Technology licensing under product differentiation

Abstract

This paper discusses the licensing of technology between rival firms in a Cournot duopoly with horizontal and vertical product differentiation. The firms produce products of different qualities (high and low) and incur different costs per unit of output produced. It is shown that technology is transferred from the firm that produces the higher quality product to the firm that produces the lower quality product via a fixed-fee if the quality difference (net of cost) and the horizontal differentiation between the two products are relatively low. Technology is transferred through royalty, for any level of quality difference (net of cost), if the horizontal differentiation between the products is relatively low. A similar result is observed for two-part tariff licensing and quota licensing, which is a combination of output quota set by the licensor coupled with a fixed-fee. It is also shown that the optimal form of contract is either two-part tariff licensing or quota licensing. Technology is never licensed from the firm that produces a lower quality product to its rival that produces a higher quality product. However, the cross-licensing of technology is sometimes possible. After licensing welfare always increases.

Appendices

Appendix 1

1.1 Appendix 1.1: Fixed-fee

From the profit functions in the main text (see Sect. 3), we get the following three reaction functions:

$$\begin{aligned} q^{F1}_{11}&= max\left[ \frac{\alpha _{1}-q^{F1}_{12}-\gamma q^{F1}_{2}-c_{1}}{2}, \,\,0\right] \\ q^{F1}_{12}&= max\left[ \frac{\alpha _{1}-q^{F1}_{11}-2\gamma q^{F1}_{2}-c_{1}}{2}, \,\,0\right] \\ q^{F1}_{2}&= max\left[ \frac{\alpha _{2}-\gamma q^{F1}_{11}- 2\gamma q^{F1}_{12}-c_{2}}{2}, \,\,0\right] . \end{aligned}$$

After licensing, having access to produce good 1, firm 2 decides whether to stop production in the old market (good 2) or not.

Let us assume that firm 2 produces both the products after licensing (call it as Case 1).²⁶ In that case, re-arranging the reaction functions we get

$$\begin{aligned} 2q^{F1}_{11}+q^{F1}_{12}+\gamma q^{F1}_{2}=\alpha _{1}-c_{1} \end{aligned}$$

(43)

$$\begin{aligned} q^{F1}_{11}+2q^{F1}_{12}+2\gamma q^{F1}_{2}=\alpha _{1}-c_{1} \end{aligned}$$

(44)

$$\begin{aligned} \gamma q^{F1}_{11}+2\gamma q^{F1}_{12}+2q^{F1}_{2}=\alpha _{2}-c_{2}. \end{aligned}$$

(45)

After solving these three equations (43, 44 and 45) simultaneously we get the equilibrium quantities in the post-licensing stage as,

$$\begin{aligned} q^{F1}_{11}&= \frac{\alpha _{1}-c_{1}}{3} , \end{aligned}$$

(46)

$$\begin{aligned} q^{F1}_{12}&= \frac{(2+\gamma ^{2})(\alpha _{1}-c_{1})- 3\gamma (\alpha _{2}-c_{2})}{6(1-\gamma ^{2})}\,\,and\,\,q^{F1}_{2}=\frac{(\alpha _{2}-c_{2})-\gamma (\alpha _{1}-c_{1})}{2(1-\gamma ^{2})} . \end{aligned}$$

(47)

1.1.1 Appendix 1.1.1: Case 1

$\Pi ^{F1}_{2}=\Bigg [\frac{(\alpha _{1}-c_{1})}{3}\Bigg ] \Bigg [\frac{(2+\gamma ^{2})(\alpha _{1}-c_{1})-3\gamma (\alpha _{2}-c_{2})}{6(1-\gamma ^{2})}\Bigg ] +\Bigg [\frac{3(\alpha _{2}-c_{2})-(\alpha _{1}-c_{1})\gamma }{6}\Bigg ] \Bigg [\frac{(\alpha _{2}-c_{2})-(\alpha _{1}-c_{1})\gamma }{2(1-\gamma ^{2})}\Bigg ]-F =\Bigg [\frac{(\alpha _{1}-c_{1})^{2}(4+5\gamma ^{2})-18\gamma (\alpha _{1}-c_{1})(\alpha _{2}-c_{2})+9(\alpha _{2}-c_{2})^{2}}{36(1-\gamma ^{2})}\Bigg ]-F.$ Therefore, as fixed-fee is charged such that $\Pi ^{F1}_{2}=\Pi ^{*}_{2}$

$$\begin{aligned} \Pi ^{F1}_{1}=\frac{(\alpha _{1}-c_{1})^{2}}{9} +\Bigg [\frac{(\alpha _{1}-c_{1})^{2}(4+5\gamma ^{2})-18\gamma (\alpha _{1}-c_{1})(\alpha _{2}-c_{2})+9(\alpha _{2}-c_{2})^{2}}{36(1-\gamma ^{2})}\Bigg ]-\Pi ^{*}_{2}. \end{aligned}$$

(48)

The total industry profit after licensing is,

$\Pi ^{F1}_{1}+\Pi ^{F1}_{2}= \Bigg [\frac{(\alpha _{1}-c_{1})^{2}(8+\gamma ^{2})-18\gamma (\alpha _{1}-c_{1})(\alpha _{2}-c_{2})+9(\alpha _{2}-c_{2})^{2}}{36(1-\gamma ^{2})}\Bigg ]$.

Therefore, fixed-fee licensing is possible only if $\Pi ^{F1}_{1}+\Pi ^{F1}_{2} > \Pi ^{*}_{1}+\Pi ^{*}_{2}$, where

$\Pi ^{*}_{1}+\Pi ^{*}_{2}=\frac{(4+\gamma ^{2})(\alpha _{1}-c_{1})^{2}-8\gamma (\alpha _{1}-c_{1})(\alpha _{2}-c_{2})+(4+\gamma ^{2})(\alpha _{2}-c_{2})^{2}}{(4-\gamma ^{2})^{2}}$. The industry profit increases after licensing, if

$$\begin{aligned} (4-\gamma ^{2})^{2}\bigg ((8+\gamma ^{2})-18\gamma \theta +9\theta ^{2}\bigg )>36(1-\gamma ^{2})\bigg ((4+\gamma ^{2})-8\gamma \theta +(4+\gamma ^{2}){\theta }^{2}\bigg ). \end{aligned}$$

(49)

The above inequality (49) holds in the shaded region or zone F as depicted in Fig. 1 of the main text. From the diagram we observe that technology is licensed in the present context through fixed-fee such that $\theta >{\gamma }$ if $\gamma$ is slightly lower than $\theta$, but $\theta \in [0.7559,1]$.

1.1.2 Appendix 1.1.2: Case 2

Given $q^{F1}_{12}>0$ and $q^{F1}_{2}=0$ for $\gamma \ge \theta$ we find that, $\Pi ^{F1}_{1} = \frac{(\alpha _{1}-c_{1})^{2}}{9}+F$ and $\Pi ^{F1}_{2}=\frac{(\alpha _{1}-c_{1})^{2}}{9}-F$ and after licensing total industry profit is $\Pi ^{F1}_{1}+\Pi ^{F1}_{2}= \frac{2(\alpha _{1}-c_{1})^{2}}{9}$, where $\Pi ^{F1}_{1}$ and $\Pi ^{F1}_{2}$ denote the profits of firm 1 and firm 2 respectively, in the post licensing stage.

As in the previous case, fixed-fee licensing is possible only if $\Pi ^{F1}_{1}+\Pi ^{F1}_{2} > \Pi ^{*}_{1}+\Pi ^{*}_{2}$ or

$$\begin{aligned} 2(4-{\gamma }^{2})^{2}-9(4+{\gamma }^{2})(1+{\theta }^{2})+72\gamma \theta > 0. \end{aligned}$$

(50)

Therefore, technology is licensed in the present context through fixed-fee such that $\theta <{\gamma }$, in the shaded region of the Fig. 2 (see the main text), if both $\gamma$ and $\theta$ are relatively higher. Technology is never licensed if either $\theta <0.5619$ or $\gamma <0.7077$. Moreover, if $0.5619<\theta <0.6$, then $\gamma$ should not be too large for licensing to take place. Otherwise, for $0.6<\theta$, technology is licensed for any $\theta$, if $\gamma$ is relatively higher as shown in the diagram.

1.2 Appendix 1.2: Royalty

From the first order conditions of profit maximization (see Sect. 4 of the main text), we obtain the following reaction functions:

$$\begin{aligned} q^{R1}_{11}&= max\left[ \frac{\alpha _{1}-q^{R1}_{12}-\gamma q^{R1}_{2}-c_{1}}{2}, \,\,0\right] \\ q^{R1}_{12}&= max\left[ \frac{\alpha _{1}-q^{R1}_{11}-2\gamma q^{R1}_{2}-(c_{1}+r)}{2}, \,\,0\right] \\ q^{R1}_{2}&= max\left[ \frac{\alpha _{2}-\gamma q^{R1}_{11}- 2\gamma q^{R1}_{12}-c_{2}}{2}, \,\,0\right] . \end{aligned}$$

The decision to enter the new market (good 1) and/or stop production in the old market (good 2) would be determined by firm 2.

Let us assume that firm 2 produces both the products after licensing (let us call it as Case 1).²⁷ In that case, re-arranging the reaction functions we get,

$$\begin{aligned} 2q^{R1}_{11}+q^{R1}_{12}+\gamma q^{R1}_{2}&= \alpha _{1}-c_{1} \end{aligned}$$

(51)

$$\begin{aligned} q^{R1}_{11}+2q^{R1}_{12}+2\gamma q^{R1}_{2}&= \alpha _{1}-c_{1}-r \end{aligned}$$

(52)

$$\begin{aligned} \gamma q^{R1}_{11}+2\gamma q^{R1}_{12}+2q^{R1}_{2}&= \alpha _{2}-c_{2}. \end{aligned}$$

(53)

Solving these three equations (51, 52 and 53) simultaneously and assuming that firm 2 operates in both the markets, the equilibrium quantities in the post licensing stage are,

$$\begin{aligned} q^{R1}_{11}&= \frac{\alpha _{1}-c_{1}+r}{3}, \end{aligned}$$

(54)

$$\begin{aligned} q^{R1}_{12}&= \frac{(2+\gamma ^{2})(\alpha _{1}-c_{1})- 3\gamma (\alpha _{2}-c_{2})-r(4-\gamma ^{2})}{6(1-\gamma ^{2})}\,\,and\,\,q^{R1}_{2}=\frac{(\alpha _{2}-c_{2})-\gamma (\alpha _{1}-c_{1}-r)}{2(1-\gamma ^{2})}. \end{aligned}$$

(55)

1.2.1 Appendix 1.2.1: Case 1

If $q^{R1}_{12}>0$ and $q^{R1}_{2}>0$, then the post-licensing profits of firm 1 and firm 2 are

$\Pi ^{R1}_{1}=(q^{R1}_{11})^{2}+r(q^{R1}_{12})=\left[ \frac{(\alpha _{1}-c_{1}+r)}{3}\right] ^{2} + r \left[ \frac{(2+\gamma ^{2})(\alpha _{1}-c_{1})- 3\gamma (\alpha _{2}-c_{2})-r(4-\gamma ^{2})}{6(1-\gamma ^{2})}\right]$ and

$\Pi ^{R1}_{2}= (p_{1}-c_{1}-r)(q^{R1}_{12})+(p_{2}-c_{2})(q^{R1}_{2})$

$=\left[ \frac{(\alpha _{1}-c_{1}-2r)}{3}\right] \left[ \frac{(2+\gamma ^{2})(\alpha _{1}-c_{1})- 3\gamma (\alpha _{2}-c_{2})-r(4-\gamma ^{2})}{6(1-\gamma ^{2})}\right] +\left[ \frac{3(\alpha _{2}-c_{2})-\gamma (\alpha _{1}-c_{1}+r)}{6}\right] \left[ \frac{(\alpha _{2}-c_{2})-\gamma (\alpha _{1}-c_{1}-r)}{2(1-\gamma ^{2})}\right]$,

respectively such that $r\in (\bar{r}, r^{max})$ as discussed in Lemma 3.

To find the optimal royalty rate $r^{*}$ that firm 1 should charge, we set $\frac{\partial \Pi ^{R1}_{1}}{\partial r}=0\,$ i.e.

$$\begin{aligned} \left[ \frac{2(\alpha _{1}-c_{1}+r)}{9}\right] +\left[ \frac{(2+\gamma ^{2})(\alpha _{1}-c_{1})- 3\gamma (\alpha _{2}-c_{2})-2r(4-\gamma ^{2})}{6(1-\gamma ^{2})}\right] =0, \end{aligned}$$

and get $r^{*}= \left[ \frac{2(10-\gamma ^{2})(\alpha _{1}-c_{1})- 18\gamma (\alpha _{2}-c_{2})}{4(10-\gamma ^{2})}\right] .$ Moreover, $r^{*}>r^{max}$.²⁸ At $r^{max}$, $\frac{\partial \Pi ^{R1}_{1}}{\partial r}>0$ and $\Pi ^{R1}_{1} = \Pi ^{*}_{1}$. However, if firm 1 charges a royalty rate greater than $r^{max}$, such that $\Pi ^{R1}_{1} > \Pi ^{*}_{1}$ (because its profit is increasing in the royalty rate r), then licensing cannot take place as $q^{R1}_{12}=0$. Therefore, licensing is never possible in this scenario as stated in the Lemma 5 of the main text.

1.3 Appendix 1.3: Two-part tariff: Case 1

If $q^{T1}_{12}>0$ and $q^{T1}_{2}>0$, then the post-licensing profits of firm 1 and firm 2 are

$$\begin{aligned} \Pi ^{T1}_{1}&=(q^{T1}_{11})^{2}+r(q^{T1}_{12})+F=\left[ \frac{(\alpha _{1}-c_{1}+r)}{3}\right] ^{2}\\&\quad + r \left[ \frac{(2+\gamma ^{2})(\alpha _{1}-c_{1})- 3\gamma (\alpha _{2}-c_{2})-r(4-\gamma ^{2})}{6(1-\gamma ^{2})}\right] +F\; and \\ \Pi ^{T1}_{2}&= (p_{1}-c_{1}-r)(q^{T1}_{12})+(p_{2}-c_{2})(q^{T1}_{2})-F\\&=\left[ \frac{(\alpha _{1}-c_{1}-2r)}{3}\right] \left[ \frac{(2+\gamma ^{2})(\alpha _{1}-c_{1})- 3\gamma (\alpha _{2}-c_{2})-r(4-\gamma ^{2})}{6(1-\gamma ^{2})}\right] \\&\quad +\left[ \frac{3(\alpha _{2}-c_{2})-\gamma (\alpha _{1}-c_{1}+r)}{6}\right] \left[ \frac{(\alpha _{2}-c_{2})-\gamma (\alpha _{1}-c_{1}-r)}{2(1-\gamma ^{2})}\right] -F, \end{aligned}$$

respectively such that $r\in (\bar{r}, r^{max})$ as discussed in Lemma 3. Further, firm 1 will set the royalty rate and the fixed-fee in the first stage of the game such that $\Pi ^{T1}_{2}=\Pi ^{*}_{2}$, as it wants to maximize its profit in the post-licensing stage. Hence, the profit function of firm 1 reduces to

$$\begin{aligned} \Pi ^{T1}_{1}&= \left[ \frac{(\alpha _{1}-c_{1}+r)}{3}\right] ^{2} + \left[ \frac{(\alpha _{1}-c_{1}+r)}{3}\right] \nonumber \\&\quad \times \left[ \frac{(2+\gamma ^{2})(\alpha _{1}-c_{1})- 3\gamma (\alpha _{2}-c_{2})-r(4-\gamma ^{2})}{6(1-\gamma ^{2})}\right] \nonumber \\&+ \left[ \frac{3(\alpha _{2}-c_{2})-\gamma (\alpha _{1}-c_{1}+r)}{6}\right] \left[ \frac{(\alpha _{2}-c_{2})-\gamma (\alpha _{1}-c_{1}-r)}{2(1-\gamma ^{2})}\right] -\Pi ^{*}_{2}. \end{aligned}$$

(56)

To find the optimal royalty rate $r^{*}$ that firm 1 should charge coupled with $F^{*}$, we set $\frac{\partial \Pi ^{T1}_{1}}{\partial r}=0\,$ and get

$$\begin{aligned} r^{*}= \left[ \frac{2(1-\gamma ^{2})(\alpha _{1}-c_{1})}{(5\gamma ^{2}+4)}\right] . \end{aligned}$$

(57)

Appendix 2: Fixed-fee licensing: Firm 2 to Firm 1

This section studies the possibility of transfer of technology from firm 2 to firm 1 under fixed-fee licensing at a fixed-fee F. After licensing, the profit functions for firm 2 and firm 1 are respectively,

$$\begin{aligned} \Pi ^{F2}_{2}= [\alpha _{2}-q^{F2}_{22}-q^{F2}_{21}-\gamma q^{F2}_{1}-c_{2}]q^{F2}_{22}+F \end{aligned}$$

and

$$\begin{aligned} \Pi ^{F2}_{1}= [\alpha _{2}-q^{F2}_{22}-q^{F2}_{21}-\gamma q^{F2}_{1}-c_{2}]q^{F2}_{21} + [\alpha _{1}-q^{F2}_{1}-\gamma q^{F2}_{22}-\gamma q^{F2}_{21}-c_{1}]q^{F2}_{1}-F, \end{aligned}$$

where $q^{F2}_{22}$ and $q^{F2}_{21}$ are the quantities of the lower quality product produced by firm 2 and firm 1 in the post-licensing stage respectively. The amount of the higher quality product produced by firm 1 after licensing is $q^{F2}_{1}$.

Let us assume that firm 1 produces both the products after licensing. As in the main text, we get the equilibrium quantities in the post licensing stage as,

$$\begin{aligned} q^{F2}_{22}&= \frac{\alpha _{2}-c_{2}}{3} , \\ q^{F2}_{21}&= \frac{(2+\gamma ^{2})(\alpha _{2}-c_{2})- 3\gamma (\alpha _{1}-c_{1})}{6(1-\gamma ^{2})}\,\,and\,\,q^{F2}_{1}=\frac{(\alpha _{1}-c_{1})-\gamma (\alpha _{2}-c_{2})}{2(1-\gamma ^{2})} . \end{aligned}$$

From the above expressions it is seen that $q^{F2}_{22}>0$ and $q^{F2}_{1}>0$. Secondly,

• i) $q^{F2}_{21}>0$ if $\theta >\frac{3\gamma }{2+\gamma ^{2}}$ or else

• ii) $q^{F2}_{21}=0$, i.e. technology is not licensed.

Licensing is possible in both these cases if and only if,

$$\begin{aligned} \Pi ^{F2}_{2}\ge \Pi ^{*}_{2} \,\,\,and\,\,\, \Pi ^{F2}_{1}\ge \Pi ^{*}_{1}. \end{aligned}$$

(58)

If $F=0$, then $\Pi ^{F2}_{2} < \Pi ^{*}_{2}$. Thus, firm 2 should set $F>0$, such that it can offset the loss in profits after licensing with the fixed-fee it receives from the licensee. We consider the case $q^{F2}_{21}>0$, or $\theta >\frac{3\gamma }{2+\gamma ^{2}}$, as otherwise licensing is not possible. Firm 2 will set F such that $\Pi ^{F2}_{21}=\Pi ^{*}_{1}$. Hence, licensing is profitable if

$$\begin{aligned} \Pi ^{F2}_{2}+\Pi ^{F2}_{1} \ge \Pi ^{*}_{2}+\Pi ^{*}_{1}. \end{aligned}$$

(59)

where $\Pi ^{F2}_{2}=\frac{(\alpha _{2}-c_{2})^{2}}{9}+F$, while firm 1’s total profit includes revenue from the both the higher quality and the lower quality product, and it is equal to

$$\begin{aligned} \Pi ^{F2}_{1}&=\Bigg [\frac{(\alpha _{2}-c_{2})}{3}\Bigg ] \Bigg [\frac{(2+\gamma ^{2})(\alpha _{2}-c_{2})-3\gamma (\alpha _{1}-c_{1})}{6(1-\gamma ^{2})}\Bigg ] \\&\quad +\Bigg [\frac{3(\alpha _{1}-c_{1})-(\alpha _{2}-c_{2})\gamma }{6}\Bigg ] \Bigg [\frac{(\alpha _{1}-c_{1})-(\alpha _{2}-c_{2})\gamma }{2(1-\gamma ^{2})}\Bigg ]-F\\&=\Bigg [\frac{(\alpha _{2}-c_{2})^{2}(4+5\gamma ^{2})-18\gamma (\alpha _{2}-c_{2})(\alpha _{1}-c_{2})+9(\alpha _{1}-c_{1})^{2}}{36(1-\gamma ^{2})}\Bigg ]-F. \end{aligned}$$

The total industry profit is,

$$\begin{aligned} \Pi ^{F2}_{2}+\Pi ^{F2}_{1}= \Bigg [\frac{(\alpha _{2}-c_{2})^{2}(8+\gamma ^{2})-18\gamma (\alpha _{1}-c_{1})(\alpha _{2}-c_{2})+9(\alpha _{1}-c_{1})^{2}}{36(1-\gamma ^{2})}\Bigg ]. \end{aligned}$$

Therefore, fixed-fee licensing is possible only if

$$\begin{aligned} \Pi ^{F1}_{1}+\Pi ^{F1}_{2} \ge \Pi ^{*}_{1}+\Pi ^{*}_{2} \end{aligned}$$

(60)

where $\Pi ^{*}_{1}+\Pi ^{*}_{2}=\frac{(4+\gamma ^{2})(\alpha _{1}-c_{1})^{2}-8\gamma (\alpha _{1}-c_{1})(\alpha _{2}-c_{2})+(4+\gamma ^{2})(\alpha _{2}-c_{2})^{2}}{(4-\gamma ^{2})^{2}}$. The above equation holds if $(4-\gamma ^{2})^{2}\bigg ((8+\gamma ^{2}){\gamma ^{*}}^{2}-18\gamma \gamma ^{*}+9\bigg )>36(1-\gamma ^{2})\bigg ((4+\gamma ^{2})-8\gamma \gamma ^{*}+(4+\gamma ^{2}){\gamma ^{*}}^{2}\bigg )$.

However, this is never possible. Hence, technology is never licensed via fixed-fee.

Appendix 3: Royalty Licensing: Firm 2 to Firm 1

This section studies the possibility of transfer of technology from firm 2 to firm 1 under royalty licensing at a rate r. After licensing, the profit functions for firm 2 and firm 1 are respectively,

$$\begin{aligned} \Pi ^{R2}_{2}= [\alpha _{2}-q^{R2}_{22}-q^{R2}_{21}-\gamma q^{R2}_{1}-c_{2}]q^{R2}_{22}+rq^{R2}_{21} \end{aligned}$$

and

$$\begin{aligned} \Pi ^{R2}_{1}= [\alpha _{2}-q^{R2}_{22}-q^{R2}_{21}-\gamma q^{R2}_{1}-c_{2}-r]q^{R2}_{21} + [\alpha _{1}-q^{R2}_{1}-\gamma q^{R2}_{22}-\gamma q^{R2}_{21}-c_{1}]q^{R2}_{1}, \end{aligned}$$

where $q^{R2}_{22}$ and $q^{R2}_{21}$ are the quantities of the lower quality product produced by firm 2 and firm 1 in the post-licensing stage, respectively. The amount of the higher quality product produced by firm 1 after licensing is $q^{R2}_{1}$.

Assuming that firm 1 operates in both the markets, the equilibrium quantities in the post licensing stage are,

$$\begin{aligned} q^{R2}_{22}&= \frac{\alpha _{2}-c_{2}+r}{3}, \end{aligned}$$

(61)

$$\begin{aligned} q^{R2}_{21}&= \frac{(2+\gamma ^{2})(\alpha _{2}-c_{2})- 3\gamma (\alpha _{1}-c_{1})-r(4-\gamma ^{2})}{6(1-\gamma ^{2})}\,\,and\nonumber \\ q^{R2}_{1}&= \frac{(\alpha _{1}-c_{1})-\gamma (\alpha _{2}-c_{2}-r)}{2(1-\gamma ^{2})}. \end{aligned}$$

(62)

As the per unit royalty rate is strictly positive, therefore $q^{R2}_{22}>0$ and $q^{R2}_{1}>0.$ From the above expression, it can be said that $q^{R2}_{21}>0$, if $r\in (0,r^{max})$, where $r^{max}= \frac{(2+\gamma ^{2})(\alpha _{2}-c_{2})- 3\gamma (\alpha _{1}-c_{1})}{(4-\gamma ^{2})}$, otherwise technology is not licensed, as $q^{R2}_{21}$ decreases if r increases.

If $q^{R2}_{21}>0$, then the post-licensing profits of firm 2 and firm 1 are

$$\begin{aligned} \Pi ^{R2}_{2}&=(q^{R2}_{22})^{2}+r(q^{R2}_{21})=\left[ \frac{(\alpha _{2}-c_{2}+r)}{3}\right] ^{2}\\&\quad + r \left[ \frac{(2+\gamma ^{2})(\alpha _{2}-c_{2})- 3\gamma (\alpha _{1}-c_{1})-r(4-\gamma ^{2})}{6(1-\gamma ^{2})}\right] \;and\\ \Pi ^{R2}_{1}&= (p_{2}-c_{2}-r)(q^{R2}_{21})+(p_{1}-c_{1})(q^{R2}_{1})\\&=\left[ \frac{(\alpha _{2}-c_{2}-2r)}{3}\right] \left[ \frac{(2+\gamma ^{2})(\alpha _{2}-c_{2})- 3\gamma (\alpha _{1}-c_{1})-r(4-\gamma ^{2})}{6(1-\gamma ^{2})}\right] \\&\quad +\left[ \frac{3(\alpha _{1}-c_{1})-\gamma (\alpha _{2}-c_{2}+r)}{6}\right] \left[ \frac{(\alpha _{1}-c_{1})-\gamma (\alpha _{2}-c_{2}-r)}{2(1-\gamma ^{2})}\right] , \end{aligned}$$

respectively such that $r\in (0, r^{max})$ .

To find the optimal royalty rate $r^{*}$ that firm 2 should charge, we set $\frac{\partial \Pi ^{R2}_{2}}{\partial r}=0\,$ i.e.

$$\begin{aligned} \left[ \frac{2(\alpha _{2}-c_{2}+r)}{9}\right] +\left[ \frac{(2+\gamma ^{2})(\alpha _{2}-c_{2})- 3\gamma (\alpha _{1}-c_{1})-2r(4-\gamma ^{2})}{6(1-\gamma ^{2})}\right] =0, \end{aligned}$$

and get $r^{*}= \left[ \frac{2(10-\gamma ^{2})(\alpha _{2}-c_{2})- 18\gamma (\alpha _{1}-c_{1})}{4(10-\gamma ^{2})}\right] .$ Moreover, as $r^{max}<r^{*}$, firm 2 will charge $r^{max}$, as at $r=r^{max}$ we have $\frac{\partial \Pi ^{R2}_{2}}{\partial r}>0\,$. Hence, technology will not be transferred from firm 2 to firm 1, as at $r=r^{max}$ we get $q^{R2}_{21}=0.$

As technology is not transferred from firm 2 to firm 1, either by fixed-fee or by royalty, hence two-part tariff licensing is also not possible in this context.