Does licensing promote innovation?

ABSTRACT

We analyse the impact of licensing on the equilibrium amount of cost-reducing innovation under several licensing mechanisms in the case of a duopoly model with heterogeneous firms. Under a wide class of licensing mechanisms, we find that as product substitutability increases, the possibility of higher innovation under licensing compared to no licensing decreases. Therefore, firms’ heterogeneity is crucial to assess whether licensing incentivizes or not R&D cost-reducing investments.

KEYWORDS:

• Patent licensing
• innovation

JEL CLASSIFICATION:

• D45

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Stefano Colombo http://orcid.org/0000-0002-9384-4530

Notes

1 Somehow differently, Roy Chowdhury (2005) evaluates the impact of the extent of patent protection on innovation both when licensing is possible and when it is not possible, and finds that patents could reduce R&D investment.

2 The upper bound to c allows restricting the analysis to non-drastic innovation, so that the non-innovating firm is never excluded from the market (for the distinction between drastic and non-drastic innovation, see Wang 1998, for example).

3 We assume that there is no uncertainty about the outcome of the R&D process. While outcome uncertainty is certainly realistic in many cases, we keep the model as simple as possible to derive clear-cut implications. For an analysis of optimal licensing in the case uncertain R&D outcomes, see Yan and Yang (2018).

4 Details are in the Appendix.

5 There is another constraint, $F\ge 0$. However, at the equilibrium values this constraint is always verified, so it can be disregarded. See later.

6 See the Appendix.

7 To fix the ideas, consider a two-firms model with linear demand functions and zero marginal costs. Routine calculations show that each firm produces $1/(2+\gamma )$, which decreases with $\gamma$.

8 See also Doganoglu, Inceoglu, and Muthers (2018), which argue that the gains from innovation under licensing increase in the quantity produced. It should be noted that there are also some second-order effects on r and F. For example, when considering the case where both constraints are slack in equilibrium, it can be observed that the equilibrium per-unit royalty decreases with product substitutability. Indeed, as the licensee produces less, the royalty is kept low to (partially) compensate the negative effect of higher $\gamma$ on the quantity produced. With regard to F, in equilibrium it is also decreasing in $\gamma$: indeed, a higher $\gamma$ reduces $x_{ss}^L\ast$, thus reducing the amount of profits that can be extracted by the patentee.

9 Pure fixed fee is used when there is scarce information about the licensee's output (Mukherjee and Mukherjee, 2013).

10 Here the subscript is just s, as there is only one constraint, $x\in [0,c]$, which might be slack or binding: if it is binding, we have simply $x^{L,f}\ast =c$.

11 It should be noted that when the constraint $r\in [0,x]$ is binding and the constraint $x\in [r,c]$ is slack, the innovation level is the same under two-part per unit royalty and pure per unit royalty, that is $x_{bs}^L\ast$. Indeed, when $r\ast =x$, we have $F=0$: therefore, the two licensing mechanisms coincide. It follows that also the relevant threshold for c is the same, that is ${\underset{\_}{c}}_3^L$.

12 In this section, we assume $\gamma \le 0.7$ in order to guarantee non drastic innovation.

13 Examples of licensing models in the case of price competition are, among the others, Wang and Yang (1999) and Yan and Yang (2018).

14 An interior solution in the case of no licensing requires that $v\ge {\underset{\_}{v}}_{1,B}^N\equiv {\displaystyle \frac{8-2\gamma -8{\gamma }^2+{\gamma }^3+2{\gamma }^4}{16-24{\gamma }^2+9{\gamma }^4-{\gamma }^6}}$and $c\ge {\underset{\_}{c}}_{1,B}^N\equiv {\displaystyle \frac{4-2\gamma -4{\gamma }^2+{\gamma }^3+{\gamma }^4}{v(16-24{\gamma }^2+9{\gamma }^4-{\gamma }^6)-\gamma (2-{\gamma }^2)}}$.

15 This amounts requiring that $v\in [{\underset{\_}{v}}_B^L,{\overline{v}}_B^L]$, where ${\underset{\_}{v}}_B^L={\displaystyle \frac{128-112\gamma +64{\gamma }^2-52{\gamma }^3-100{\gamma }^4+46{\gamma }^5+17{\gamma }^6-8{\gamma }^7-{\gamma }^8}{128-32{\gamma }^2-168{\gamma }^4+82{\gamma }^6-10{\gamma }^8}}$ and ${\overline{v}}_B^L={\displaystyle \frac{64-80\gamma +64{\gamma }^2-60{\gamma }^3-18{\gamma }^4+31{\gamma }^5-6{\gamma }^6-{\gamma }^7}{64\gamma +32{\gamma }^2-96{\gamma }^3-48{\gamma }^4+36{\gamma }^5+18{\gamma }^6-4{\gamma }^7-2{\gamma }^8}}$, and $c\ge {\underset{\_}{c}}_B^L$, with ${\underset{\_}{c}}_B^L={\displaystyle \frac{128-96\gamma +64{\gamma }^2-40{\gamma }^3-100{\gamma }^4+36{\gamma }^5+17{\gamma }^6-8{\gamma }^7-{\gamma }^8}{32\gamma +24{\gamma }^3-10{\gamma }^5+v(256-64{\gamma }^2-336{\gamma }^4+164{\gamma }^6-20{\gamma }^8)}}$.

16 Indeed, ${\displaystyle \frac{\mathrm{\partial }{q}_2\left(p_1\right(r_B^L\ast ,x_B^L\ast ),p_2(r_B^L\ast ,x_B^L\ast \left)\right)}{\mathrm{\partial }\gamma }<0}$.

17 The expression of d* is quite long. Therefore it is not reported here.

18 Numerical comparisons of the equilibrium profits under different licensing schemes, suggest that ad-valorem royalties is preferred by the patentee to other licensing methods when product substitutability is sufficiently high. This is in line with the result obtained by San Martín and Saracho (2015).

19 Details are available on request.

20 For other triopoly models in patent licensing literature, see Fosfuri and Roca (2004) and Colombo and Filippini (2013).

21 See however Filippini and Vergari (2017) for a related analysis.

22 In Figure 1, we set $c=7/20$.

23 By expressing this threshold in terms of $\gamma$, if, for example, $c=0.45$ and $v=0.7$, we have that ${\hat{x}}^N\ast \ge (\le )x_{ss}^L\ast$ if $\gamma \ge (\le )0.88$.

24 In Figure 2, we set $c=0.4$.

25 $x^{L,v}\ast$ indicates the equilibrium innovation level under ad valorem licensing.