Compulsory licensing and access to drugs

Abstract

Compulsory licensing allows the use of a patented invention without the owner’s consent, with the aim of improving access to essential drugs. The pharmaceutical sector argues that, if broadly used, it can be detrimental to innovation. We model the interaction between a company in the North that holds the patent for a certain drug and a government in the South that needs to purchase it. We show that both access to drugs and pharmaceutical innovation depend largely on the Southern country’s ability to manufacture a generic version. If the manufacturing cost is too high, compulsory licensing is not exercised. As the cost decreases, it becomes a credible threat forcing prices down, but reducing both access and innovation. When the cost is low enough, the South produces its own generic version and access reaches its highest value, despite a reduction in innovation. The global welfare analysis shows that the overall impact of compulsory licensing can be positive, even when accounting for its impact on innovation. We also consider the interaction between compulsory licensing and the strength of intellectual property rights, which can have global repercussions in other markets beyond the South.

Appendix

Proof of Proposition 1

In region (i), if F is high, the outside option is ineffective; thus, we obtain the same result as in the unregulated benchmark case, that is, $u^{{\rm CL}}=u^{\ast }$, which therefore is valid as long as $Fk/(a^{4}\sigma ^{2})>3(u^{\ast })^{2}/32$. The other limiting case is in region (iii) when F is very low, so that the firm never sells in the South; then π _S  = 0, and quality is $u=u^{N}<u^{\ast }$. This candidate solution is valid as long as Fk/(a ⁴ σ ²) < 37(u ^N)²/512. The analysis is more involved in region (ii), valid for intermediate values of F, the constraint identified by the threat of the government to use the outside option binds, which is re-written as

$$CS_{S}=\frac{u^{2}a^{4}\sigma ^{2}}{8k}-F\Rightarrow 4(au-p_{S})^{3}p_{S}\sigma ^{2}-u^{2}(a^{4}\sigma ^{2}u^{2}-8Fk)=0.$$

(11)

In the first stage, the firm solves

$$\frac{{\rm d}\varPi }{{\rm d}u}=\frac{\partial \varPi }{\partial u}+\frac{\partial \varPi }{ \partial p_{S}}\frac{\partial p_{S}}{\partial u}=0.$$

(12)

By means of implicit differentiation of (11), it is

$$\frac{\partial p_{S}}{\partial u}=\frac{\frac{\partial \frac{ u^{2}a^{4}\sigma ^{2}}{8k}}{\partial u}-\frac{\partial CS_{S}}{\partial u}}{ \frac{\partial CS_{S}}{\partial p_{S}}}=\frac{ (au)^{4}-2(au-p_{S})^{2}p_{S}(au+2p_{S})}{2u(au-p_{S})^{2}(au-4p_{S})}.$$

(13)

The numerator of (13) is always positive, as the expression 2(au − p _S )² p _S (au + 2p _S ) is single peaked in p _S for p _S  < au and even at the maximum can never exceed in absolute value (au)⁴. The denominator is always negative as we are in a range of prices between the unregulated benchmark and the preferred price for the South: au/4 < p _S  < au/2. Thus, $\frac{\partial p_{S}}{\partial u}<0,$ as an increase in u increases the value of the outside option relatively more; hence, the price must be decreased in order to make the South accept the offer. Since in (12) it is also $\frac{\partial \varPi }{\partial p_{S}}=0$ for $p_{S}=p_{S}^{\ast }$, and $\frac{\partial \varPi}{\partial p_{S}}>0$ for all $p_{S}<p_{S}^{\ast }$, we can then conclude that quality starts at $u^{\ast }$ when $Fk/(a^{4}\sigma ^{2})=3(u^{\ast })^{2}/32$ and then decreases monotonically as F becomes smaller and $p_{S}<p_{S}^{\ast }$. The solution is found by looking, in the {p _S , u} space and within the admissible cone $au/4<p_{S}<p_{S}^{\ast }$, at the highest isoprofit curve of the monopolist satisfying the constraint (11). It is easy to prove that there is always an interior solution strictly inside the cone. Although we omit the expressions for space limitation, this result is obtained because the isoprofit curves are vertical at $p_{S}=p_{S}^{\ast }$ and then convex for lower prices, while instead the constraint is vertical at p _S  = au/4 and then concave for higher prices; thus, there is always a tangency point. We can also prove that the monopolist chooses this solution as long as it makes at least the same amount as $\frac{u^{N}}{4}-C(u^{N})$. In fact, as $\left. \frac{\partial \varPi}{\partial u}\right\vert _{u=u^{N}}<0$ for low enough p _S , we can show that it is optimal for the monopolist: (1) to push the interior solution for values of Fk/(a ⁴ σ ²) up to a limit value L strictly lower than 37(u ^N)²/512, where L is defined by the isoprofit curve $\varPi =\frac{u^{N}}{4}-C(u^{N})$ subject to (11), and (2) it necessarily offers a quality below u ^N for values of Fk/(a ⁴ σ ²) approaching L from above.

Results on coverage follow easily by noting that in region (iii) it is x ^CL = u ^N a ² σ/2k. When instead we are in the intermediate region (ii), then from (4) it is $x^{{\rm CL}}<x^{\ast }=u^{\ast }a^{2}\sigma /4k$. As we have just established that, when Fx/(a ⁴ σ ²) approaches L from above, u < u ^N, for sure coverage jumps up when Fx/(a ⁴ σ ²) is further reduced and CL is exercised in region (iii) (with autarky it would be x ^N = 0). In fact, it may even be that coverage in region (iii) is higher than in region (i) where it is $x^{{\rm CL}}=u^{\ast }a^{2}\sigma /4k$. This arises when u ^N a ² σ/2k > $u^{\ast }a^{2}\sigma /4k$, or $u^{\ast }<2u^{N}$, which depends on the convexity of the cost function C(u) and on the value of k. The inequality is always satisfied when k is large enough, as in the limit $u^{N}\rightarrow u^{\ast }$. □

Proof of Proposition 2

When $C(u)=\frac{u^{2}}{2}$, it is $u^{\ast }=\frac{4k}{16k-a^{4}\sigma ^{2}}$, ¹⁸ as well as $p^{N}=\frac{u^{\ast }}{2},p^{S}=\frac{au^{\ast }}{2}$ and $x^{\ast }=\frac{u^{\ast }a^{2}\sigma }{4k}$. These can be substituted in (9) to obtain

$$W^{\ast }=\frac{k(32k-3a^{4}\sigma ^{2})}{2(16k-a^{4}\sigma ^{2})^{2}}.$$

In region (iii), it is $u^{N}=\frac{1}{4}$, and $p^{N}=\frac{u^{N}}{2},p^{S}=0, x^{{\rm CL}}=\frac{u^{N}a^{2}\sigma }{2k}$. From (10) we compute

$$W^{{\rm CL}}=\frac{8k+a^{4}\sigma ^{2}}{128k}-F.$$

When $F \rightarrow 0$ it is always $W^{{\rm CL}}>W^{\ast }$ for all admissible values of k/(a ⁴ σ ²) > 1/16. The result is in fact stronger and extends to the entire region (iii). This is because, from Proposition 1, in region (iii) it must be Fx/(a ⁴ σ ²) < L < 37(u ^N)²/512, and therefore F is bounded above by 37(u ^N)² a ⁴ σ ²/(512k). Even at this higher threshold, it is immediate to show that $W^{{\rm CL}}>W^{\ast }$ for all admissible values of k/(a ⁴ σ ²). □

Proof of Proposition 3

Let us start with the restricted case, where goods produced in the South under CL cannot be reimported, whereas those supplied by the monopolist can. If F is high, the outside option is ineffective, $p=p^{\ast }$ is set everywhere; thus, we obtain the same result as without parallel trade (identical to the unregulated benchmark case). If F is very low, the firm never sells in the South (which recurs to CL) but can set $p_{N}=p^{\ast },$ so that π _S  = 0 and π _N  = u/4, and quality is again as in the case without parallel trade. If F is intermediate, then π _N  = p _S (1 − p _S /u) and π _S  = p ² _S (u − p _S )²/2ku ². Like in the proof of the previous proposition, the solution is found by looking at the highest isoprofit curve (8) satisfying the constraint (11). The constraint is the same, with and without parallel trade, and it is still characterized by ∂p _S /∂u < 0. Since it is easy to prove that parallel trade reduces the marginal revenue, ¹⁹ we thus obtain that the effect of parallel trade is to reduce investment and increase the price cap. Another difference with the case without parallel trade is that, while the monopolist still pushes this interior solution for values of Fk lower than 37(u ^NC)²/512, now profits when both countries are supplied are strictly lower than without parallel trade (they coincide only when $p_{S}=p^{\ast }$); thus, the monopolist stops supplying the South for values below $\widetilde{L}$, which is strictly higher than L, and again defined by the isoprofit curve $\varPi =u^{N}/4-C(u^{N})$ subject to (11).

We now turn our analysis to the case of “unrestricted parallel trade.” The good manufactured in the South will be exported and traded everywhere, also under the compulsory licensing regime (i.e., by means of the gray market). Cases (i) and (ii) are unchanged and do not need to be analyzed again. The difference is that now the monopolist will never withdraw from the South; hence, CL will never be used along the equilibrium path. If it did so, then the price in the South would be zero but would apply everywhere, and the monopolist would not invest at all. Therefore, the validity of region (ii) is now extended also for all values below $\widetilde{L}$. Investment approaches zero, as well as prices and coverage, only as $F\rightarrow 0$. □