical integration of successive oligopolists. American Economic Review 69(1):137–141

    Google Scholar 

  • Gu Y, Wenzel T (2009) A note on the excess entry theorem in spatial models with elastic demand. Int J Ind Organ 27:567–571

    Article  Google Scholar 

  • Higgens RS (1999) Competitive vertical foreclosure. Manag Decis Econ 20:229–237

    Article  Google Scholar 

  • Hwang H, Mai CC, Wu SJ (2017) Tariff escalation and vertical market structure. The World Economy 40:1597–1613

    Article  Google Scholar 

  • Kabiraj A, Kabiraj T (2017) Tariff induced licensing contracts, consumer surplus and welfare. Econ Model 60:439–447

    Article  Google Scholar 

  • Kabiraj T, Marjit S (2003) Protecting consumers through protection: the role of tariff-induced technology transfer. Eur Econ Rev 47:113–124

    Article  Google Scholar 

  • Kamien MI, Tauman Y (1986) Fee versus royalties and the private value of a patent. Quart J Econ 101:471–491

    Article  Google Scholar 

  • Kamien MI, Tauman Y (2002) Patent licensing: the inside story. Manch Sch 70(1):7–15

    Article  Google Scholar 

  • Lin YJ, Lin YS, Shih PC (2022) Welfare reducing vertical licensing in the presence of complementary inputs. J Econ 137:121–143

    Article  Google Scholar 

  • Mukherjee A, Pennings E (2006) Tariff, licensing and market structure. Eur Econ Rev 50:1699–1707

    Article  Google Scholar 

  • Nagaoka S (2005) Determinants of high-royalty contracts and the impact of stronger protection of intellectual property rights in Japan. J Japanese Int Econ 19:233–254

    Article  Google Scholar 

  • Poddar S, Sinha UB (2010) Patent licensing from a high-cost firm to a low-cost firm. Econ Rec 86:384–395

    Article  Google Scholar 

  • Salinger MA (1988) Vertical mergers and market foreclosure. Quart J Econ 103:345–356

    Article  Google Scholar 

  • Salop SC, Scheffman DT (1987) Cost-raising strategies. The Journal of Industrial Economics, pp 19–34

  • Salop SC, Scheffman DT (1983) Raising Rivals’ cost. Am Econ Rev 73:267–271

    Google Scholar 

  • Schrader A, Martin S (1998) Vertical market participation. Rev Ind Organ 13:321–331

    Article  Google Scholar 

  • Sinha UB (2010) Strategic licensing, export, FDI, and host country welfare. Oxf Econ Pap 62:114–131

    Article  Google Scholar 

  • Smith PJ, Anti SJ (2022) How does TRIPs compliance affect the economic growth of developing countries? application of the synthetic control method. World Economy, on Line,. https://doi.org/10.1111/twec.13282

    Article  Google Scholar 

  • Spencer BJ, Jones RW (1992) Trade and protection in vertically related markets. J Int Econ 32:31–55

    Article  Google Scholar 

  • Tsao KC, JL Hu, H Hwang, and YS Lin (2023) More licensed technologies may make it worse: a welfare analysis of licensing vertically two-tier foreign technologies. Journal of Economics, forthcoming

  • Vishwasrao S (2007) Royalties vs. fees: how do firms pay for foreign technology? Int J Ind Organ 25:741–759

    Article  Google Scholar 

  • Wang H (1998) Fee versus Royalty Licensing in a Cournot duopoly model. Econ Lett 60:55–62

    Article  Google Scholar 

  • Wang XH (2002) Fee versus royalty licensing in a differentiated cournot duopoly. J Econ Bus 54(2):253–266

    Article  Google Scholar 

  • Wang H, Yang BZ (1999) On licensing under bertrand competition. Aust Econ Pap 38:106–119

    Article  Google Scholar 

  • Wang XH, Yang B (2004) On technology transfer to an asymmetric cournot duopoly. Economics Bulletin 4(14):1–6

    Google Scholar 

  • Wang KCA, Koo HW, Chen TJ (2011) Domestic trade protection in vertically related markets. Econ Model 28:1595–1603

    Article  Google Scholar 

  • Wang KCA, Liang WJ, Chou PS (2013) Patent licensing under cost asymmetry among firms. Econ Model 31:297–307

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Economics, National Dong Hwa University, Shoufeng, Hualien, 97401, Taiwan, ROC

    Dang-Long Bui & Wen-Jung Liang

  2. Center for General Education, Chang Gung University, Taoyuan, Taiwan, ROC

    Kuang-Cheng Andy Wang

  3. Department of International Business, Ming Chuan University, Taipei, Taiwan, ROC

    Yi-Jie Wang

  4. Department of Business and Management, Ming Chi University of Technology, New Taipei City, Taiwan, ROC

    Kuang-Cheng Andy Wang

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Correspondence to Wen-Jung Liang.

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We are indebted to the editor and anonymous referees for inducing us to improve our exposition and for offering several suggestions leading to improvements in the substance of the paper. Valuable comments from the participants of NTU Trade Workshop are acknowledged. The fourth author would like to thank for the financial support from National Science and Technology Council, Taiwan (MOST 108-2410-H-259-005). The usual disclaimer applies.

Appendix

Appendix

1.1 A. The equilibria under no licensing with an exogenous tariff

The equilibria under no licensing with an exogenous tariff can be solved by backward induction, beginning with the final stage. In stage 3, by substituting (1) into (2) and differentiating \({\pi }_{i}^{N}\)(i = D, I) with respect to \({q}_{D}^{N}\) and \({q}_{I}^{N},\) respectively, and letting them equal zero, we can obtain:

$$q_{I}^{N} = \frac{1}{3}\left\{ {1 - 2c_{U} + MC_{D}^{N} - 2t} \right\},{ }q_{D}^{N} = \frac{1}{3}\left\{ {1 + c_{U} - 2MC_{D}^{N} + t} \right\},$$
(A.1.1)
$${w}^{N}={-c}_{D}+\frac{1}{2}\left\{1{+c}_{U}+t-3{q}_{D}^{N}\right\}, {q}_{D}^{N}={x}_{U}^{N}+{x}_{I}^{N}.$$
(A.1.2)

where the superscript “N” denotes variables in the no licensing case.

In stage 2, by using the derived demand in (A.1.2), i.e., \({q}_{D}^{N}={x}_{U}^{N}+{x}_{I}^{N}\), and differentiating \({\pi }_{i}^{N}\)(i = I, U) with respect to \({x}_{I}^{N}\) and \({x}_{U}^{N}\), respectively, we obtain the following first-order conditionsFootnote 39:

$$\frac{d{\pi }_{I}^{N}}{d{x}_{I}^{N}}=\frac{d{\pi }_{I}^{UN}}{d{x}_{I}^{N}}+\frac{d{\pi }_{I}^{DN}}{d{x}_{I}^{N}}=\left({w}^{N}-{c}_{U}-\frac{3}{2}{x}_{I}^{N}\right)-{ q}_{I}^{N}=0,$$
(A.2.1)
$$\frac{d{\pi }_{U}^{N}}{d{x}_{U}^{N}}=\left(\frac{\partial {\pi }_{U}^{N}}{\partial {x}_{U}^{N}}\right)+\left(\frac{\partial {\pi }_{U}^{N}}{\partial {w}^{N}}\right)\left(\frac{d{w}^{N}}{d{x}_{U}^{N}}\right)={w}^{N}-{c}_{U}-\frac{3}{2}{x}_{U}^{N}=0.$$
(A.2.2)

Substituting (A.1.2) into (A.2) gives:

$${x}_{I}^{N}=-\frac{1}{12}+\frac{1}{12}{c}_{U}-\frac{1}{3}{c}_{D}+\frac{5}{12}t, \mathrm{and }{x}_{U}^{N}=\frac{5}{24}-\frac{5}{24}{c}_{U}-\frac{1}{6}{c}_{D}-\frac{1}{24}t.$$
(A.3.1)

According to (2), (A.1), (A.2), (A.3.1), and the profit-maximizing conditions, we obtain:

$${q}_{D}^{N}=\frac{1}{8}-\frac{1}{8}{c}_{U}-\frac{1}{2}{c}_{D}+\frac{3}{8}t, {\mathrm{and} q}_{I}^{N}=\frac{7}{16}-\frac{7}{16}{c}_{U}+\frac{1}{4}{c}_{D}-\frac{11}{16}t,$$
(A.3.2)
$${w}^{N}=\frac{5}{16}+\frac{11}{16}{c}_{U}-\frac{1}{4}{c}_{D}-\frac{1}{16}t,$$
(A.3.3)
$${\pi }_{U}^{N}=\frac{3}{2}{\left({x}_{U}^{N}\right)}^{2}=\frac{3}{2}{\left(\frac{5}{24}-\frac{5}{24}{c}_{U}-\frac{1}{6}{c}_{D}-\frac{1}{24}t\right)}^{2},$$
(A.3.4)
$${\pi }_{D}^{N}={\left({q}_{D}^{N}\right)}^{2}={\left(\frac{1}{8}-\frac{1}{8}{c}_{U}-\frac{1}{2}{c}_{D}+\frac{3}{8}t\right)}^{2}.$$
(A.3.5)

1.2 B. The optimal tariff

By substituting \({r}^{*}=2{c}_{D}\) into (11), we can solve for the optimal tariff under royalty licensing as followsFootnote 40:

$${t}^{*Lr}=\frac{11}{761}(23-23{c}_{U}+4{c}_{D}),$$
(B.1)

where the superscript “Lr” denotes variables associated with the case under royalty licensing.

By taking into account the non-drastic innovation and the non-prohibitive tariff restrictions, we observe from Fig. 1 that the optimal tariff under royalty licensing is confined by 0 \(<{t}^{*Lr}<\overline{t}\) for \({c}_{D}\le {c}_{1},\) and by 0 \(<{t}^{*Lr}\le \widetilde{t}\) for \({c}_{1}<{c}_{D}\le {c}_{0}.\) By manipulating, we obtain the optimal tariff under royalty licensing as followsFootnote 41:

$$\left\{\begin{array}{c}{t}^{*Lr}=\frac{11}{761}\left(23-23{c}_{U}+4{c}_{D}\right) for {c}_{D}<{c}_{6}\equiv \frac{291}{1703}(1-{c}_{U}), \\ {t}^{*Lr}=\widetilde{t}\equiv \frac{53}{49}\left(1-{c}_{U}-4{c}_{D}\right) for {c}_{6}\le {c}_{D}\le {c}_{0}.\end{array}\right.$$
(B.2)

Next, by substituting (9) into (11), we can solve for the optimal tariff under mixed licensing as followsFootnote 42:\({t}^{*Lm}=\frac{41923}{96727}-\frac{41923}{96727}{c}_{U}-\frac{33708}{96727}{c}_{D}\), (B.3)where the superscript “Lm” denotes variables associated with the case under mixed licensing.

Similarly, by taking into account the non-drastic innovation and the non-prohibitive tariff restrictions, we observe from Fig. 2 that the optimal tariff under mixed licensing is confined by \(\widetilde{t}<{t}^{*Lm}<\overline{t}\) for \({c}_{1}<{c}_{D}\le {c}_{0},\) and by \(\underline{t}<{t}^{*Lm}<\overline{t}\) for \({c}_{0}<{ c}_{D}<(1-{c}_{U}).\) By manipulating, we obtain the optimal tariff under mixed licensing as followsFootnote 43:

$$\left\{\begin{array}{c}{t}^{*Lm}=\widetilde{t}\equiv \frac{53}{49}\left(1-{c}_{U}-4{c}_{D}\right)\mathrm{ for} {c}_{1}<{c}_{D}\le {c}_{7}\equiv \frac{3623}{22234}\left(1-{c}_{U}\right), \\ {t}^{*Lm}=\frac{41923}{96727}-\frac{41923}{96727}{c}_{U}-\frac{33708}{96727}{c}_{D} \mathrm{for }{c}_{7}<{c}_{D}<{c}_{3}, \\ \begin{array}{c} \\ {t}^{*Lm}\le \underline{t}\equiv -\frac{1}{3}\left(1-{c}_{U}\right)+\frac{4}{3}{c}_{D} for {c}_{3}\le { c}_{D}<\left(1-{c}_{U}\right). \end{array}\end{array}\right.$$
(B.4)

Recall that the non-drastic innovation requires that \(t>\underline{t}\). It follows that the case where \({t}^{*Lm}\le \underline{t}\) in (B.4) can never occur. Note that both royalty and mixed licensing contracts can occur, when the innovation size lies in \({c}_{1}<{c}_{D}\le {c}_{0}\). In what follows, we compare the welfare between these two licensing contracts to determine the optimal tariff.

First, provided that \({c}_{1}<{c}_{D}\le {c}_{7}\), we have proved that the optimal tariff under royalty licensing is \({t}^{*Lr}\) while that under mixed licensing is \(\widetilde{t}\). We also learn that \({SW}^{Lr}\left(t=\widetilde{t}\right)={SW}^{Lm}\left(t=\widetilde{t}\right).\) It follows that \({SW}^{Lr}\left(t={t}^{*Lr}\right)>{SW}^{Lr}\left(t=\widetilde{t}\right)={SW}^{Lm}\left(t=\widetilde{t}\right).\) As a result, the host government will select a tariff \({t}^{*}={t}^{*Lr}\) such that the foreign licensor will choose royalty licensing.

Second, given \({c}_{7}<{c}_{D}<{c}_{6},\) we have shown that the optimal tariff under royalty licensing is \({t}^{*}={t}^{*Lr},\) while that under mixed licensing is \({t}^{*}={t}^{*Lm}.\) Accordingly, we can obtain that \({SW}^{Lm}\left(t={t}^{*Lm}\right)-{SW}^{Lr}\left(t={t}^{*Lr}\right)\le 0,\) if \({c}_{7}<{c}_{D}\le {c}_{2}.\) Thus, the host government will set a tariff \({t}^{*Lr}\) such that the foreign licensor will choose royalty licensing. Moreover, we obtain \({SW}^{Lm}\left(t={t}^{*Lm}\right)-{SW}^{Lr}\left(t={t}^{*Lr}\right)\ge 0,\) if \({c}_{2}\le {c}_{D}<{c}_{6}\). The government will set a tariff \({t}^{*Lm}\) such that the foreign licensor will choose mixed licensing. Accordingly, the optimal tariff will jump from \({t}^{*Lr}\) to \({t}^{*Lm}\) at \({c}_{2}\), because the optimal licensing switches from royalty to mixed licensing.

Third, for \({c}_{6}\le {c}_{D}\le {c}_{0}\), we have obtained that the optimal tariff under royalty licensing is \(\widetilde{t}\) under royalty licensing, while that under mixed licensing is \({t}^{*Lm}.\) Since \({SW}^{Lr}\left(t=\widetilde{t}\right)={SW}^{Lm}\left(t=\widetilde{t}\right),\) and \({SW}^{Lm}\left(t={t}^{*Lm}\right)>{SW}^{Lm}\left(t=\widetilde{t}\right),\) we can derive \({SW}^{Lm}\left(t={t}^{*Lm}\right)>{SW}^{Lr}\left(t=\widetilde{t}\right)\). Thus, the host government will choose a tariff \({t}^{*Lm}\) such that the foreign licensor will choose mixed licensing.

1.3 C. The welfare difference

1.3.1 C.1. The calculations of the four effects in (15) under royalty licensing

By substituting \({t}^{*Lr}\) and \({r}^{*}=2{c}_{D}\) into (6.1) and (6.2) and substituting \({t}^{*N}\) into (A.3.4) and (A.3.5), we can calculate that \({\pi }_{U}^{L}\left({t}^{*Lr}\right)-{\pi }_{U}^{N}\left({t}^{*N}\right)=\frac{3}{2}\left[{{x}_{U}^{Lm}\left({t}^{*Lm}\right)}^{2}-{{x}_{U}^{N}\left({t}^{*N}\right)}^{2}\right]=\frac{3}{2}\left[{x}_{U}^{L}\left({t}^{*Lr}\right)+{x}_{U}^{N}\left({t}^{*N}\right)\right]\left[-\frac{2}{3}{c}_{D} -\frac{1}{24}\left({t}^{*Lr}-{t}^{*N}\right)\right]<0\) and that \(\left[{\pi }_{D}^{L}\left({t}^{*Lr}\right)-{\pi }_{D}^{N}\left({t}^{*N}\right)\right]=\left[{{q}_{D}^{N}\left({t}^{*Lr}\right)}^{2}-{{q}_{D}^{N}\left({t}^{*N}\right)}^{2}\right]=\left[{q}_{D}^{L}\left({t}^{*Lr}\right)+{q}_{D}^{N}\left({t}^{*N}\right)\right] \left[\frac{3}{8}\left({t}^{*Lr}-{t}^{*N}\right)\right]>0,\) as \({t}^{*Lr}-{t}^{*N}>0.\) Thus, the difference in the host upstream firm’s profit effect is negative, while the difference in the host downstream firm’s profit effect is positive. By Proposition 6, we can derive that \(C{S}^{L}\left({t}^{*Lr}\right)-C{S}^{N}\left({t}^{*N}\right)=\frac{1}{2}[{{Q}^{L}\left({t}^{*Lr}\right)}^{2}-{{Q}^{N}\left({t}^{*N}\right)}^{2}]=\frac{1}{2}\left[{Q}^{L}\left({t}^{*Lr}\right)+{Q}^{N}\left({t}^{*N}\right)\right][-\frac{5}{16}\left({t}^{*Lr}-{t}^{*N}\right)]<0.\) Therefore, the difference in the consumer surplus effect is negative. Finally, the difference in the tariff revenue effect is ambiguous, because \({t}^{*Lr}{q}_{I}^{L}\left({t}^{*Lr}\right)-{t}^{*N}{q}_{I}^{N}\left({t}^{*N}\right)=\frac{{c}_{D}}{{\left(761\right)}^{2}}\left(11120{c}_{D}-956\left(1-{c}_{U}\right)\right).\) By combining the above four effects, we can derive the welfare difference in (15.1).

1.3.2 C.2. The calculations of the four effects in (15) under mixed licensing

By substituting \({t}^{*Lm}\) and \({r}^{*}=\frac{1}{2}\left(1-{c}_{U}-\frac{49}{53}{t}^{*Lm}\right)\) into (6.1) and (6.2) and substituting \({t}^{*N}\) into (A.3.4) and (A.3.5), we can calculate that \({\pi }_{U}^{L}\left({t}^{*Lm}\right)-{\pi }_{U}^{N}\left({t}^{*N}\right)=(\frac{3}{2})\left[{x}_{U}^{L}\left({t}^{*Lm}\right)+{x}_{U}^{N}\left({t}^{*N}\right)\right][\frac{8315634}{73609247}{c}_{D}-\frac{9499988}{73609247}\left(1-{c}_{U}\right)]<0,\) and that \(\left[{\pi }_{D}^{L}\left({t}^{*Lm}\right)-{\pi }_{D}^{N}\left({t}^{*N}\right)\right]=\left[{q}_{D}^{L}\left({t}^{*Lm}\right)+{q}_{D}^{N}\left({t}^{*N}\right)\right]\left[\frac{3}{8}\left({t}^{*Lm}-{t}^{*N}\right)\right]>\left(<\right) 0,\) because \({t}^{*Lm}-{t}^{*N}>\left(<\right) 0 if {c}_{2}\le {c}_{D}<\left(>\right){c}_{4}.\) Thus, the difference in the host upstream firm’s profit effect is negative, while the difference in the host downstream firm’s profit effect is ambiguous. By Proposition 6, we can derive that \(C{S}^{L}\left({t}^{*Lm}\right)-C{S}^{N}\left({t}^{*N}\right)=\frac{1}{2}\left[{Q}^{L}\left({t}^{*Lm}\right)+{Q}^{N}\left({t}^{*N}\right)\right][\frac{24331714}{73609247}{c}_{D}-\frac{5079438}{73609247}\left(1-{c}_{U}\right)]>\left(<\right) 0,\) if \({c}_{D}>\left(<\right){c}_{5}.\) Therefore, the difference in the consumer surplus effect is indeterminate because \({c}_{3}>{c}_{5}>{c}_{2}\). Finally, the difference in the tariff revenue effect is also ambiguous, because \({t}^{*Lm}{q}_{I}^{L}\left({t}^{*Lm}\right)-{t}^{*N}{q}_{I}^{N}\left({t}^{*N}\right)=-\frac{452233890883176}{5418321243907009}{c}_{D}^{2}-\frac{177391205813694}{5418321243907009}{c}_{D}\left(1-{c}_{U}\right)+\frac{39255126179194}{5418321243907009}{\left(1-{c}_{U}\right)}^{2}\). By combining the above four effects, we obtain the welfare difference in (15.2).

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Wang, KC.A., Bui, DL., Wang, YJ. et al. International licensing under an endogenous tariff in vertically-related markets. J Econ 139, 93–123 (2023). https://doi.org/10.1007/s00712-023-00824-z

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