Compulsory licenses in the pharmaceutical industry: Pricing and R&D strategies

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Innovative Applications of O.R

Compulsory licenses in the pharmaceutical industry: Pricing and R&D strategies Archita Sarmah a, Domenico De Giovanni b, Pietro De Giovanni c,∗ a

Department of Management, Technology and Economics, ETH Zürich, Zürich, Switzerland Depatrment of Economics, Statistics an Finance Giovanni Anania University of Calabria, Arcavacata di Rende, Italy c Department of Business and Management, LUISS Guido Carli University, Rome, Italy b

a r t i c l e

i n f o

Article history: Received 9 February 2019 Accepted 15 October 2019 Available online xxx Keywords: Game theory Compulsory license Pricing R&D Pharmaceutical industry

a b s t r a c t A pharma manufacturer enters a developing country with a new drug after investing some R&D in the first period. The firm can be subjected to a compulsory license mechanism that allows a generic manufacturer to produce an imitated version of the patented product in exchange of a fixed royalty. When the patent expires, a traditional price competition ensues between the patent-holder and the generic manufacturer. We compare two deterministic scenarios wherein the patent-holder has full information regarding the compulsory license. We identify the conditions under which the license is socially and economically beneficial. Our analyses suggest that the patent-holder is seldom economically better-off. We next model a stochastic compulsory license decision rule whereby the patent-holder is exposed to a certain probability that the compulsory license is issued. We show that uncertainty renders the patentholder more willing to operate in that market. © 2019 Elsevier B.V. All rights reserved.

1. Introduction One of the plaguing challenges for health policy is the worldwide access to pharmaceutical medicines. For developing countries, the issue of access is particularly pertinent as they lack not only the financial and manufacturing prowess of the developed nations to produce innovative drugs but also the negotiating power to buy those drugs at affordable prices. To address this issue, in 2001, the members of the World Trade Organization signed the TRIPS (Trade-Related Aspects of Intellectual Property Rights) and Public Health declaration at Doha. The adoption of this declaration attested to the sovereign right of governments to undertake measures for public health protection. This declaration was groundbreaking because for the first time developing and least developed countries obtained a strong negotiating tool to issue compulsory licenses for pharmaceutical drugs.1 A compulsory license (henceforth CL) is a government authorized non-voluntary license from a patent-holder to a third party. When granted, the third party can use the patented technology without the patent-holder’s consent. The third party would

∗

Corresponding author. E-mail address: [email protected] (P. De Giovanni). 1 “Declaration on the TRIPS agreement and public health”. World Trade Organization. http://www.wto.org/english/thewto_e/minist_e/min01_e/mindecl_trips_ e.htm. Last accessed: March 2, 2018.

nevertheless be required to pay an appropriate remuneration (a royalty) to the patent-holder and the license would be valid only in the domestic market.2 To assess both the social and the economic implications of compulsory licensing, we set up and analyze a dynamic game between a patent-holder, H, and a generic producer, G. The actions linked to our game evolve as displayed in Fig. 1. In t0 , a patent-holder invests in R&D. By R&D we refer to the effort made by H to develop, produce and sell a new drug in a developing country. In t1 , the patent-holder is granted a patent for the drug in the developing country, fixes the price of the patented drug and acts as a monopolist because a CL cannot be issued for a period of three years after the grant of the patent. In t2 , he is exposed to a CL. A generic drug manufacturer obtains access to the Intellectual Property (IP) and enters as a competitor in that market. As the patent is still active, the patent-holder continues to own the IP and receives a royalty as a counterpart (e.g., Aoki & Small, 2004). In t3 , the patent expires. Firms then play a traditional price competition in the drug industry between a patent-holder and a generic manufacturer, as in Grabowski and Vernon (1992).3

2 “TRIPS Article 31(f)”. World Trade Organization. http://www.wto.org/english/ docs_e/legal_e/27-trips_04c_e.htm. Last accessed: January 21, 2019. 3 The modeling structure of our paper is influenced by the details of the 1995 TRIPS Agreement and the follow up 2001 Doha Ministerial Declaration on TRIPS and Public Health. These agreements enabled governments to allow (under certain

https://doi.org/10.1016/j.ejor.2019.10.021 0377-2217/© 2019 Elsevier B.V. All rights reserved.

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Fig. 1. Explored scenarios.

We compare the dynamic game described above with scenario B (summarized in Fig. 1). The comparison between these two scenarios allows us to identify the conditions under which the CL yields higher profits for the patent-holder. It additionally helps us evaluate, from the patent holder’s perspective, the trade-offs inherent in entering a market subject to the CL mechanism as well as the patient welfare. The latter is measured through sales, which uniquely describe the number of people who can afford to purchase the drug.4 Further, we model a stochastic game (Scenario S depicted in Fig. 1) in which the patent-holder enters the market with partial information about the issuance of a CL. In particular, the CL will be probably issued if customers (patients) cannot afford the patent-holder’s drug as it is priced too high. In fact, in many real world cases,5 a CL is initiated only when a small amount of people living in developing countries are able to afford or gain access to a certain drug during the first period.6 In this scenario, the patent-holder is somehow responsible for the presence of competition in the second period due to a CL. This way of linking the CL to the firms’ decisions is completely new in the literature, which instead mainly focuses on the threat of a CL and the choice of entry vs. voluntary licensing (e.g., Bond & Saggi, 2014). Our approach

conditions) another firm to produce a drug under patent protection without the consent of the patent holder. This legislation applies to all member countries of WTO. 4 This approach is quite common in health care research (see, for instance, Bokhari and Fournier, 2013; Chatterjee, Kubo, and Pingali, 2015; Duso, Herr, and Suppliet, 2014, where sales is a proxy of patient welfare). 5 For example, Efavirenz in Brazil (anti-retroviral drug), Tarceva (anti-cancer drug) in Thailand, Sorafenib (anti-cancer drug) in India, Sofosbuvir (anti-Hepatitis C Virus drug) in Malaysia etc. 6 “Declaration on the TRIPS agreement and public health”. World Trade Organization. http://www.wto.org/english/thewto_e/minist_e/min01_e/mindecl_trips_ e.htm. Last accessed: March 2, 2018.

builds on the literature on R&D management under incomplete information (e.g., Grishagin, Sergeyev, & Silipo, 2001). Our findings show that a sufficiently high royalty always makes the CL efficient. H invests more efforts in R&D, increases the price of his goods to exploit the monopolist position in t1 as much as possible and enjoys high royalty in the future periods. These directions persist according to consumers’ sensitivity towards H’s goods. High values of this parameter along with high royalty always lead to higher R&D and low prices under a CL. This package makes the CL socially sustainable in the first and the third periods, as a larger population can afford the drug when the CL is issued. However, in the second period, the presence of competition together with the royalty makes consumers insecure with respect to purchasing from either G or H; thus, their decisions mainly depend on their private sensitivity to the price. When the royalty is low, H finds the CL not economically convenient. However, for the society, CL remains an efficient mechanism as more people are able to afford the drug. For G, there are already few accrued economic benefits in the second period, though she has to pay a royalty fee to H. Thus, if the royalty is not adequately fixed, the CL can be socially sustainable and economically unfeasible at the same time. Finally, our findings show that the threat of a CL linked to a stochastic rule enlarges the area in which people have a better access to the drug, thus being more socially sustainable. The paper is organized as follows. In Section 2, we provide a literature review, identify the existing research gaps and highlight our contribution. In Section 3, we introduce the game under investigation and present the equilibria of the game in the deterministic scenario. We present the comparison of equilibrium quantities between scenarios D and B in Section 4. In Section 5, we introduce the stochastic scenario S and in Section 6 we compare the scenarios S and B (benchmark scenario). Section 7 concludes. An online appendix contains further results from our analyses and a presents set of experiments to assess the robustness of our findings.

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2. Literature review Our paper is related to the stream of literature that investigates the role of compulsory licensing in facilitating access to essential drugs for people living in developing countries. Tandon (1982) develops a static decision theory model in which a firm can be subject to CL. The fee corresponds to a fraction of the cost reduction that the patented firm experiences, which always leads to an increased social welfare. The decisions are modeled as based on pricing with the assumption (for simplicity) that the patent lasts infinitely; thus, the decisions are then analyzed at the steady state. Seifert (2015) evaluates innovation, pricing, production and compulsory licensing decisions at different stages albeit within a static framework. These decisions are taken by approximating a firm’s choice of hazard rate (that is, its instantaneous innovation probability, conditional on no firm having innovated up to that point) by its competitive threats. Licensing can be either voluntary or compulsory. It is voluntary when firms bargain on its amount, thus the fee is fixed between the maximum that the generic would offer and the minimum that the patented firm would receive. When licensing is compulsory, the fee is fixed by both firms at the same level. The finding is that compulsory licensing reduces industrywide innovation incentives. Aoki and Small (2004) model a static game with one patented firm and one generic producer, both producing and selling homogeneous goods in the market. The CL can be either fee-based or royalty-based. In the former case, the generic producer transfers a fixed and margins-independent fee to the patented firm. In the latter case, the generic producer transfers a royalty that depends on the quantity produced. When analyzing the Cournot equilibria, Aoki and Small (2004) find that a royaltybased mechanism is preferable. Further, using a Nash bargaining approach, they demonstrate that the static efficiency of both the CL mechanism can be replicated in a dynamic setting. However, they do not model any dynamics to demonstrate this result. Our paper fills this gap in the literature by considering a dynamic model of compulsory licensing. Contrary to Seifert (2015), Aoki and Small (2004) and Tandon (1982) who focus on static analyses, we develop a four-period model. Our motivations to model a four-period game are based on the dynamic effects of a CL, which evolve over time: The pre-sales period consists of product development. The CL cannot be issued in the first period of sales(lasting three years in real life), might be issued in the second period of sales (lasting from years 4 to 20), and cannot be issued in the third period of sales (after year 20) as the patent expires. Our analysis contributes to the literature by providing an investigation of economic and social value created according to the CL state. To our knowledge, this type of dynamic modeling has been fully disregarded by the literature. Bertran and Turner (2017) model a static game with different stages in which an inventor develops an invention at a certain fixed cost and sells the product in the market at a certain price. Later, the imitator enters the market through a two-part tariff CL mechanism, which is composed of a fixed part and a variable part. The latter depends on the imitator’s production rate. When the imitator enters the market, the innovator enjoys the royalty by fully stopping the production and sale of the branded drug. This particular form of royalty is very new in the literature. Nevertheless, it is also based on a rather simplifying assumption because pharmaceutical firms that spend high amounts in R&D seldom renounce their business in a market to allow a generic to be the leader (Bruce, 2003; Hess & Litalien, 2005). Relaxing the assumption made by Bertran and Turner (2017), we allow the patent-holder to be active in the market and compete against the generic while receiving royalty payments from it for the license. Finally, we also model the moment in which the patent expires and the royalty is not due any more.

3

Grabowski and Vernon (1992) study the implications of the Drug Price Competition and Patent Term Restoration Act, 1984 that facilitated generic entry upon patent expiration. Using econometric methods and data on products that witnessed patent expiration, the authors show how pioneers and generic manufacturers prices and market shares changed post generic entry. The empirical results of this paper underscore change in prices of drugs post generic entry. We use the insights to inform our model and set up a traditional pricing game between the branded and the generic firms in the final period. Our model differs from Grabowski and Vernon (1992) in two key aspects. First, we use game theory to study the implications of compulsory licensing issuance upon innovator and generic manufacturers prices and strategies. Grabowski and Vernon (1992) perform an econometric analysis to study the impact of the Drug Price Competition and Patent Term Restoration Act, 1984 (that facilitated generic entry) on pioneers’ and generic manufacturers’ prices and market shares post generic entry. Second, in our setup the innovator and the generic manufacturer can compete in the market even when the innovator’s patent is active, due to compulsory licensing. Grabowski and Vernon (1992) however consider generic entry upon patent expiration of the innovator. This is unlike the case in our paper where generic entry is the result of compulsory licensing. Bond and Saggi (2014) analyze the effect of compulsory licensing when a focal market government exercises some control on the price that a patent-holder firm fixes for that market. This approach ensures greater consumer access to the patented drug. However, since the government has a certain influence on the patent-holder’s price, the latter has to cede some control on its business in that market and is thereby confronted by the tradeoffs facing its market entry decision. Ramani and Urias (2015) compared CL to an alternative instrument-negotiation with patentholder for price reduction of the patented product. This alternative can be availed by governments of developing countries to improve local access of foreign manufactured and patent-protected drugs. They develop a game theoretic model and their findings suggest that the CL would not be issued under complete information. However, under incomplete information and sufficient local manufacturing capabilities or import access, a CL could be issued by the government of a developing country. Our model differs from those proposed in Bond and Saggi (2014) and Ramani and Urias (2015) since in our case, the patent-holder sets and exercises complete control on the price of the patented product. In addition, we model the dynamic competition between the patent-holder and a generic product producer according to the CL state. Stavropoulou and Valletti (2015) go a step forward by focusing on the developing country’s manufacturing ability in producing the generic drug of the patented product, as measured by manufacturing cost. When manufacturing costs are high, compulsory licensing has no effect on drug access because the threat of a CL issue is not credible. For intermediate manufacturing costs, the threat of compulsory licensing becomes credible and the patent-holder reduces the price. However, this does not increase the market access as the monopolist covers a small population. When manufacturing costs are very low, CL appears in equilibrium and access to the patented drug is the highest. Further, the authors find that the overall welfare effects of compulsory licensing are positive even when accounting for innovation effects, contingent on the commitment of the use of license only for the domestic market. Our paper introduces competition between the patent-holder and a generic producer after the CL’s issue. This allows us to deepen the analyses presented in Stavropoulou and Valletti (2015) by studying the effects of various levels of royalties as well as the elasticity of demand with respect to both the patented and the generic products. Avagyan, Esteban-Bravo, and Vidal-Sanz (2014) suggest the use of licenses as a strategy to speed up the sales and diffusion of new

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products. This is doable by commercializing the innovation and licensing the technology to one or more competitors. They show that licensing can be beneficial for the innovator, who increases its cash-flow through the royalties in spite of renouncing its monopolistic power. Avagyan et al. (2014) model a dynamic game of a classical supply chain. Differently, we consider the dynamics of CL within the context of the health care sector. In view of the preceding literature review, we summarize below the different elements of the research gap that we seek to address in this paper: 1. All published papers investigate compulsory licensing as a static phenomenon. However, compulsory licensing is clearly a dynamic phenomenon and should be studied as such. For example, in accordance with the provisions of law on compulsory licensing, the CL cannot be issued in the first three years following the patent grant; from year four to year twenty, the CL can be issued; after year twenty, the patent expires and the CL naturally ceases its effects. The literature misses a model that contains these dynamics. Therefore, we aim at developing a game theory model that captures this evolution. 2. All published papers till date focus on controls (e.g., pricing strategies) that do not consider the CL dynamics. Since compulsory licensing is a dynamic phenomenon, firms adjust their strategies based on the effects of the issuance of the CL in the preceding years. We seek to incorporate this consideration in a dynamic game model, wherein firms adjust their pricing and R&D strategies according to the CL state. 3. Extant work in this area has investigated the implications of CL without explicitly accounting for the factors driving that decision. Performance of the patent-holder in the market could be a driving factor. For example, by the Provisions of Law on compulsory licensing, a CL can be issued when the patented invention: (a) Does not satisfy the reasonable requirements of the public; (b) Is not available to the public at a reasonably affordable price; and (c) Is not worked within the territory. While the literature fully disregards these elements, we seek to develop a stochastic rule to capture the previous points (a) and (b). Specifically, a CL becomes active if and only if a minority of people in a certain market can access the patent-holder’s drugs in the first three years. 3. A dynamic game of compulsory licensing-deterministic and benchmark scenarios 3.1. Deterministic scenario In this section, we describe our dynamic game of compulsory licensing. Two firms engage in a price competition between two differentiated drugs to maximize their profits: a patent-holder (e.g., Bayer), referred to as firm H or he, selling his patented drug and a generic manufacturer (e.g., Natco Pharma), referred to as player G or she, who is granted a CL on H’s patent to manufacture and sell the generic version of the patented drug. Firm H enters the market knowing that a CL will be issued in the second period and aware of the royalty that the CL guarantees. It is worth noting that in our game framework, we leave aside any endogenous action from the regulator (government). This modeling choice is driven by several reasons. First, H is not subject to the CL in t1 and thus, in this period, the regulator would not have any role. Second, the government would have played no role in t3 as well, when the patent expires. Therefore, we seek to investigate a framework in which companies can decide whether to go and sell medicines in a country where the CL will be never issued, or where the CL can be issued according to some targets, or in countries in which the CL will be definitely issued. Several previous studies such as,

Table 1 Notations used. Notation

Description

i = H, G Z = D, B, S t = 1, . . . , 3 SiZt

Patent-holder and generic firm Deterministic, benchmark and stochastic scenarios Time Sales of firm i in time t and scenario Z Marginal profits of firm i in time t and scenario Z Marginal production cost of firm i H’s price in period t and scenario Z G’s price in period t and scenario Z Market potential Consumers’ sensitivity to H’s and G’s prices Consumers’ sensitivity to H’s R&D Discount factor R&D in scenario Z R&D investment efficiency Profits of firm i in time t and scenario Z Royalty Percentage of consumers who purchase from H in period t Percentage of consumers that H loses when G enters the market in period t Cumulative distribution function of sales with parameters a and b

πiZt ci pZt

ωtZ α βH, βG θ δ AZ l

Zit φ rt kt F S (x )

Table 2 Parameter values used in the numerical analysis.

α

θ

cH

cG

κ2

κ3

l

δ

3

0.2

0.1

0.1

1

1

1

0.9

Xiong, Zhou, Li, Chan, and Xiong (2013), Ma, Hu, Dai, and Ye (2018), and Wang, Zhang, Zhang, Bai, and Shang (2015), model the regulator’s actions exogenously; for example, the regulator provides some incentives to support some policies but without playing any role in the game. This allows for a genuine analysis of the firms’ strategies, profits and social outcomes and represents exactly the type of framework that we wish to implement. We will refer to this situation as the deterministic scenario and summarize the notation used in Table 1.7 Although the timing of actions in Fig. 1 shows that firm H’s investments in R&D8 and pricing occur in two distinct instants of time, namely t0 and t1 , the resulting decision problem is equivalent to a three-period game in which the patent-holder invests in R&D and sets the first-period price at the same instant of time, say in t1 . In fact, applying backward induction, one can observe that firms’ pricing decisions do not interact across time and that the only factor that links equilibrium quantities through time is firm H’s effort in R&D. According to this simplified scheme, in t = 1, H optimally sets the price for his patented product, pD1 , invests in R&D, AD , and incurs a constant marginal production cost, cH . Thus, πHD1 = pD1 − cH represents H’s marginal profits in t = 1. The sales in period 1 take the following form:





SHD1 pD1 , AD = α − βH pD1 + θ AD ,

(1)

where α is the market potential and shows the number of potential people who can be treated by H’s drug and are potentially interested in purchasing it; β H is the consumers’ elasticity to price and describes the change in the number of consumers with respect to changes in prices; θ is the consumers’ sensitivity to R&D that H invests for the development and production of the new drug, and shows the additional number of consumers who recognize the effectiveness of the drug reflected in the R&D. In the first period, 7 As far as notation is concerned, we use the superscript D to indicate that sales, strategies and profits are all linked to the deterministic scenario. 8 Throughout the paper, by effort in R&D we mean all the firm’s activities related to development, production and selling of the new drug.

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H invests in R&D and develops an IP that is protected through a patent. R&D costs take the classical quadratic form (e.g., Bernstein & Mohnen, 1998):

 D

C A

 2

=

l AD 2

High values of l signify that each dollar invested in R&D substantially penalizes H’s profits. Then, H’s first period profit function takes the following form:

   lAD2 DH1 = SHD1 pD1 , AD pD1 − cH − .

(2)

2

In t = 2, the government issues a CL, which grants G the right to produce the generic version of H’s product. G optimally sets the selling price ω2 , by playing a Nash game with H. With the issued CL, G can use H’s patent without investing in R&D. As counterpart, G pays a royalty to H, which takes exogenous values. Further, G faces a marginal production cost, cG , such that her marginal profits are πGD = ω2D − cG − φ . This marginal revenue entails a trade-off 2 for G, who can access H’s patent on the condition of a fixed royalty payment to H. Also, H is challenged to compete with G while receiving a royalty from G as a counterpart. The introduction of the generic product can erode the economic profits that H could have garnered from his patented product and we assume that the cannibalization effect is a function of the product prices and takes the following forms:







r2 pD2 , ω2D = 1 − k2 pD2 − ω2D







 D





SHD2 AD , pD2 , ω2D = SGD2 AD , pD2 , ω2 =

α r2 − βH pD2 + θ AD









SHD3 AD , pD3 , ω3D = SGD3 AD , pD3 , ω3D =

α (1 − r2 ) − βG ω2D + θ AD ,

 (A, p2 , ω2 ) =

D

A ,

pD2 ,

α r3 − βH pD3 + θ AD

(8)

α (1 − r3 ) − βG pD3 + θ AD

(9)

     DH3 AD , pD3 , ω3D = SHD3 AD , pD3 , ω3D pD3 − cH

(10)

     DH3 AD , pD3 , ω3D = SGD3 AD , pD3 , ω3D ω3D − cG ,

(11)

In sum, the firms’ objective functions over the entire planning horizon are given by:

DH = maxD DH1 + δ DH2 + δ 2 DH3

(12)

DG = max δ DG2 + δ 2 DG3 , D

(13)

AD ,pt

with δ being the discount factor. 3.1.1. Firms’ strategies The decision problem described above is solved by backward induction and leads to the following equilibrium: pD3 ∗ (AD∗ ) =

θ AD∗ (2βG + 3α k3 ) + ((βG + k3 α )(2α + 2cH (βH + α k3 ) + α k3 cG )) (4βH βG + k3 α (4(βG + βH ) + 3α k3 )) (14)

ω3D∗ (AD∗ ) 

θ AD∗ (2βH + 3α k3 ) + 2cG βG βH +k3 α ( (2cG (βH + βG ) + cH βH ) + α (2k3 cG + k3 cH + 1 )) (4βH βG + k3 α (4(βG + βH ) + 3α k3 ))

ω

D 2





ω − cG − φ . D 2

 (15)

(4) pD2 ∗ (AD∗ )



=

AD∗ θ (2βG + 3α k2 ) +(βG + α k2 )(2α + 2cH (βH + k2 α ) + 3αφ k2 + α k2 cG )

(4βH βG + k2 α (4(βG + βH ) + 3α k2 ))

 (16)

ω2D∗ (AD∗ )   θ AD∗ (2βH + 3α k2 ) + α 2 k2 + (2βH βG + k2 α (2βH + 2βG + 3α k2 ))φ +2cG (βH + α k2 )(βG + α k2 ) + cH k2 α (βH + α k2 ) = (4βH βG + k2 α (4(βG + βH ) + 3α k2 )) (17)

     DH2 (A, p2 , ω2 ) = φ SGD2 AD , pD2 , ω2D + SHD2 AD , pD2 , ω2D pD2 − cH



(7)

thus, the firms’ objective functions in the third period are:

=

SGD2



where r3 represents the fraction of the market potential that purchases a patented product over a generic good. As in t = 2, the attraction rate in the third period considers both the difference between the patented product price and the generic product price. Similarly, the sales functions also take a similar shape as in the second period, specifically:

(3)

where β G is the consumers’ sensitivity to the price of a generic drug. In Eqs. (3) and (4) we are assuming that: (1) firm H’s investments in product development has an impact on sales and (2) The impact of H’s R& D investment is identical on the two competitor. The first assumption is in line with the classical theory of product innovation (see, for instance, Aoki, 1991). The second assumption appears in line with the spirit of CL. Under a CL, the generic producer has full access to the patent. Consequently, she enjoys the same benefits as those of her competitor. G will entirely lose the market when pD2 = ω2D , the condition p2 < ω2 always applies to make the game feasible for G. The firms’ profit functions in the second period are:

D G2



ωt

r2 represents the fraction of potential consumers who seek to purchase from H, even when G operates in the market (De Giovanni & Ramani, 2018). The amplitude of r2 mainly depends on the price difference between patented and generic products. So, if pD2 = ω2D , consumers do not have any interest in purchasing from a generic. As discussed in Grabowski and Vernon (1992) and analyzed by Ching (2010), the price difference between patented and generic drugs is a key driver of consumers’ buying decision in the pharmaceutical industry. Thus, we assume that no consumer purchases from a generic if the patented drug is sold at the same price as that of the generic drug. k2 is a scaling parameter that represents the fraction of consumers that H loses when his price differs from G’s price. Therefore, the sales functions for both the patented and the generic drug in the second period take the following form:



In t = 3, H’s patent expires and the CL mechanism ceases to exist. Thus, G accesses H’s IP, competes with him but without paying the royalty any more. H’s attraction rate will then be given by:

r3 pD3 , ω3D = 1 − k pD3 − ω3D .

5

α + AD∗ θ + βH cH , 2βH

(5)

pD1 ∗ (AD∗ ) =

(6)

where AD∗ is the optimal R&D strategy that solves H’s problem (details on the computations are available in Appendix A.1). Although

(18)

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H has full information on the compulsory license, his decisions are very complex because the game is solved backward and R&D link the pricing strategies in all the three time periods. Interestingly, all strategies are positively influenced by R&D. Thus, firms can charge higher prices when H invests more in R&D, independently of the amplitude of competition (e.g., k2 and k3 ).

Proposition 3. H always sets a larger price than G. Proof. It suffices to compute the differences to get the result: pD2 − ω2D



2θ A(βG − βH ) + k2 α 2 (k2 (cH − cG ) + 1 ) + φ (k2 α (βG − 2βH ) − 2βH βG ) + βG α ( k 2 ( 2 c H − c G ) + 2 ) + k 2 β H α ( c H − 2 c G ) + 2 βG β H ( c H − c G )

=

Lemma 1. The amplitude of the R&D does not impact the amplitude of the price competition when βH = βG . ∂ pD ∂ pD θ 2β +3α k Proof. One can easily check that ∂ AD2 = ( G D 2 ) > 0, ∂ AD3 = DEN[ p2 ]

∂ωD ∂ωD 2β +3α k θ 2β +3α k > 0, ∂ AD2 = ( H D 2 ) > 0, and ∂ AD3 = H D 3 > 0, DEN[ω2 ] DEN[ω3 ] ∂ pD3 ∂ω3D ∂ pD2 ∂ω2D thus ∂ AD = ∂ AD and ∂ AD = ∂ AD when βH = βG .  2βG +3α k3 DEN[ pD ] 3

According to Lemma 1, the key parameters driving competition are consumers’ sensitivity to price in both the market segments. When β H < β G , G is in a weaker position and hence H can charge a larger price when investing more in R&D without losing his competitive position. As shown in Grabowski and Vernon (1992) and Frank and Salkever (1997), this is most likely the case in the pharma industry, in which consumers exert a lower willingness to pay for a generic drug. Exceptions are generic products manufactured by branded producers. Grabowski and Vernon (1992) show that these drugs can be priced considerably higher than the lowest price in that market category but can still obtain significant market share.

pD3

( 4 β H β G + k 2 α ( 4 ( β G + βH ) + 3 α k 2 ) )

−ω



=

D 3

2βG (α − βH cG + βH cH ) − 2Aθ (βH − βG ) + k 3 α ( α − c G ( 2 β H + βG ) + c H ( β H + 2 βG ) + k 3 α ( c H − c G ) ) 4 β H β G + 4 k 3 α ( β H + βG ) + 3 α k 3

Proof. Assume

for

simplicity

that

k2 = k3 ,

then

pD1 − pD2 =

α k2 (θ AD (3α k2 −2βH +4βG )+2βG (2α −βH (3φ +cG ))+k2 α (3α −βH (6φ +2cG +cH )) ) >0 2βH (4βH βG +4k2 α (βH +βG )+3α 2 k22 ) β + α k αφ k ( ) and pD2 − pD3 = 34β β +k Gα (4(2β +β 2 )+3α k ) > 0. Thus the relationship H G 2 G H 2

pD1 > pD2 > pD3 always holds.



This finding highlights the competitive environment in which H works. In t1 , when he is the monopolist in the market, he fixes a high price because consumers do not have any alternative to be treated. Further, he has heavily invested in R&D, so the higher price is justified by the needs of recovering that investment. In t2 , G enters the market with a compulsory license, pushing H for a price reduction. The competitive pressure is also followed by the presence of a royalty payment from G, which could help H offset his loss from the price reduction. Finally, H needs to substantially reduce the price in the third period because he no longer receives the royalty payment and decreasing the price is the only way to compete in the market place.

>0

 > 0.

 H is always able to price more in both periods, thus providing support to few of the findings in Grabowski and Vernon (1992) and Frank and Salkever (1997) according to which the patent-holder firm always prices more than a generic firm when operating in a segmented market of price sensitive and brand loyal consumers. This is due to two main factors. First, H has a competitive advantage as the originator of the drug. Thus, investing in R&D creates market awareness regarding his presence and enables him to satisfy consumers in the first period as well as compete against generics later. Second, H has to recover the investments in R&D. Hence, the overall convenience of entering into a certain market with a new and innovative drug pushes up the prices. D > SD and SD > SD always hold. Lemma 2. SH G H G 3

Proposition 1. H’s pricing strategies over the three periods compare ∗ ∗ ∗ as pD1 > pD2 > pD3 .



3

3

3

D − SD = Proof. Use the results in Proposition 3 to show that SH G 2     2 α − βH pD2 − ω2D > 0 and SHD3 − SGD3 = α − βH pD3 − ω3D > 0. 

Finally, Lemma 2 highlights the idea that H always gains higher market shares than G. This result can be explained by the fact that H is always recognized as the developer of a certain drug and hence continues to sell more even when generic alternatives are available in the market. 3.2. Benchmark scenario Our benchmark scenario is one in which a CL is not issued. For this scenario, we derive the firms’ optimal strategies, which leads to a classical pricing game between a brand holder and a generic. The profit functions are given as follows:9

BH = maxB BH1 + δ BH2 + δ 2 BH3 AB ,pt

BG = max δ 2 BG3 , B ω3

and the optimal strategies are given by: Proposition 2. G’s pricing strategies over two periods compare as ∗ ∗ ω2D > ω3D . Proof. Assume for simplicity that k2 = k3 , then ω2D − ω3D =

(

2 (βH +α k2 )(βG +α k2 )+α 2 k22

)φ > 0. Thus the relationship ωD > ωD al2 3

4βH βG +k2 α (4 (βG +βH )+3α k2 )

ways holds.



The presence of a compulsory license and the related royalty forces G to price more in the second period. This is mainly due to the economic convenience of exploiting the compulsory license. G transfers the royalty to H, which represents a pure cost for her. Thus, G’s pricing strategy is very much comparable to a traditional distribution channel in which a buyer transfers a certain wholesale price to the seller and charges the price accordingly (e.g., Cachon & Lariviere, 2005). Hereby, the logic is equivalent: G sets a lower price only in the last period when the royalty disappears.





pB3∗ AB∗ =

θ A B ∗ ( 2 βG + 3 α k 3 ) + ( ( βG + k 3 α ) ( 2 α + 2 c H ( βH + α k 3 ) + α k 3 c G ) ) , ( 4 β H β G + k 3 α ( 4 ( β G + βH ) + 3 α k 3 ) )

  ω3B∗ AB∗  B∗  θ A (2βH +3α k3 ) +2cG βG βH +k3 α ((2cG (βH + βG ) +cH βH ) + α (2k3 cG +k3 cH +1 )) = , ( 4 β H β G + k 3 α ( 4 ( β G + βH ) + 3 α k 3 ) ) B∗     α + A θ + βH cH pB1∗ AB∗ = pB2∗ AB∗ = , 2 βH

where AB∗ is the optimal R&D when the compulsory license is not issued (we refer to Appendix A.2 for all computational details). We avoid discussing the outcomes that we obtained in the benchmark scenario as they are aligned Bond and Saggi (2014).

9 We use the superscript B to indicate that profits and equilibrium quantities refer to the benchmark scenario.

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Table 3 D∗ B∗ for different triples (β H , β G , φ ). The equilibrium values for benchmark scenario are reported in Appendix C. The remaining The table shows the relative difference A A−A D∗ parameter values are those displayed in Table 2.

βH

φ = 1.3 βG

0.1 0.3 0.5 0.7

φ = 1.5

φ = 1.7

0.1

0.3

0.5

0.7

0.1

0.3

0.5

0.7

0.1

0.3

0.5

0.7

−93.2% −21.8% 0.1% 12.3%

−95.4% −24.0% −1.9% 10.6%

−97.2% −26.0% −3.7% 9.0%

−98.8% −27.7% −5.2% 7.6%

−89.9% −17.7% 4.3% 16.5%

−92.1% −19.9% 2.4% 14.8%

−94.0% −21.8% 0.6% 13.3%

−95.7% −23.5% −0.9% 12.0%

−86.7% −13.8% 8.2% 20.4%

−89.0% −16.0% 6.3% 18.7%

−91.0% −18.0% 4.6% 17.2%

−92.6% −19.7% 3.1% 15.9%

∗

∗

Fig. 2. The solid surface indicates where AD > AB .

4. Comparison between scenarios D and B In this section, we analyze the effects of a CL on firms’ strategies and profits as well as on drug access (patient welfare), by comparing equilibria in scenarios D and B. We perform this task by exploiting a mix of numerical analyses and analytical results. In the numerical analysis, we use the parameter values displayed in Table 2 and let the parameters that measure the elasticities of demand and the royalty vary in the following ranges: β H , β G ∈ [0, 1] and φ ∈ [0, 2]. These ranges as well as the baseline parameter values have been set according to our calibration performed on real world data that we detail in Appendix B. 4.1. R&D strategies



∗

∗



In Fig. 2, we identify the region DB = (βH , βG , φ ) : AD > AB in which firm H invests more in R&D when a CL is issued. Compulsory licensing makes firm H willing to increase his effort in R&D in a few cases, specifically when both the royalty and the consumers’ sensitivity to H’s product price (β H ) are sufficiently high. However, the consumers’ sensitivity to G’s price (β G ) plays a marginal role. While the literature shows that firms subjected to a compulsory license invest less in R&D (Stavropoulou & Valletti, 2015), we find contrasting results. First, Fig. 2 shows that high royalty guarantees some rewards to H, which could be attractive and incentivize him to increase his investments in R&D. Intuitively, a low royalty reduces H’s profit due to the price competition with G. Thus, H reduces the R&D accordingly. The presence of a compulsory license penalizes innovative drug development and consequently the chance to treat people. Governments and policy makers should be aware of this result and ensure that a sufficiently high royalty payment is fixed so as to enhance H’s R&D. The spirit of a CL is favoring the diffusion of a drug in a certain country by making it accessible through a competitive price, e.g., pG . Our results demonstrate that this mechanism is a necessary but not a sufficient condition

for diffusing treatments because the royalty amplitude plays a key role. Also, in the long-term there are effects on future innovative drug launches in the market (Bond & Saggi, 2014). This trade-off is offset only when a high royalty amount can be negotiated between the patent-holder and the generic manufacturer. However, it is worth noting real-world evidence, which suggests that royalties paid to the patent-holders are rather low (Scherer & Watal, 2002). Second, if a CL exists, H invests more in R&D additionally when consumer price sensitivity is high. This is quite convincing because H aims to compensate the negative price effect with a positive R&D effect: Consumers’ sensitivity to price can be mitigated when H produces very effective treatments. Interestingly, H spends more in R&D when the compulsory license exists. This comes from the pressure linked to competition, which exerts a negative impact on H’s business overall. Finally, the price sensitivity of G’s consumers plays a role only in a few cases, e.g., when the royalty is low vs. medium and consumers’ sensitivity to the price of H’s product is medium vs. high. In such circumstances, when H knows that the license will be surely issued, he has no need to invest more in R&D because consumers will most likely prefer to purchase from H. Table 3 gives a more detailed picture by reporting the relative variation of the optimal R&D in the Scenario D with respect to Scenario B. We observe that the three parameters have different abilities to impact on the amount of R&D. For instance, an increase of 0.2 of the value of β G impacts in a change of variation approximately between −2% and −3% while and increase of the value of the royalty of 0.2 implies a change in variation approximately between 2% and 4%. The effect of β H is much more dramatic. When the price sensitivity of G’s consumer is very low, H’s investment reduction in the deterministic scenarios is heavy. Our simulations report optimal R&D levels that are almost halved. In this cases, an increase of the royalty does not alleviate the negative impact of β H . Intuitively, very low levels of β H discourages investment in R&D since the share of customers that the firm can attract is not enough to compensate for the investment’s cost. This is especially true in the deterministic scenarios where price competition is stronger. However, a sufficiently high level of β H boosts investments: In such cases firm H’s desire is to challenge his competitor by offering a product of better quality, thus trying to capture a large market share. In such cases, the larger the royalty the stronger the challenge, as firm H might use the extra money to finance investment costs. 4.2. Pricing strategies Pricing and the R&D strategies are very much interlinked, as shown below. ∗

∗

∗

∗

Proposition 4. Inside DB , it results that pD1 > pB1 , pD3 > pB3 and ∗ ∗ ω3D > ω3B . Proof. Compute the difference between pD1 ∗

∗

that pD1 − pB1 =

(

AD∗ −AB∗ 2 βH

∗

∗

and pB1

to show

)θ ≥ 0 inside DB . Similarly, compute

the difference between pD3

∗

∗

and pB3

∗

∗

to show that pD3 − pB3 =

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∗

∗

Fig. 3. Solid surface indicates where pD2 > pB2 .

θ (2βG +3α k3 )(AD∗ −AB∗ ) ≥ 0 inside (4βH βG +k3 α (4(βG +βH )+3α k3 ))

difference between ω3D θ( )(2βH +3α k3 ) 4βH βG +k3 α (4 (βG +βH )+3α k3 ) AD∗ −AB∗

∗

and ω3B

∗

DB . Finally, compute the ∗ ∗ to show that ω3D − ω3B =

≥ 0 inside DB .



When the compulsory license plays a minor role (e.g., in t1 and t3 ), the pricing strategies are simply updated according to the R&D and set to balance positive and negative effects of a CL. In t1 , for the Scenario D, H knows that the CL will be issued, thus he accounts for this information and exploits fully his monopolist position. In t3 , both firms react to the patent expiration by increasing prices across all cases displayed in Fig. 2. Indeed, ∗ ∗ AD > AB signifies that drugs are of superior quality and effectiveness, thus a larger price for both firms is largely justified. This result is in line with prior work which suggests that generic entry leads patent holders to charge high prices (Frank & Salkever, 1997; Grabowski & Vernon, 1992). The finding that branded drug manufacturers increase drug prices upon generic entry is known as the “Generic Competition Paradox” (Scherer, 1993). It holds for a segmented market in the prescription drug industry that comprises of two consumer groups – 1. price sensitive and 2. quality sensitive (Frank & Salkever, 1992) and has been empirically tested and confirmed (Frank & Salkever, 1997). With generic entry and availability of generics, the price sensitive segment of consumers switches to generics. The result is that demand for the branded drug becomes more inelastic and this in turn enables the branded drug manufacturer to increase the prices. Brand loyalty accrued by the patentholder as first mover advantage (Regan, 2008) also enables it to charge high prices even in the face of increasing competition. Further, generic drugs are priced above the long-run marginal cost until further competitors enter the market (Reiffen & Ward, 2005). So, the CL mechanism reduces prices only in the medium term (e.g., t2 ).  In Fig. 3, we identify the region DB = {(βH , βG , φ ) : ∗ ∗ D B DB p2 > p2 } ⊂ . The presence of competition in the second period imposes a price reduction on H when a compulsory license is in place. There exists only a small region of parameters in which this result does not hold true, which mainly depends on the royalty: when the royalty is high, H sets a higher price to lose as many consumers as possible while gaining a high royalty. So, H suffers from the presence of a compulsory license in most of the cases and reacts by lowering his price. This finding is similar to that in Stavropoulou and Valletti (2015), who demonstrate that when compulsory license in a market is a credible threat, the patent-holder reduces the price of its drug in that market.

Fig. 4. The solid surface indicates where SHD2 > SHB 2 .

To get a feeling of the price change due to the introduction of the CL, we present in Table 4 the relative price variation in each time step. The introduction of a compulsory license has the effect of reducing prices only at time 2. This implies that the price reduction, although impressive especially at low levels of β H , is temporary. 4.3. Drug access The analysis of sales allows us to investigate people’s drug access, which reflects the findings obtained so far for pricing and R&D. D > SB , SD > SB and Proposition 5. Inside DB , it results that SH H H H 1

SGD > SGB . 3

1

3

3

3

D Proof. Compute the difference between SH D − SB = that SH H 1

1

θ (AD −AB ) 2

1

≥0

B and SH

1

and show

inside DB . Similarly, compute

D and SB the difference between SH H3 3 D −AB 2 β β + k α β +4 β +3 α k θ A ( H G 3 ( H ) ≥0 G 3 )) ( (4βH βG +k3 α (4(βG +βH )+3α k3 )) the difference between SGD and SGB 3 3 (2βH βG +k3 α (βH +4βG +3α k3 ))θ (AD −AB ) ≥ 0 (4βH βG +k3 α (4(βG +βH )+3α k3 ))

D − SB = and show that SH H 3

3

inside DB . Finally, compute and show that SGD − SGB = 3

inside DB .

3



In t1 and t3 , H sells more when a CL is definitely issued in t2 according to his R&D. Consequently, in the same periods, higher investments in R&D improve drug access. The comparison of firm H’ssales is presented in Fig. 4, where D > SB we identify the region DB = (βH , βg , φ ) : SH H 2

2

. When the

royalty is low, H’s sales in the second period are larger with a CL as firms substantially decrease their prices while the royalty only has a marginal effect. However, when the royalty is high, the amplitude of both β H and β G determines whether H sells more in t = 2 under a compulsory license regime. When consumers show a very large sensitivity to G’s price, they will most likely purchase from H. Since H knows that the greatest amount of consumers will purchase his product, he increases the price thus damaging the overall sales. Consumers do not purchase from G as they perceive her product to be too expensive for a generic product, and they do not purchase from H due to the high price. Under these circumstances, the compulsory license is marginally effective as the amount sold by H is lower than the case without CL. Note that when consumers’ sensitivity to price is low for both firms, which signifies that consumers weight more the positive drug effect than its price, H sells more in the second period in presence of a compulsory license,

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Table 4 ∗ ∗ pD −pB Each column shows ( i pD∗ i ), i = 1, 2, 3, for different triples (β H , β G , φ ). The equilibrium value for benchmark scenario are reported in Appendix C. The remaining parameter i

values are those displayed in Table 2.

βH

φ = 1.3 βG

φ = 1.5

φ = 1.7

0.1

0.3

0.5

0.7

0.1

0.3

0.5

0.7

0.1

0.3

0.5

0.7

0.1

−23.5% −990.8% −28.2%

−23.8% −1014.5% −28.2%

−24.0% −1035.5% −28.1%

−24.2% −1054.3% −28.0%

−22.9% −904.7% −27.6%

−23.2% −926.3% −27.6%

−23.5% −945.5% −27.5%

−23.7% −962.6% −27.4%

−22.4% −831.2% −26.9%

−22.7% −851.1% −26.9%

−23.0% −868.6% −26.9%

−23.2% −884.3% −26.9%

0.3

−2.5% −203.8% −3.1%

−2.7% −209.6% −3.3%

−2.9% −214.6% −3.5%

−3.0% −219.1% −3.6%

−2.1% −178.0% −2.6%

−2.3% −183.2% −2.8%

−2.5% −187.8% −3.0%

−2.6% −191.9% −3.2%

−1.7% −156.2% −2.1%

−1.9% −161.0% −2.4%

−2.1% −165.3% −2.6%

−2.3% −169.0% −2.7%

0.5

0.0% −89.3% 0.0%

−0.2% −92.5% −0.2%

−0.3% −95.4% −0.4%

−0.4% −98.0% −0.5%

0.4% −73.0% 0.5%

0.2% −76.0% 0.3%

0.1% −78.6% 0.1%

−0.1% −80.9% −0.1%

0.8% −59.3% 1.0%

0.6% −62.0% 0.7%

0.4% −64.4% 0.5%

0.3% −66.6% 0.3%

0.7

0.9% −43.2% 1.1%

0.7% −45.4% 0.9%

0.6% −47.5% 0.7%

0.5% −49.2% 0.6%

1.2% −30.8% 1.5%

1.1% −32.9% 1.3%

0.9% −34.7% 1.1%

0.8% −36.4% 1.0%

1.6% −20.4% 2.0%

1.4% −22.3% 1.7%

1.3% −24.0% 1.5%

1.1% −25.5% 1.4%

Table 5 Each column shows (

D B SH −SH i

i

D SH

), i = 1, 2, 3, for different triples (β H , β G , φ ). The equilibrium value for benchmark scenario are reported in Appendix C. The remaining parameter

i

values are those displayed in Table 2. Scenario D

βH

φ = 1.3 βG

φ = 1.5

φ = 1.7

0.1

0.3

0.5

0.7

0.1

0.3

0.5

0.7

0.1

0.3

0.5

0.7

0.1

−23.6% 13.2% −31.2%

−23.9% 8.1% −31.3%

−24.1% 3.3% −31.3%

−24.3% −1.3% −31.2%

−23.1% 13.3% −30.5%

−23.3% 7.9% −30.5%

−23.6% 2.6% −30.6%

−23.8% −2.4% −30.6%

−22.5% 13.5% −29.7%

−22.8% 7.6% −29.8%

−23.1% 1.9% −29.9%

−23.3% −3.5% −29.9%

0.3

−2.6% 18.8% −3.6%

−2.8% 13.9% −3.8%

−2.9% 9.3% −4.0%

−3.1% 4.8% −4.2%

−2.1% 18.5% −3.0%

−2.4% 13.0% −3.3%

−2.5% 7.8% −3.5%

−2.7% 2.8% −3.6%

−1.7% 18.1% −2.4%

−2.0% 12.1% −2.7%

−2.1% 6.3% −2.9%

−2.3% 0.6% −3.1%

0.5

0.0% 15.5% 0.0%

−0.2% 10.5% −0.2%

−0.3% 5.7% −0.4%

−0.4% 1.1% −0.6%

0.4% 14.6% 0.6%

0.2% 9.0% 0.3%

0.1% 3.5% 0.1%

−0.1% −1.7% −0.1%

0.8% 13.6% 1.1%

0.6% 7.4% 0.8%

0.4% 1.3% 0.6%

0.3% −4.8% 0.4%

0.7

0.9% 11.6% 1.3%

0.8% 6.5% 1.1%

0.6% 1.6% 0.9%

0.5% −3.1% 0.7%

1.3% 10.1% 1.8%

1.1% 4.3% 1.6%

1.0% −1.4% 1.3%

0.9% −6.8% 1.2%

1.6% 8.5% 2.3%

1.5% 2.0% 2.0%

1.3% −4.5% 1.8%

1.2% −10.8% 1.6%

independently of the royalty (Table 5). In sum, our findings provide support to prior literature that has underscored the role of appropriate royalty amount in creating social welfare (Aoki & Small, 2004; Bertran & Turner, 2017). We extend the literature by highlighting the important role of consumer price sensitivity in ascertaining whether the social welfare goals of the compulsory license mechanism are achieved. Lemma 3. SGD − SGB > 0 always holds as SGB = 0. 2

2

2

SGD 2

Note the positivity assumption > 0; thus, in all cases in which H sells more under a compulsory license, people have greater access to the drug, meeting thereby the compulsory license target of making the drug affordable for a larger population. 4.4. Profits Proposition 6. Inside DB , it results that D > BH H 3

BG . 3

3

and D > G 3

− Proof. Use the results in Propositions 4 and 6 to show that D H

BH ≥ 0 and DG − BG ≥ 0 inside DB .  3

3

3

3

Trade-off emerges between first and third period profits. When summing up Fig. 5(a) and (c), we obtain a full solid cube, leading to a trade-off between profits in t1 and t3 : when a compulsory

license is issued, H can optimize either the first or the third period profits, but not both at the same time (Stavropoulou & Valletti, 2015). Nevertheless, we numerically checked (see the Online D Appendix) that D H1 > H3 always holds. Thus H should prefer a compulsory license in all cases when the first period profits are improved due to the presence of this mechanism. G will have a preference for a compulsory license in the same cases when H is also economically better-off with a CL (the cube representing the B area D G3 > G3 is the same of the cube 5c). So, the presence of a compulsory license aligns the firms’ motivations in the third period, as they prefer the compulsory license to be issued under the same conditions. The area in which H is in favor of a compulsory license corresponds to the case in which the compulsory license makes all firms economically better-off. H’s second period profits are highly impacted by the presence of a compulsory license, which turns out to be a convenient mechanism in a small region where the royalty is very large, consumers’ sensitivity to H’s product is high (thus H makes the attempt to reduce the price) and consumers’ sensitivity to G’s price is low (then G attempts to increase the price in most of the cases). Overall, if H knows that a CL will be surely issued, he should aim for a policy to mitigate the negative effects of this mechanism. This policy will be composed of three key ingredients: high royalty, consumers’ high sensitivity to H’s price and low sensitivity to G’s price. First, H should aim towards negotiating a high royalty amount with G in t = 2. Second, H should convince the market in t1 that his product

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Table 6 D −B The table shows ( HD H ) for different triples (β H , β G , φ ). The equilibrium value for benchmark scenario are reported in Appendix C. The remaining parameter values are H

those displayed in Table 2.

βH

φ = 1.3 βG

0.1 0.3 0.5 0.7

φ = 1.5

φ = 1.7

0.1

0.3

0.5

0.7

0.1

0.3

0.5

0.7

0.1

0.3

0.5

0.7

−94.7% −30.1% −12.7% −4.7%

−98.2% −35.0% −18.4% −10.9%

−101.3% −39.7% −24.0% −17.3%

−104.2% −44.3% −29.6% −24.0%

−90.8% −26.3% −9.7% −2.5%

−94.9% −32.0% −16.3% −10.1%

−98.6% −37.6% −23.1% −18.1%

−102.0% −43.2% −30.1% −26.7%

−87.2% −22.9% −7.1% −1.0%

−91.8% −29.5% −14.8% −10.1%

−96.1% −36.1% −23.0% −20.0%

−100.1% −42.7% −31.7% −31.1%

findings extend the literature that has previously demonstrated the negative effects of compulsory license on the patent-holder’s future innovation efforts (Seifert, 2015). We show that CL could have additional negative implications for the patent-holder’s economic value appropriation in the short-term. 5. Stochastic scenario In this section, we relax the assumption that H has full information on the existence of a CL. Due to the lack of sufficient information, H faces a decision problem under uncertainty. We refer to this situation as a stochastic CL game and use the label Scenario S. We suppose that the government will not issue the compulsory license if the sales of the branded product in the first period, SH1 , will be at least equal to the sales target, say S˜.10 In our modeling framework, although H is aware that the government’s decision is based on realized sales in the first period, the precise value of S˜ is not publicly available. In other words, at time t = 1, H sees S˜ as a random variable. Let FS˜ (x ) ≡ Prob (S˜ ≤ x ) denote the cumulative distribution function of S˜. This distribution function may be interpreted as firm H’s belief about government’s target at time t = 1. In this setup, as seen at t = 1 by firm H, FS˜ (SH1 ) measures the probability that the CL will be issued and 1 − FS˜ (SH1 ) describes the probability that the compulsory license will not be issued. Thus, before knowing the government’s decision, H’s profits in the second period is actually a subjective expected profit given by: Fig. 5. The solid surfaces indicate where: (a) DH1 > BH1 ; (b) DH2 > BH2 ; (c) DH3 >

BH3 ; (d) DH > BH .

is of high quality to increase consumers’ willingness to purchase his product. Thus, in t1 , H can enjoy the monopolist position by setting a very high price. Later, in t2 , people will continue to purchase from G as the price is lower. To make this happen, H needs a third condition: consumers should be fully aware of the presence of a competitor and the lower price she charges. Thereby, consumers can perceive the difference between the two pricing policies and favor G’s business. People will purchase more from G while H will enjoy receiving the royalty (Table 6). From her side, G will always be happy to get the license, by means of which she can produce and sell her product already in period 2. In fact, her profits are always positive by construction as it is null in the case without compulsory licensing. From a drug access point of view, H sells more units under a compulsory license mechanism. Thus more people can access H’s drug and get a proper treatment at a fair price. Nevertheless, higher sales seldom lead to higher profits. Intuitively, competition destroys economic value and higher social performance does not pay back. Further, the presence of a compulsory licensing mechanism necessitates more R&D (Table 7). This highlights that a compulsory license mechanism may create social value in the shortterm through higher sales. However, it reduces the private value that the patent-holder can appropriate in terms of profits. Our

E [H2 ] = FS˜ (SH1 )(SH2 ( p2 − cH ) + SG2 φ )





+ 1 − FS˜ (SH1 ) SH2 ( p2 − cH )

(19)

while the profit functions in the first and the third periods remain the same as those of the Scenario D. The problem faced by H cannot be solved in closed form. Though in Appendix A.3 we characterize the equilibrium prices and R&D, we base our conclusions on numerical evidence based on the parametric setup presented in Table 2. We model H’s belief about government’s target as a log-logistic random variable, 1 with cumulative distribution function FS˜ (x ) = x −b , where we 1+( a )

fix a = 2 and b = 2. With this parametric setup, firm H ’s expected target level is E[S˜] = 3.14. The choice of this probabilistic model to represent H’s beliefs is motivated by the following. First, S˜ is positive. This excludes all probabilistic models with non-positive support. Second, the log-logistic distribution possesses the ability to model a wider variety of patterns, when compared to classical probabilistic model with positive support (exponential, log-normal, weibull, etc.). Third, the log-logistic distribution has been successfully used to model beliefs in decision problems (see for instance, Phan, 2013). We choose the value of the parameters a and b so that the patent-holder’s beliefs about the government’s target is realistic, when compared to the numerical setup we used. Nevertheless, 10 WIPO, Committee on Development and Intellectual Property (CDIP), Annex I and II (CDIP/5/4 “Provisions of Law on Compulsory Licensing”). http://www.wipo. int/edocs/mdocs/mdocs/en/cdip_5/cdip_5_4_rev-annex1_V2.pdf.

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11

Table 7 S∗ B∗ for different triples (β H , β G , φ ). The equilibrium values for benchmark scenario are reported in Appendix C. The remaining The table shows the relative difference A A−A S∗ parameter values are those displayed in Table 2. Scenario D

βH

φ = 1.3 βG

0.1 0.3 0.5 0.7

φ = 1.5

φ = 1.7

10.0%

30.0%

50.0%

70.0%

10.0%

30.0%

50.0%

70.0%

10.0%

30.0%

50.0%

70.0%

−10.6% −4.9% 2.5% 8.7%

−10.7% −5.0% 2.2% 8.4%

−10.7% −5.1% 2.1% 7.9%

−10.8% −5.1% 2.1% 47.5%

−10.5% −3.7% 4.4% 11.2%

−10.5% −3.8% 4.3% 11.0%

−10.6% −3.8% 4.2% 39.6%

−10.6% −3.8% 20.7% 64.5%

−10.4% −2.6% 6.4% 13.7%

−10.4% −2.6% 6.3% 14.2%

−10.4% −2.5% 6.2% 58.7%

−10.3% −2.3% 45.4% 73.2%

B∗ S∗ B∗ Fig. 7. The solid surfaces indicate where: (a) pS∗ 1 > p1 ; (b) p2 > p2 .

∗

∗

Fig. 6. The solid surface indicates where AS > AB .

we have performed a series of robustness analyses over a range of alternative parameter values and show that the relevant quantities are sufficiently insensitive to changes of those values. We report these numerical experiments in the online appendix. 6. Comparison between Scenarios S and B In this section, we investigate the effects of uncertainty on compulsory licensing. The objective of this analysis is to gain insights on how strategies, profits, and sales change when firms seek to establish their business in a country in which a CL will be probably issued based on the sales in the first period. 6.1. R&D strategies



∗

∗



In Fig. 6, we identify the region SB = (βH , βG , φ ) : AS > AB in which firm H invests more in R&D in the scenario S. When H faces the uncertainty linked to the issue of a compulsory license, he invests more in R&D as there is a chance to avoid it by encouraging sales in t1 . As mentioned, the royalty and the consumer’s sensitivity to H s product influence the R&D decisions: the higher they are, the higher the R&D. However, contrary to Fig. 2, high price sensitivity to G’s product (β G ) pushes H to raise his R&D: high β G reduces H’s willingness to invest in R&D as consumers will purchase H’s product anyway. Nevertheless, H has still a strong incentive to increase his R&D to reduce the probability of the compulsory license being issued. 6.2. Pricing strategies Differently from the findings displayed in Proposition 4, the presence of a stochastic rule linked to the compulsory licensing makes the pricing strategies less linked to the R&D. In fact, the decision maker has a chance to reduce the probability of a compul-

sory licensing being issued. Therefore, the first period price does not follow the R&D. We refer to Table 8 for the order of magnitude of the effect shown in Fig. 7. In t = 1, H has to solve the trade-off between exploiting his monopolist position and increase his first period sales to reduce the probability (the risk) that the compulsory li panel in Fig. 7 plots the region SB = cense is issued.S∗ The Bleft ∗

(βH , βG , φ ) : p1 > p1 . Intuitively, with low levels of royalty, the introduction of a compulsory license implies a substantial reduction of firm H’s profit in subsequent period. In this case, H has an incentive to increase the first period sales and reduce the risk of incurring the compulsory license. However, if the royalty is sufficiently high, H can also increase the price when the consumers’ sensitivity to G ’s product is large, as consumers prefer to purchase from H even in case of high price. The comparison between the second period prices in the DScenario and B-Scenario mainly reflects the comparison as in the right panel of Fig. 7; as inferred from the figures, the uncertainty linked to compulsory license considerably influences the first period pricing strategy while having no effect on the future pricing strategies. Following the findings in Proposition 4, the pricing and the R&D strategies are very much interlinked in the third period as the ∗ ∗ compulsory license does not play any role. Therefore, pS3 > pB3 and ∗ ∗ ∗ ∗ S B S B ω3 > ω3 in the same cases when A > A . ∗

∗

∗

∗

Proposition 7. Inside SB , it results that pS3 > pB3 and ω3S > ω3B . Proof. See the Proof of Proposition 4.



Firms’ pricing strategies in t = 3 follow the shape of H’s R&D displayed in Fig. 6. Consequently, any time H invests high R&D, the drug will be of superior quality and its higher price is always justified in the consumers’ eyes.

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Table 8 ∗ ∗ pS −pB Each column shows ( i pS∗ i ), i = 1, 2, 3, for different triples (β H , β G , φ ). The equilibrium value for benchmark scenario are reported in Appendix C. The remaining parameter i

values are those displayed in Table 2. Scenario D

βH

φ = 1.3 βG

φ = 1.5

φ = 1.7

0.1

0.3

0.5

0.7

0.1

0.3

0.5

0.7

0.1

0.3

0.5

0.7

0.1

−0.22 −8.88 −0.05

−0.22 −9.10 −0.05

−0.22 −9.30 −0.05

−0.23 −9.48 −0.04

−0.21 −8.19 −0.05

−0.22 −8.39 −0.04

−0.22 −8.57 −0.04

−0.22 −8.73 −0.04

−0.21 −7.58 −0.04

−0.21 −7.77 −0.04

−0.22 −7.93 −0.04

−0.22 −8.08 −0.04

0.3

−0.11 −2.01 −0.01

−0.12 −2.06 −0.01

−0.13 −2.11 −0.01

−0.14 −2.16 −0.01

−0.10 −1.76 −0.01

−0.11 −1.81 −0.01

−0.13 −1.85 −0.01

−0.14 −1.89 −0.01

−0.08 −1.55 0.00

−0.10 −1.59 0.00

−0.12 −1.63 0.00

−0.13 −1.67 0.00

0.5

−0.05 −0.89 0.00

−0.07 −0.92 0.00

−0.09 −0.95 0.00

−0.10 −0.97 0.00

−0.04 −0.73 0.00

−0.06 −0.76 0.00

−0.08 −0.78 0.00

−0.08 −0.79 0.03

−0.02 −0.59 0.01

−0.05 −0.62 0.01

−0.08 −0.64 0.01

−0.02 −0.62 0.08

0.7

−0.01 −0.43 0.01

−0.04 −0.46 0.01

−0.06 −0.48 0.01

−0.02 −0.46 0.06

0.00 −0.31 0.01

−0.03 −0.33 0.01

−0.02 −0.33 0.05

0.04 −0.31 0.12

0.01 −0.21 0.01

−0.03 −0.23 0.01

0.03 −0.20 0.10

0.09 −0.18 0.17

6.3. Drug access When comparing the sales in the S − Scenario and the B − Scenario in t1 , we obtain the same qualitative results as in the left panel of Fig. 7. Note that this result is less sensitive to the consumers’ price sensitivity to H ’s product and the royalty. For example, medium levels of price sensitivity and royalty values are sufficient to guarantee larger sales in any period under stochastic compulsory license. S ∗ > S B∗ S ∗ > S B∗ Claim 1: SH and SH holds for all triples H H 2

2

3

3

(β H , β G , φ ). Interestingly, in Scenario S, H will always sell more than in S∗ > SB∗ and the case without compulsory license. Consequently, SH H ∗

∗

3

3

2

2

S > SB always hold for any triple (β , β , φ ). This is mainly SH H G H

due to the combination of higher R&D and lower prices. When the compulsory license is an uncertain event, H’s sales are not impacted at all. In contrast, H finds more effective strategies to ensure larger sales. These results suggest that whenever there is a threat of a compulsory license, H’s sales are positively influenced as compared to a situation when the compulsory license is certain. In fact, H pushes for solving the compulsory licensing issue as early as possible through the first period sales. These results are in line with prior work that suggests that a threat of compulsory license may be an effective mechanism to create desired social value through increased sales ((Bond & Saggi, 2014). When His threatened by an uncertain CL mechanism, more people gain access to H’s drug owing to lower price in most of the cases (Table 9).

Fig. 8. The solid surfaces indicate where: (a) SH1 > BH1 ; (b) SH2 > BH2 ; (c) SH3 >

BH3 ; (d) SH > BH .

6.4. Profits Claim 2: Inside SB , it results that SH > BH , while SH >

BH



3

3

2

holds inside SB . 2 Profits’ comparison displayed in Fig. 8 shows a certain trade-off between the sales and related profits in the first period (Fig. 8(a)). In t = 1, H sells more to avoid the compulsory license; nevertheless, these larger sales are not economically viable (Table 10). The higher profits that H can get in the second and the third period due to a stochastic compulsory license are not sufficient to cover the lower profits generated in the first period due to low margins and high R&D expenses. In fact, Fig. 8(d) shows that H gains more under a stochastic compulsory license only in a very small area where the royalty should be fixed at a very high value and consumers’ sensitivity to price for H’s product is very high.

Overall, this finding highlights the contrasting outcomes that H obtains when comparing social and economic performance. From a social point of view, the uncertainty surrounding the compulsory license pushes H to do better; the related optimal strategies are socially sustainable. From an economic perspective, the presence of a stochastic compulsory license seldom leads to value creation. Thus, the threat of a compulsory license does not resolve the trade-off between social value creation through increased sales and private value appropriation through higher economic profits. Previous studies have found a negative effect of a compulsory license threat on innovation incentives of the patent-holder (Bond & Saggi, 2014). We additionally identify negative effects of a compulsory license threat on the patent-holder’s profits.

Please cite this article as: A. Sarmah, D. De Giovanni and P. De Giovanni, Compulsory licenses in the pharmaceutical industry: Pricing and R&D strategies, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.10.021

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A. Sarmah, D. De Giovanni and P. De Giovanni / European Journal of Operational Research xxx (xxxx) xxx Table 9 Each column shows (

S B SH −SH

i

i

S SH

13

), i = 1, 2, 3, for different triples (β H , β G , φ ). The equilibrium value for benchmark scenario are reported in Appendix C. The remaining parameter

i

values are those displayed in Table 2. Scenario D

βH

φ = 1.3 βG

φ = 1.5

φ = 1.7

0.1

0.3

0.5

0.7

0.1

0.3

0.5

0.7

0.1

0.3

0.5

0.7

0.1

9.3% 30.9% −5.0%

9.5% 27.3% −4.9%

9.6% 24.0% −4.9%

9.8% 20.8% −4.9%

9.1% 30.7% −4.9%

9.3% 26.9% −4.9%

9.5% 23.3% −4.8%

9.7% 19.8% −4.8%

8.9% 30.5% −4.9%

9.2% 26.5% −4.8%

9.4% 22.6% −4.8%

9.6% 18.8% −4.7%

0.3

8.0% 21.1% −0.9%

8.8% 16.6% −0.9%

9.6% 12.3% −0.9%

10.2% 8.3% −0.9%

7.4% 20.4% −0.7%

8.4% 15.5% −0.7%

9.4% 10.7% −0.7%

10.2% 6.1% −0.7%

6.8% 19.7% −0.5%

8.1% 14.3% −0.5%

9.3% 9.0% −0.5%

10.3% 3.9% −0.4%

0.5

5.0% 15.8% 0.3%

6.5% 11.0% 0.3%

7.8% 6.5% 0.3%

9.0% 2.1% 0.2%

4.1% 14.6% 0.6%

6.1% 9.2% 0.5%

7.8% 4.1% 0.5%

10.7% 2.2% 3.0%

3.4% 13.4% 0.8%

5.9% 7.4% 0.8%

7.9% 1.5% 0.8%

14.3% 6.7% 8.9%

0.7

2.6% 11.2% 0.9%

4.8% 6.2% 0.8%

6.7% 1.4% 0.8%

12.0% 5.4% 7.3%

1.9% 9.4% 1.2%

4.7% 3.7% 1.1%

9.4% 4.0% 5.5%

16.0% 10.3% 13.7%

1.4% 7.6% 1.5%

4.9% 1.2% 1.5%

12.8% 8.3% 11.2%

19.6% 14.7% 19.3%

Table 10 D −B The table shows ( HD H ) for different triples (β H , β G , φ ). The equilibrium values for benchmark scenario are reported in Appendix C. The remaining parameter values are H

those displayed in Table 2.

βH

φ = 1.3 βG

0.1 0.3 0.5 0.7

φ = 1.5

φ = 1.7

0.1

0.3

0.5

0.7

0.1

0.3

0.5

0.7

0.1

0.3

0.5

0.7

−27.5% −14.9% −7.2% −2.9%

−28.2% −16.9% −10.1% −6.4%

−28.8% −18.8% −12.8% −9.8%

−29.3% −20.6% −15.4% −19.1%

−26.7% −13.2% −5.5% −1.6%

−27.5% −15.7% −9.0% −5.9%

−28.2% −17.9% −12.3% −12.6%

−28.8% −20.1% −16.1% −46.5%

−25.9% −11.7% −4.1% −0.7%

−26.8% −14.6% −8.3% −6.0%

−27.6% −17.3% −12.3% −26.3%

−28.4% −19.8% −23.8% −140.3%

7. Conclusions and future research directions The compulsory license mechanism can be an effective tool to enhance the diffusion of a drug in developing countries. Our study shows that its adoption favors sales, signifying that a greater portion of people has access to the drug. This result is mainly led by the R&D that a patent-holder invests in developing, producing and selling a new drug in a developing country. When the royalty that the patent-holder receives is sufficiently high, he has an incentive to invest more in R&D. Interestingly, when the royalty is sufficiently large, the presence of a competitor favors the patentholder who diverges his interest from selling his products to boosting the generic’s sales and enjoying the royalty. Further, the patentholder would operate in a market where consumers are highly sensitive to its price rather than to its competitor’s price: Once again, this allows the patent-holder to enjoy the royalty. From our study, it clearly emerges that consumers are reluctant to purchase from the generic even when the patent-holder charges a high price and receives a royalty. This happens most likely because the patent-holder acquires a certain IP through R&D, and this becomes common knowledge. Thus, people recognize that the patent-holder is the incumbent, they can trust its products and a larger price does not necessarily discourage people from purchasing its drug. In fact, our results show that, when a compulsory license is active, consumers mainly decide according to the difference in their sensitivity to the patent-holder’s and the generic’s product (period 2). However, when the compulsory license is not active (periods 1 and 3), they decide their purchases according to the R&D (and then the drug effectiveness). Although the compulsory license fulfills its main objective to make drugs more affordable for people, its implementation seldom leads to improved economic performance for the patentholder. As expected, the presence of a competitor hurts its business profitability. At the same time, high royalty is a necessary but not a sufficient condition to make the patent-holder economically better off. We identified a small region in which the

patent-holder’s profits improve with the implementation of a compulsory license, which consists not only of high royalty, but also of consumers’ high sensitivity to the price of the patent-holder’s drug and low sensitivity to the price of the generic’s drug. In sum, consumers must be very sensitive to price and be willing to shift from purchasing patented drugs to purchasing generic drugs. In such a case, the patent-holder makes the drug formula appealing for people by investing more in R&D while pushing consumers to purchase from a generic at the same time. By doing so, the patent-holder protects his business as the royalty sets a lowercase limit below which the generic drug price cannot go. Thus, the larger the royalty is, the lower the generic’s capability to fix a low price to attract consumers to its portfolio. In fact, when the compulsory license in the third period is not active any more (the royalty is not due anymore), firms compete a classical pricing game. Further, we characterize a stochastic compulsory license mechanism, in which the patent-holder is responsible for a compulsory license being issued. In particular, we model a decision rule according to which the probability that a compulsory license is issued depends on the cumulative sales in the first period. This way of modeling the compulsory license mechanism is in line with the real motivations that push governments to issue a compulsory license. For example, a compulsory license is activated when the patent-holder prices a drug considerably high, thus limiting people’s access to the treatment. When the realized sales are not satisfactory, governments have the option to issue the compulsory license. In our model, the larger the sales in the first period are, the lower the exposure to a compulsory license is. We show that the patent-holder spends more in R&D and prices less in the first period to avoid the compulsory license. Since under a stochastic rule the patent-holder is responsible for the compulsory license activation, it can better set its strategies to mitigate the related negative consequences. We demonstrate that a stochastic rule is more socially feasible and more economically viable than the certain event linked to the compulsory license being issued.

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Finally, we compare the compulsory license game equilibria to the stochastic compulsory license equilibria to show the impact of uncertainty on strategies and profits. We identify a region in which the patent-holder invests more in R&D and charges a lower price to avoid the compulsory license. This implies lower profits in the short-run (first period) but higher profits in the long-run (subsequent periods). In contrast, in the complementary region, the patent-holder gives up as it is too costly trying to raise the first period sales. In this region, we observe a substantial loss of both social welfare (low number of people with access to the drug) and economic welfare (lower profits). In conclusion, this research highlights the trade-offs inherent in the compulsory license mechanism. It is however not free of limitations that we mention here to inspire future research in the same field. We model a compulsory license mechanism that is issued in the second period and it is not revised in the future. In reality, after a compulsory license is issued, governments can step back and withdraw the compulsory license or the compulsory license term may expire before the patent expiration. Future research could account for this by considering an intermediate stage where the patent-holder regains its monopoly status and retains it until its patent expires. Next, a patent-holder can have multi-project objectives when planning to sell drugs in a certain developing country, which we do not account for in this paper. Future work can introduce this assumption and set up a game where a patent-holder can adjust the introduction of new drugs in the same country according to the past outcomes of compulsory licensing in the same



AD

∗

To find the optimal solution of the game, we solve the game backward by starting from the third period. The firms’ strategies in t = 3 are given by: pD3 =

ω3D =



θ AD (2βG + 3α k3 ) + ((βG + k3 α )(2α + 2cH (βH + α k3 ) + α k3 cG )) (4βH βG + k3 α (4(βG + βH ) + 3α k3 )) θ AD (2βH +3α k3 ) +2cG βG βH +k3 α ((2cG (βH + βG ) +cH βH ) + α (2k3 cG +k3 cH +1 )) (4βH βG +k3 α (4(βG + βH ) +3α k3 ))



These strategies do not have any direct interface with the second period strategies. Thus, plugging them into the firms’ objective functions and maximizing for the second period strategies gives:

 pD2 =



AD θ (2βG + 3α k2 ) + (βG + α k2 )(2α + 2cH (βH + k2 α ) + 3αφ k2 + α k2 cG )

 ω2D =

( 4 βH βG + k 2 α ( 4 ( βG + βH ) + 3 α k 2 ) )

θ AD (2βH + 3α k2 ) + α 2 k2 + (2βH βG + k2 α (2βH + 2βG + 3α k2 ))φ +2cG (βH + α k2 )(βG + α k2 ) + cH k2 α (βH + α k2 ) ( 4 βH βG + k 2 α ( 4 ( βG + βH ) + 3 α k 2 ) )



Substituting these strategies in H’s objective functions, results in the following pricing strategy for H in the first period:

pD1 =

α + AD θ + βH cH 2βH

and AD is the optimal R&D that solves H’s problem and is given by:



−G1 (α − βH L1 )−(θ − βH G1 )(L1 − cH )−δφ (θ − F2 βG + α k2 (−F2 + G2 ) ) −δ G2 (−βH L2 + α (−k2 (L2 − D2 ) + 1 ) )−δ (L2 − cH )(θ − βH G2 − α k2 (−F2 + G2 ) ) 2 2 −δ G3 (−βH L3 + α (−k3 (L3 − D3 ) + 1 ) )−δ (L3 − cH )(θ − βH G3 − α k3 (−F3 + G3 ) ) = −l + 2G1 (θ − βH G1 ) + 2δ G2 (θ − βH G2 − α k2 (−F2 + G2 ) ) + 2δ 2 G3 (θ − βH G3 − α k3 (−F3 + G3 ) )

(A.1)

where: country. We also assume that the government does not play an active role in this game. Future research can consider the government as an additional player that maximizes a certain welfare function. In such a case, we can formulate a game in which the government plays an important role not only in the decision of compulsory license issuance but also in the eventual royalty negotiation process between the patent-holder and the generic manufacturer. Finally, we make the firms maximize their profit functions and show that the maximization of social outcomes does not necessarily match with the maximization of the economic rewards. Future research can develop multi-criteria objective functions to simultaneously maximize social and economic performance.

α + βH cH θ (2βG + 3α k2 ) , G2 = , 2βH (4βH βG + k2 α (4(βG + βH ) + 3α k2 )) θ (2βG + 3α k3 ) (βG + α k2 )(2α + 2cH (βH + k2 α ) + 3αφ k2 + α k2 cG ) L2 = , G3 = , (4βH βG + k2 α (4(βG + βH ) + 3α k2 )) (4βH βG + k3 α (4(βG + βH ) + 3α k3 ))

G1 =

L3 =

θ

2βH

, L1 =

θ (2βH + 3α k2 ) ( (βG + k3 α )(2α + 2cH (βH + α k3 ) + α k3 cG )) , F2 = , (4βH βG + k3 α (4(βG + βH ) + 3α k3 )) (4βH βG + k2 α (4(βG + βH ) + 3α k2 ))



 α 2 k2 + (2βH βG + k2 α (2βH + 2βG + 3α k2 ))φ + 2cG (βH + α k2 )(βG + α k2 ) + cH k2 α (βH + α k2 ) , (4βH βG + k2 α (4(βG + βH ) + 3α k2 )) θ (2βH + 3α k3 ) F3 = , (4βH βG + k3 α (4(βG + βH ) + 3α k3 )) (2cG βG βH + k3 α ((2cG (βH + βG ) + cH βH ) + α (2k3 cG + k3 cH + 1 )) ) D3 = . (4βH βG + k3 α (4(βG + βH ) + 3α k3 )) D2 =

Substituting the first period strategies in the second and third period strategies as well as in the firms’ objective functions, we get the optimal strategies as well as the optimal profits ∗H and ∗G .

Appendix A. Equilibria A.1. Equilibria in the Scenario D

A.2. Equilibria in the Scenario B

When a compulsory licensing is issued, the firms’ objective functions turn out to be:

When a compulsory licensing is not issued, the firms’ objective functions turn out to be:

⎛

⎞   D2 α − βH pD1 + θ AD pD1 − cH − lA2      ⎟ ⎜ ⎟ ⎜ + δ α 1 − k2 pD2 − ω2D − βH pD2 + θ AD ⎟ DH = ⎜         ⎟ ⎜ D 2 D D D D D ⎝ p 2 − c H + δ α 1 − k 3 p 3 − ω 3 − βH p 3 + θ A p 3 − c H ⎠   D   +δ α k2 p2 − ω2D + θ AD − βG ω2D φ        δ 2 α k3 pD3 − ω3D + θ AD − βG ω3D ω3D − cG D G =      +δ α k2 pD2 − ω2D + θ AD − βG ω2D ω2D − cG − φ 

⎛

   lAB2 α − βH pB1 + θ AB pB1 − cH − 2 ⎜    BH = ⎜ +δ α 1 − k2 ( pB2 − ω2B − βH pB2 + θ AB )( pB2 − cH ) ⎝       +δ 2 α 1 − k3 pB3 − ω3B − βH pB3 + θ AB pB3 − cH      BG = δ 2 α k3 pB3 − ω3B + θ AB − βG ω3B ω3B − cG

⎞ ⎟ ⎟ ⎠

To find the optimal solution of the game, we solve the game backward by starting from the third period. The firms’ strategies in t3 are given by:

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15

θ AB (2βG + 3α k3 ) + ((βG + k3 α )(2α + 2cH (βH + α k3 ) + α k3 cG )) (4βH βG + k3 α (4(βG + βH ) + 3α k3 ))  B  θ A (2βH + 3α k3 ) + 2cG βG βH + k3 α ((2cG (βH + βG ) + cH βH ) + α (2k3 cG + k3 cH + 1 )) B ω3 = (4βH βG + k3 α (4(βG + βH ) + 3α k3 )) pB3 =

These strategies do not have any direct interface with the second period strategies. Thus, plugging them into the firms’ objective functions and maximizing for the second period gives the following price for H:

pB2 =

α + AB θ + βH cH 2βH

Substituting these strategies in H’s objective functions, H’s pricing strategy in the first period is given by:

pB1 =

α + AB θ + βH cH 2βH

and AB is the optimal R&D to be obtained as the procedure above. Substituting the first period strategies in the second and the third period strategies as well as in the firms’ objective functions, we get the optimal strategies as well as the optimal profits BH∗ and BG∗ .

2. For each drug, we searched for the price of the patented drug as well as the total sales by using the website http://www. pharmacompass.com/. From the same website, we also identified the name of the branded company producing the drug. 3. To obtain the R&D of the branded company, we use the firms’ names in Step 2 and searched for the firms’ R&D on the website www.statista.com. 4. We used several sources to derive the royalty applied for each drug. From data collected in Table 11, we derived the benchmark parameter values by proceeding as follows: 1. We set the demand parameter values α , β H and θ linked to our demand function D1 = α − βH p + θ A and, after normalizing the data, we estimated the following regression model:

A.3. Equilibria in the Scenario S

H H    SalesH 1 = α1 − βH1 price1 + θ1 R&D + ε1

We proceed by backward induction. Let AS denote the optimal R&D effort in the S-Scenario. We have:

where ε1H is the error. From our regression analysis, it results  = 0.39 and θ = 0.36.  = 2.4, β that α H

(B.1)

1. At time t = 3 the firms’ equilibrium strategies are

θ AS (2βG + 3α k3 ) + ((βG + k3 α )(2α + 2cH (βH + α k3 ) + α k3 cG )) (4βH βG + k3 α (4(βG + βH ) + 3α k3 ))  S  θ A (2βH + 3α k3 ) + 2cG βG βH + k3 α ((2cG (βH + βG ) + cH βH ) + α (2k3 cG + k3 cH + 1 )) ω3S = (4βH βG + k3 α (4(βG + βH ) + 3α k3 )) pS3 =

2. At time t = 2, if the compulsory license has been issued, then the firms’ equilibrium strategies are



pS2

=



AS θ (2βG + 3α k2 ) + (βG + α k2 )(2α + 2cH (βH + k2 α ) + 3αφ k2 + α k2 cG )



ω2S =

(4βH βG + k2 α (4(βG + βH ) + 3α k2 ))

θ A (2βH + 3α k2 ) + α 2 k2 + (2βH βG + k2 α (2βH + 2βG + 3α k2 ))φ +2cG (βH + α k2 )(βG + α k2 ) + cH k2 α (βH + α k2 ) (4βH βG + k2 α (4(βG + βH ) + 3α k2 )) S

3. At time t = 2, if the compulsory license has not been issued, then H s optimal pricing strategy is

pS2 ==

α + AS θ + βH cH ; 2βH

4. Finally, at time t = 1, H  s optimal pricing strategy, denoted pS1 , solves the following implicit equation:

    ¯H =0 α + AS θ + βH cH − 2βH pS1 − δβH f S1H pS1 , AS H2 −  2

where f(·) is the density function associated with the cumulative distribution function of the patent-holder’s beliefs, and H 2 ¯ H ) are the optimal profits in the second period in case the ( 2 compulsory license has been issued (has been not issued). Appendix B. Parameters calibration This appendix provides specific details about the data used and the criteria applied when selecting the parameter value in used in this paper throughout. We use the following procedure: 1. Data regarding drugs subject to CL was collected starting from Hill and Pozniak (2016), who give details on the lowest prices of generic drugs.



2. We set the demand parameter value for β G by investigating the demand functions in the second period. To do so, we first need to set the parameter k2 , which represents the link between H’s and G’s prices in determining consumers’ willingness to purchase from H and G respectively. We measure this link through the correlation between ω2 and p2 , which turns out to be k2 = 0.93. Using this parameter value, we estimate the demands in the second period: H H    SalesH 2 = α2 r2 − βH2 price2 + θ2 R&D + ε2

(B.2)

 priceG + θ R&D + ε G 2 (1 − r2 ) − β SalesG2 = α 2G 2 2 2

(B.3)

ε2i

where r2 = 1 − k2 ( p2 − ω2 ) and with i = H, G being the regression error (Tables 12–15). The previous regression gives  = 0.45, β  = 0.36 and θ = 0.18. The same type 2 = 3.1, β α H2 2G 2 of analysis is conducted for the third period. Considering all estimates, we calibrate the demand parameter values accordingly, setting α = 3, βH = 0.5, βG = 0.4, θ = 0.2 and ki = 1. 3. The royalty has been estimated by using the information linked to each drug we analyzed and the royalty applied in some instances. In Table 11, we report some sources linked to each

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Table 11 Data used for calibration purposes. Drug

Sales (106 kilogram)a

Average price (USD)

Global lowest for genericb

R&D(106 USD)c

Royalty (%)

Source for royalty

Abacavir Lamivudine Tenofovir Zidovudine/3TC Nevirapine Efavirenz Rilpivirine Etravirine Atazanavir Lopinavir/r Darunavir/r Dolutegravir Raltegravir Elvitegravir

14,481 1,946,247 774,305 608,049 459,669 459,669 38 71 14,476 6531 31,310 1464 734 203

554 92 14,464 10,536 187 120 3815 3704 946 532 1309 2382 3259 28,137

123 18 3,9 4,6 28 38 0,4 0,438 219 243 0,658 0,6 0,973 4,904

4980 4476 3734 4476 3078 4820 1848 660 3900 3900 2590 4476 7560 4980

4 2 5 4 2,5 5 5 5 3 1 4 5 1 4

https://www.keionline.org/22041 Viljoen and Precious (2007) Bond and Saggi (2014) Love (2007) http://www.cptech.org/ip/health/cl/recent-examples.html Bond and Saggi (2014) Beyer (2013) https://khn.org/morning-breakout/dr00050350/ Cherian (2016) Bond and Saggi (2014) Raju (2017) http://www.ip-watch.orgd www.bardehle.com e Cherian (2016)

a

Source: http://www.pharmacompass.com/. Source: Hill and Pozniak (2016). Latest R&D expenditure in billions of USD of the major drug manufacturer. Source:www.statista.com. d http://www.ip-watch.org/2017/05/24/beyond-obvious-direct-indirect-territorial-coverage-mppviiv-voluntary-license-dolutegravir/ e http://www.bardehle.com/ip- news- knowledge/ip- news/news- detail/german- federal- court- of- justice- confirms- the- compulsory- license- granted- by- way- of- apreliminary-inju.html b c

Table 12 ∗ The table shows AB for different triples (β H , β G , φ ). The remaining parameter values are those displayed in Table 2.

Table 15 The table shows B for different triples (β H , β G , φ ). The remaining parameter values are those displayed in Table 2.

βG

βG βH

0.1

0.3

0.5

0.7

βH

0.1

0.3

0.5

0.7

0.1 0.3 0.5 0.7

9.79 2.41 1.41 1.01

9.75 2.39 1.40 1.00

9.71 2.38 1.39 1.00

9.68 2.37 1.38 0.99

0.1 0.3 0.5 0.7

72.24 17.33 9.97 7.03

72.01 17.27 9.93 6.99

71.83 17.22 9.89 6.97

71.67 17.17 9.86 6.94

Table 13  ∗ Each column shows pBi , i = 1, 2, 3 for different triples (β H , β G , φ ). The remaining parameter values are those displayed in Table 2.

βG βH

0.1

0.1

24.84 24.84 1.34 5.85 5.85 0.81 3.33 3.33 0.71 2.34 2.34 0.64

0.3

0.5

0.7

0.3 24.80 24.80 1.30 5.85 5.85 0.80 3.33 3.33 0.69 2.34 2.34 0.63

0.5 24.76 24.76 1.27 5.84 5.84 0.78 3.33 3.33 0.68 2.34 2.34 0.62

0.7 24.73 24.73 1.24 5.84 5.84 0.77 3.33 3.33 0.67 2.33 2.33 0.61

Table 14   Each column shows SHB i , i = 1, 2, 3 for different triples (β H , β G , φ ). The remaining parameter values are those displayed in Table 2.

βG βH

0.1

0.3

0.5

0.7

0.1

2.47 2.47 3.85 1.73 1.73 2.36 1.62 1.62 2.12 1.57 1.57 2.01

2.47 2.47 3.73 1.72 1.72 2.30 1.62 1.62 2.08 1.57 1.57 1.97

2.47 2.47 3.62 1.72 1.72 2.25 1.61 1.61 2.04 1.56 1.56 1.93

2.46 2.46 3.54 1.72 1.72 2.21 1.61 1.61 2.00 1.56 1.56 1.90

0.3

0.5

0.7

of the drug investigated and the related percentage of royalty. Then, we computed our royalty as follows:

= φ

 j

φ j ∗ generic price j

T otal number o f drugs analyzed

(B.4)

 = 1.25. For calibration purposes, from our data, it results that φ we fix φ = 1.2. 4. The production cost has been estimated according to the paper by Hill, Barber, and Gotham (2018), which makes global evaluation of the variable cost associated with the production of a drug, independently of whether it is a generic drug. After several simulations, we calibrate the production cost at cH = cG = 0.1 (e.g., Genc & De Giovanni, 2018) to guarantee positive margins for both companies in each of the investigated parameters and consequently, guarantee that profits are always positive. 5. The discount factor δ and the R&D efficiency l have been fixed according to the game theory literature carrying out multiperiod investigation as in Martín-Herrán and Sigué (2017) and De Giovanni and Zaccour (2019). Appendix C. Equilibrium values for Scenario B This appendix supplements the illustration provided in the main text by reporting the equilibrium values for the benchmark scenario. Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ejor.2019.10.021.

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Please cite this article as: A. Sarmah, D. De Giovanni and P. De Giovanni, Compulsory licenses in the pharmaceutical industry: Pricing and R&D strategies, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.10.021