Innovation, loyalty and generic competition in pharmaceutical markets

Antoñanzas, Fernando; Juárez-Castelló, Carmelo; Rodríguez-Ibeas, Roberto Article Innovation, loyalty and generic competition in pharmaceutical markets SERIEs - Journal of the Spanish Economic Association Provided in Cooperation with: Spanish Economic Association Suggested Citation: Antoñanzas, Fernando; Juárez-Castelló, Carmelo; Rodríguez-Ibeas, Roberto (2011) : Innovation, loyalty and generic competition in pharmaceutical markets, SERIEs - Journal of the Spanish Economic Association, ISSN 1869-4195, Springer, Heidelberg, Vol. 2, Iss. 1, pp. 75-95, https://doi.org/10.1007/s13209-010-0032-5 This Version is available at: https://hdl.handle.net/10419/77759 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/2.0/ SERIEs (2011) 2:75–95 DOI 10.1007/s13209-010-0032-5 ORIGINAL ARTICLE Innovation, loyalty and generic competition in pharmaceutical markets Fernando Antoñanzas ·Carmelo Juárez-Castelló · Roberto Rodríguez-Ibeas Received: 8 September 2009 / Accepted: 23 September 2010 / Published online: 12 October 2010 © The Author(s) 2010. This article is published with open access at Springerlink.com Abstract We analyze how a manufacturer with a brand-name drug close to patent expiration decides to launch a product-line extension (an upgrade through innovation) before it faces generic competition. There are two types of physicians: loyal physicians always prescribe first the product-line extension and then either the off-patent drug or a generic drug while the non-loyal ones prescribe taking into account the prices of the drugs. We consider a two-stage game. In the first stage, the incumbent firm decides the level of innovation. In the second-stage, all firms decide sequentially their prices, with the incumbent firm acting as a Stackelberg leader. We find two equilibria in the pricing-decision game. For relatively large levels of innovation, the incumbent firm competes for the price-sensitive physicians. However, for low levels of innovation, the incumbent firm prefers to exploit the loyal physicians and to charge the monopoly price. The equilibrium level of innovation exhibits an inverted U-shaped behaviour with respect to the degree of loyalty. Keywords Brand-name drug ·Line extension ·Generic competition · Pharmaceutical markets ·Innovation JEL Classification I10 ·L13 The authors thank two anonymous referees for their helpful comments and suggestions. The usual disclaimer applies. F. Antoñanzas ·C. Juárez-Castelló ·R. Rodríguez-Ibeas (B ) Departamento de Economía y Empresa, Universidad de La Rioja, Cigüeña 60, 26004 Logroño, Spain e-mail: [email protected] F. Antoñanzas e-mail: [email protected] C. Juárez-Castelló e-mail: [email protected] 123 76 SERIEs (2011) 2:75–95 1 Introduction When brand-name drugs go off patent, they face generic competition. Historically, pharmaceutical firms have been able to maintain high prices after patent-expiration due to the existence of barriers to entry. During the eighties, new legislation to overcome entry barriers in pharmaceutical markets was enacted in the European Union and the US. In particular, the 1984 Drug Price Competition and Patent Term Restoration Act eased market entry for generic drugs as they only needed to prove bioequivalence to the brand-name drugs. Since then, the number of generic drugs entering the market has been continuously growing. The economic literature has mainly focused on analyzing the effect of generic entry in the prices of brand-name drugs. However, less attention (specially from a theoretical perspective) has been paid to other strategies that pharmaceutical firms can use to cope with generic competition and protect their market shares and revenues. In particular, brand-name drug firms can limit generic entry by launching the so called ‘pseudogenerics’ (Kong 2009) or even by taking over generic firms (European Commision 2009). Alternatively, pharmaceutical firms can launch product-line extensions (Hong et al. 2005). A line extension is a variation of an existing product or a modification of an existing molecular entity. This strategy allows brand-name drug firms extend the original drug with a new modification, and shifts demand from the original brand to the new extension. These line extensions of off-patent drugs allow the firm to introduce a differentiated product to compete with the generic drugs, and they often substitute for the original drug.1 Product line extensions require, compared to blockbuster drugs, lower R&D expenditures.2According to global figures, most of the new drugs are not based on really new chemical ingredients. Grabowski and Wang (2006) reported that worldwide and for the period 1982–2003, 919 new chemical ingredients were introduced, being only 115 of them first in class, i.e. strictly new.3Hong et al. (2005) report that only 15% of the 1,035 new drugs approved by the US Food and Drug Administration from 1989 through 2000 were innovative drugs. In Spain, according to the Ministry of Health and Consumption (2005), 782 new specialities were registered in 2004, from which 747 were based in already known chemical ingredients (about 300 were generic drugs) and 35 in new ones. These figures indicate that new drugs (new registered products with their corresponding patents) come basically from developments of known chemical 1According to a recent classification for drug innovations suggested by Caprino and Russo (2006), productline extensions could be considered as type D (new chemical ingredients structurally related to a chemical class already described for a similar therapeutic indication); type E (known pharmaceutical products generated through biotechnology or other very innovative techniques); type F (known pharmaceutical products with either new characteristics or more relevance due to its pharmaceutical form or administration path) and type G (known pharmaceutical products that have new characteristics or that have less relevance due to its pharmaceutical form, administration path, safety or handling improvements). 2DiMasi et al. (2003) found a cost of over 800 million US dollars to produce a new pharmaceutical product, including the whole R&D process to discover the new chemical ingredient. 3Among the 804 minor innovations, there are also follow-on drugs produced by competing firms and product-line extensions (upgrades) produced by leading firms. New registered drugs based on minor innovations overwhelmingly exceed new first-in-class drugs. 123 SERIEs (2011) 2:75–95 77 ingredients and only a low proportion (less than 5% in Spain) are derived from really new ones. The pharmaceutical industry seems to have opted for the strategy of introducing new drugs based on minor innovations (upgrades and product-line extensions). The new drug is usually protected by additional periods of patent exclusivity.4The original drug, the new product (line-extension) and the generics may coexist in the market. In this paper, we consider a manufacturer of a brand-name drug close to patent expiration that must decide whether to introduce a product-line extension (upgrade) of the existing brand-name drug to cope with generic competition. Specifically, we focus on the situation of a leading firm that produces a successful drug (a star product) with high loyalty from prescribers and close to patent expiration. The firm can, of course, try to get another first-in-class drug after a long and expensive R&D process. However, the firm can also take advantage of brand-loyalty and, with low R&D expenses, obtain a new differentiated product (through a minor innovation, upgrading or just designing a line extension) before generic drugs similar to the off-patent drug enter the market and price competition takes place. This situation is frequently observed when the off-patent drug has been very successful in the market.5 To simplify the analysis, we assume that the outcome of R&D expenditures is deterministic, and that there is free pricing for new drugs.6We consider a standard model of vertical product differentiation where the incumbent firm faces the competition of ngeneric drug firms. The demand for drugs is determined by physicians. We consider that there are two types of doctors. On the one hand, a proportion of physicians prefers the innovative drug and prescribes it first as long as the patient’s net utility is non-negative. In case the line extension prescription causes negative net utility to a patient, these doctors may prescribe her either the off-patent drug or any of the available generic drugs. Throughout the paper, we call these doctors “loyal” physicians. These physicians prefers the new drug produced by the incumbent firm to any other drug available. Here, loyalty means that these physicians always prescribe the line extension regardless of the prices as long as the patient’s net utility is non-negative. They remain loyal to the firm that initially developed the off-patent drug, and now, they prefer first to prescribe the upgraded drug (the line extension) produced by the same firm. On the other hand, the remaining physicians base their prescribing decisions on efficiency considerations and take into account both the effectiveness and the prices of the drugs. 4Whentheline-extensionisbased ona new chemicalentity,it isalwaysgrantedpatent protection.However, it may be the case that slight modifications of existing drugs are not protected by a patent or are protected by weak patents. Manufacturers of line-extensions seek the highest protection for their new product through a patent to replace sales lost when the original drug goes off patent. 5For instance, the firm Lundbeck that produced “citalopram”, a band-name drug for moderate–severe depression with patent protection, developed, before patent expiration, a minor innovation “escitalopram” to reduce the immediate generic competition. 6In European countries, pharmaceutical markets are highly regulated, and reference pricing systems are used. Our model considers free pricing, and therefore it is better applied to a situation like the US market. See the conclusion section for some comments on how our results might change if a reference pricing system is in place. 123 78 SERIEs (2011) 2:75–95 We model the interaction between firms as a two-stage game. In the first stage, the producer of the brand-name drug decides the degree of product differentiation through innovation. In the second stage, firms sequentially set their prices. We assume that the producer of the brand-name drug acts as a Stackelberg leader in the pricing game. Brand-name drug firms often introduce the line extension previously to the generic entry to shift demand through brand-loyalty. So, the choice of Stackelberg leadership seems to be a good approximation to real world. Our goal is to characterize the equilibrium prices and the equilibrium level of innovation. We also analyze how both the equilibrium level of innovation and the sales of generic drugs behave when the proportion of loyal physicians changes. We find that, for a sufficiently high proportion of loyal physicians, the equilibrium price of the line extension is such that the line extension is only prescribed by the loyal physicians. When the proportion of loyal physicians is low, however, both type of physicians prescribe in equilibrium the line extension. We also find that the higher the proportion of price-sensitive physicians, the larger the sales of generic drugs are. Regarding the level of innovation, we find that the equilibrium level of innovation exhibits an inverted U-shaped behavior with respect to the proportion of loyal physicians: itincreases when this proportion is low and decreases when it is high. From a social welfare perspective, the level of innovation chosen by the firm is too low. When the proportion of loyal physicians is relatively high, it seems to be desirable to reduce it. The level of innovation would be higher as well as the social welfare. Health authorities could act on detailing as, after all, pharmaceutical firms rely on this marketing activity to generate loyalty to their products. Our paper contributes to the theoretical literature of pharmaceutical markets and generic competition. Theoretical analysis of competition in pharmaceutical markets began with the seminal paper by Frank and Salkever (1992). They focused on analyzing the effect of generic competition on the prices of off-patent drugs to explain the puzzling increase in the prices of brand-name drugs after generic entry (‘Generic Competition Paradox’). They concluded that one possible explanation was that generic entry could make the demand of brand-name drugs steeper. (See also Lexchin 2004; Regan 2008;Mestre 1999;Kong 2009;Ferrara and Kong 2008 for alternative explanations of the ‘Generic Competition Paradox’.) Innovation in pharmaceutical markets has been recently analyzed by Bardey et al. (2010). They focus on how reference pricing affects the type of innovations brought to the market, but they do not analyze the relationship between innovation and generic entry. Another strand of research has focused on advertising as the mechanism by which pharmaceutical firms try to influence the prescribing behavior of physicians. (See Königbauer 2006,2007) These papers endogenize the proportion of price-sensitive physicians by determining the optimal level of advertising expenditures, leaving as exogenous the level of product differentiation (the quality of the brand-name and the generic drug). On the contrary, our paper takes as exogenous the proportion of price-sensitive physicians and endogenizes the degree of product differentiation through innovation. Finally, other papers (Brekke et al. 2007;Cabrales 2003;Kyle 2007;Ekelund and Peersson 2003;Dalen et al. 2006;Danzon and Chao 2000) have analyzed the effects of market regulation in drug prices. To the best of our knowledge, our paper is the first attempt to analyze, from 123 SERIEs (2011) 2:75–95 79 a theoretical perspective, the role that product-line extensions play in pharmaceutical markets following generic entry. The paper is structured as follows. Section 2describes the model, and in Sect. 3, the equilibrium prices are determined. Section 4characterizes the equilibrium level of innovation, and analyses its behavior with the proportion of brand-loyal physicians, variablethroughwhichpublicpolicycaninfluenceprescriptioninourmodel.InSect.5, welfare analysis is carried out and public policy considerations are described. Finally, conclusions are presented in Sect. 6. 2 The model We consider a standard model of vertical product differentiation applied to pharmaceuticals markets.7There are n+1 firms where firm 1 is the incumbent firm producing a brand-name drug and a line extension while the remaining nfirms produce generic drugs. Firm 1 faces patent expiration for its brand-name prescription drug and must decide the degree of product differentiation through innovation for its new drug (a line extension) before nfirms enter the market with generic drugs. There is a continuum of consumers (patients) with the same illness that can be treated with either of the three drugs.8Consumers are indexed by θ, which is uniformly distributed in the interval [0,1]with density one. The parameter θmeasures the severity of the illness. Each consumer is assumed to buy (to be prescribed) at most, one unit of the product (drug). In the case of drugs, patients do not choose directly the drug but this is, instead, prescribed by physicians. So, the demand for drugs is determined by physicians. We assume that there is a population of physicians of size one. Physicians, when they treat a patient, observe the parameter θand decide which drug to prescribe. Let sibe the quality of the good (drug) from firm iperceived by all physicians, i∈{1,2,...,n+1}. Perceived quality is a combination of baseline and innovative attributes, such that we can write si=fi+ki, where fiand kidenote respectively the baseline and innovative attributes of the drug produced by firm i.From the physicians perspective, patients’ utility depends on the severity of their illness and on the prescribed drug so that we can write θsias the gross utility that a patient with a severity of illness θobtains when she is prescribed the good (drug) from firm i.9 All drugs have the same baseline attribute, fi=f∀i, but only firm 1 produces a drug (the line extension) with the innovative attribute: k1=kand ki=0fori= 1 For the sake of simplicity and without loss of generality, we will assume that f=1. Firm 1 can produce the innovative attribute k≥0 at a cost C(k), with C(0)=0, C(k)>0,C(k)>0 and C(0)=0. To simplify the analysis, we assume that marginal production costs for all firms and drugs are zero. The line extension is assumed to be protected by the patent system. 7See Gabszewicz and Thisse (1979), Mussa and Rosen (1978)andShaked and Sutton (1982) for standard product differentiation models. 8Throughout the paper, we use the terms consumers and patients interchangeably. 9The parameter θcan be also interpreted as the patients’ marginal valuation for quality. 123 80 SERIEs (2011) 2:75–95 Among the population of physicians, a proportion α∈(0,1)consists of physicians who take only into account the innovative attribute, and therefore, they prescribe first the drug with the highest perceived innovative quality (the line extension) regardless of its price as long as the patients’ net utility is non-negative.10 Type αphysicians also consider that the off-patent drug is therapeutically equivalent to the generic drugs. For patients whose net utility was negative if they were prescribed the line extension, these physicians may prescribe either the off-patent or a generic drug. The off-patent drug and the generic drugs have the same chemical entity. In some countries, doctors prescribe taking into account the chemical entity, and for them, both types of drugs are therapeutically equivalent. Alternatively, we could think that drugs are dispensed by pharmacists that take into account the chemical entity of the drugs. Both situations fit well within the framework of the model. Type αphysicians are loyal to the innovation, although they may also prescribe the off-patent drug or a generic drug. The remaining doctors take into account both the total perceived quality and the prices, and prescribe the drug for which patients’ net utility is higher. We assume that they prescribe generic drugs instead of the off-patent drug for equal prices. They prescribe the line-extension if it provides a higher net utility. Net utility when the line extension is prescribed is given by θs1−p1=θ(1+k)−p1, where p1is the price of the line extension. For generic drugs, the net utility is θsi−pgi =θ−pgi, where pgi is the price for the generic drug of firm i. When the off-patent drug is prescribed, the net utility is θ−pb, where pbis the price of the off-patent drug. All physicians prescribe as long as the net utility is non-negative.11 We consider a two-stage game. In the first stage, firm 1 decides the level of innovation k. In the second stage, firms set their prices. We assume that firm 1 acts as a Stackelberg leader and chooses its prices (p1,pb)first.12 The remaining firms, taking as given the level of innovation kand firm 1s prices decide simultaneously their prices pgi. We solve the game by backward induction, and find the subgame perfect equilibrium. Our goal is to characterize the equilibrium prices and the equilibrium level of innovation. Note that, in equilibrium, the price of the line extension p1must be higher than both the price of the off-patent drug and the price of the generic drugs. If p1≤pgi for any i,firmihas incentives to lower its price as otherwise, it would end up with zero demand. If p1≤pb, firm 1 can increase its profit by increasing p1. 10 These physicians have been prescribing the off-patent drug during patent protection, and inertia makes them keep on prescribing the same drug improved with the innovation (the line-extension). See Coscelli (2000). 11 Based on the 1989 National Ambulatory Medical Care Survey for the US, Hellerstein (1998) points out that almost all physicians prescribe both brand-name drugs and generic drugs, although some physicians tend to prescribe more often brand-name drugs while others prescribe generic drugs instead. 12 From a theoretical point of view, the decision on prices could have been modelled as a simultaneous move game. However, in real world, pharmaceutical firms often introduce the line extension previously to the generic entry to shift demand through loyalty. So, the choice of Stackelberg leadership seems to be a good approximation to real world. Kong (2009), Ferrara and Kong (2008)andFrank and Salkever (1992) also model competition between brand-name drugs and generics as a sequential price-setting game. 123 SERIEs (2011) 2:75–95 81 3 The determination of the equilibrium prices 3.1 The second stage subgame Given kand (p1,pb), the firms producing generic drugs simultaneously decide their prices. As all the generics drugs are equivalent, price competition drives their prices to zero, the marginal cost, for all kand (p1,pb). Firm 1, taking the level of innovation as given, set its prices (p1,pb)to maximize its profits taking into account that the price of the generic drugs is equal to zero. Firm 1 will set pbequal to zero.13 As the off-patent drug is considered to be equal to the generic drug by both types of physicians, firm 1’s profits from the sales of the off-patent drug are equal for any pb.Ifpb>0, the off-patent drug is not sold, and the profits are zero. Likewise, if pb=0, the off-patent drug is sold but the profits are zero. So, the relevant problem for firm 1 is to determine the price of the line extension p1that maximizes its profits. Let us first derive the firm 1’s demand function for the line extension. As the price of the generic drugs is zero, the consumer indifferent between the line extension and the generic drug has a severity of illness ˆ θthat satisfies: ˆ θ(1+k)−p1=ˆ θ Therefore, ˆ θ=p1 k. If the consumer with a severity of illness ˆ θobtains a non negative net utility (ˆ θ(1+k)−p1≥0), he buys (is prescribed) the line extension, and so do all consumers with a higher severity of illness. All physicians prescribe the line extension to these patients. Let ˜ θ=p1 1+kbe the severity of illness of the consumer indifferent between the line extension and not buying. Note that ˆ θ>˜ θ. When p1<k,itfollowsthat ˆ θis lower than 1 and firm 1, on the one hand, sells to all consumers whose severity of illness is greater or equal to ˆ θ. Thus, its demand is 1 −ˆ θ, as both types of physicians prescribe the line extension. These consumers obtain a strictly positive net utility. On the other hand, there are consumers (those with severity of illness between ˜ θand ˆ θ) that would obtain a higher net utility consuming the generic drug, although their net utility is also positive if they are prescribed the line extension. Therefore, a proportion αof physicians (physicians that do not compare prices) prescribe the line extension to these consumers, and firm 1’s demand is α(ˆ θ−˜ θ). Then, firm 1’s total demand is 1 −ˆ θ+α(ˆ θ−˜ θ). When p1≥k, it follows that ˆ θ≥1. In this case, firm 1’s total demand is given by α(1−˜ θ): the line extension is only prescribed by the loyal physicians as long as patients’ net utility is non-negative. The demand for the line-extension is depicted in Fig. 1. The demands for the off-patent drug and the generics are respectively α˜ θand (1−α)˜ θ. 13 Given the assumptions of the model, the price of the off-patent drug must be equal to the price of generic drugs. Although we have considered a free pricing framework, a possible justification for this price equalization can be found in the reference pricing system in place in most European countries. For example, in Spain, the reference pricing implies that off-patent drugs are priced similarly to generic drugs. 123 82 SERIEs (2011) 2:75–95 Fig. 1 The demand for the line extension Thus, firm 1’s demand for the line extension is given by: D1(p1,α,k)=α(1−˜ θ) if p1≥k 1−ˆ θ+α(ˆ θ−˜ θ) if p1<k Firm 1 can follow any of two available strategies. On the one hand, it may choose a high price such that the line extension is prescribed only by the loyal physicians. Alternatively, it may choose a lower price such that both types of physicians prescribe the line extension. If the line extension is prescribed only by the loyal physicians, the price that maximizes firm 1’s profits is the solution to: max αp1(1−˜ θ) =αp11−p1 1+k s.t.p1∈[k,1+k] It is easy to see that the solution to this problem is p1=1+k 2if 1 >k. Otherwise, the solution is p1=k.Letl 1(α, k)denote firm 1’s profits when the line extension is prescribed only by the loyal physicians. By taking into account the solution to the above problem, we have: l 1(α, k)=⎧ ⎪ ⎨ ⎪ ⎩ α(1+k) 4if 1 >k αk 1+kif 1 ≤k(1) Alternatively, firm 1 can maximize its profits by choosing a price such that ˆ θ<1. In this case, it solves the problem: 123 SERIEs (2011) 2:75–95 89 of our model, as it happens in real world, innovation increases product differentiation, reduces competition and increases revenues for firm 1. So, firm 1 has incentives to choose a relatively high level of innovation. From a social perspective, a high level of innovation makes competition be less intense. As a result, the price of the line extension is higher but the size of the covered market remains unchanged. However, some of the treated patients benefit from the innovation and enjoy higher levels of utility. In aggregate, consumers’ surplus increases. From a social perspective, as well as in the case of the firm, there are incentives to choose a high level of innovation. However, innovation is costly. Thus, there are two forces pulling in different directions. The relationship between both levels of innovation will depend on how both firm 1’s revenues and consumers’ surplus change with innovation. Proposition 4 The levelof innovation thatmaximizessocialwelfareks(α, c)ishigher than the level of innovation k∗(α, c)chosen by firm 1∀(α, c). Proof (See Appendix.)  From a social perspective, the firm invests in innovation less than the level that it would be desirable. From this result, some policy implications arise. Health authorities can act on αgiven the relationship between k∗and αfound in the previous section. Health authorities have traditionally tried to promote generic prescription, not only to reduce the pharmaceutical expenses, but also because it is perceived that new brandname drugs incorporate minor innovations that do not bring about health outcomes substantially better than those from generic drugs, and brand-name drugs are much more expensive. Within the framework of our model, the promotion of generic prescription is formally equivalent to a reduction in α, and, a priori, it is not clear if this policy is socially desirable. Proposition 5 For α∈α∗(c), 1, social welfare decreases with αif 1+k∗(α, c)−α(1+2k∗(α, c))≥0.Forα∈0,α∗(c), social welfare increases with αif 1+k∗(α, c)−α(1+2k∗(α, c))≤0. Proof (See Appendix.)  When the level of loyal physicians is sufficiently high, policies that reduce loyalty are desirable as they increase social welfare. However, these policies can also be counterproductive when the proportion of loyal physicians is relatively low. In this case, they induce less innovation and may lower welfare. While ∂W ∂α <0, ∂W ∂k dk∗(α) dαis positive and the sign of dW dαis ambiguous. If the direct effect of αdominates the indirect effect in welfare through k∗, then reductions in αwould lead to higher levels of welfare. Otherwise, welfare would be smaller, and the correct policy would be to encourage loyalty. 6 Conclusions In this paper, we have considered a model of product differentiation applied to a pharmaceutical market with n+1 firms. In particular, we have analyzed how a 123 90 SERIEs (2011) 2:75–95 manufacturer of a brand-name drug whose patent is close to expiration decides the degree of product differentiation of a line extension through innovation before it faces generic competition. Decisions on innovation and prices are taken within an environment characterized by loyalty to innovative drugs, where some physicians prescribe first the line extension while others base their prescribing decisions on efficiency considerations. We have considered a game with two stages. In the first stage, the brand-name drug manufacturer decides the level of innovation. In the second stage, all firms play a sequential pricing game, where the brand-name drug firm acts as a Stackelberg leader. Depending on the level of innovation and the degree of loyalty, we have found two equilibria in the pricing game. For high enough levels of innovation, it is optimal for the incumbent firm to compete for the price-sensitive physicians. However, for levels of innovation lower than the degree of loyalty, such firm prefers to exploit the loyal physicians and charges the higher monopoly price. Regarding the level of innovation, we have characterized the equilibrium level of innovation and analyzed how it changes with the proportion of loyal physicians. We find that the equilibrium level of innovation exhibits an inverted U-shaped behavior with respect to the proportion of loyal physicians. For high levels of loyalty, the equilibrium level of innovation grows when loyalty is reduced. Alternatively, for low levels of loyalty, the equilibrium level of innovation decreases when loyalty is reduced. From a social perspective, the firm chooses a level of innovation that is too low. When the proportion of brand-loyal physicians is sufficiently high, we find that public policies that reduce loyalty are socially desirable. However, these policies can have a negative effect on social welfare if the proportion of loyal physicians is relatively low. In this case, the right policy would be to promote loyalty. The basic model can be used to analyze several extensions. We have implicitly assumed that the patients pay the full price of drugs. An extension of the model would be to introduce a co-payment system and analyze its relationship with the level of innovation. Co-payment would affect the patient’s net utility, but qualitatively, the analysis would not be very different. We have also considered that pricing decisions are taken sequentially. Another possible extension could consist of carrying out the analysis when pricing decisions are taken simultaneously. We have also assumed a deterministic innovation process to simplify the analysis. In reality, the outcome of the innovation effort is random. While considering stochastic innovation outcomes adds realism to the model, it also complicates its analytical tractability. The model can be used to analyze also the marketing decisions that endogenize the proportion of loyal physicians as pharmaceutical firms devote an important proportion of their sales revenues to marketing activities to generate loyalty to their products. We have carried out our analysis within a framework of free pricing. In many European countries, pharmaceutical markets are highly regulated and a reference pricing system operates. The introduction of such a system in our model would not bring about qualitatively different results. If the reference pricing system is based on bioequivalence principles (as in Spain), the off-patent drug and the generics would be subject to such a system, but the line extension would remain out of it. Price competition would drive the prices of the off-patent drug and the generics to zero and the line extension would be priced freely, as in our model. If the reference pricing system is based on therapeutical equivalence principles (as in Germany), besides the off-patent 123 SERIEs (2011) 2:75–95 91 drug and the generics, line extensions that incorporate low levels of innovation would likely be also subject to such a system, and the results of our model could change. We hope to explore these issues in further research. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited. Appendix Concavity of the profit function of firm 1 The profit function of firm 1is strictly concave for k ≥α. d1 dk k≥α=(1+k)2−α(1+2k) 4(1+k−α)2−ck d21 dk2k≥α=− α(1−α) 2(1+k−α)3−c<0 Proof of Proposition 2The Lagrangian function for problem P1is: L=k(1+k) 4(1+k−α)−ck2 2+λ1(k−α) where λ1is the Lagrange multiplier. The first order conditions are: dL dk =(1+k)2−α(1+2k) 4(1+k−α)2−ck +λ1=0(A.1) plus the restrictions and the complementary slackness conditions. In the solution, the restriction k≥αcannot be binding. If it were binding, λ1≥0, and from (A1),it should be that d1 dk |k=α≤0. However: d1 dk k=α=1+α(1−α)−4cα 4>0∀α∈[0,1]if c<0.25 Therefore, ∀α∈[0,1]and c<0.25, the solution to the optimization problem is interior: k∗(α, c)>α  Proof of Proposition 3As k∗(α, c)is interior, it satisfies d1 dk k≥α=0: 1+k∗(α, c)2−α1+2k∗(α, c)=4ck∗(α, c)1+k∗(α, c)−α2(A.2) 123 92 SERIEs (2011) 2:75–95 Differentiating this expression with respect to αyields: dk∗(α, c) dα=1+2k∗(α, c)−8ck∗(α, c)1+k∗(α, c)−α 2[1+k∗(α, c)−α][ 1−2c(1+k∗(α, c)−α)−4ck∗(α, c)] (A.3) From (A.3): dk∗(α, c) dαα=1=1+2k∗(1,c)−8ck∗(1,c)2 2k∗(1,c)[1−6ck∗(1,c)] where k∗(1,c)is the equilibrium level of innovation for α=1. From (A.2), it follows that k∗(1,c)=1 4c. Therefore: dk∗(α, c) dαα=1=−4c<0 It remains to shows that dk∗(α, c) dα|α=0>0.From (A.3), it follows: dk∗(α, c) dαα=0=1+2k∗(0,c)−8ck∗(0,c)1+k∗(0,c) 2[1+k∗(0,c)][1−2c(1+k∗(0,c)) −4ck∗(0,c)] where k∗(0,c)is the equilibrium level of innovation for α=0. From (A.2), it follows that k∗(0,c)=1 4c. Therefore: dk∗(α, c) dαα=0=4c (1+4c)2>0 Thus, dk∗(α, c) dα|α=0>0 and dk∗(α, c) dα|α=1<0. Therefore, there exists a value α∗(c)∈(0,1)such that dk∗(α, c) dα=0. From (A.3),itfollows: 1+2k∗α∗(c),c−8ck∗α∗(c),c1+k∗α∗(c),c−α∗(c)=0 Differentiating with respect to cyields: dα∗(c) dc =1+k∗(α∗(c), c)−α∗(c) c>0  Lemma 1 From a social perspective, the equilibrium level of innovation must be above α. 123 SERIEs (2011) 2:75–95 93 Proof It suffices to show that the social welfare function is strictly increasing for k<αand that ∂W ∂k|k=α>0. When k<α, social welfare can be written as: W(k,α)|k<α =3αk 8+α−1 2−ck2 2 and its derivative is: ∂W(k,α)|k<α ∂k=3α 8−ck This expression is strictly positive for all k<αand c≤0.25. When k≥0.5α, social welfare is: W(k,α)|k≥α=0.5(1−α)ˆ θ∗2−˜ θ∗2+0.5˜ θ∗2 +0.5(1+k)1−ˆ θ∗2+αˆ θ∗2−α˜ θ∗2−0.5ck2 =0.51+3k 4−kα(1−α) 4(1+k−α)2−ck2(A.4) and its derivative is: ∂W(k,α)|k≥α ∂k=3 8−ck −α(1−α) 8(1+k−α)4(1+k−α)2−2k(1+k−α) =3 8−ck −α(1−α) (1−2k) 8(1+k−α)3(A.5) It follows that: ∂W(k,α)|k≥α ∂kk=α=3 8−cα−α(1−α) (1−2α) 8=3−8cα−α(1−α)(1−2α) 8 This expression evaluated at c=0.25 is strictly positive: ∂W(k,α)|k≥α ∂kk=α,c=0.25 =3−α[2+(1−α)(1−2α)] 8>0  Proof of Proposition 4Recall that both ks(α, c)and k∗(α, c)are above α. In particular, k∗(α, c)satisfies: ∂1 ∂k=0⇔(1+k∗(α, c))2−α(1+2k∗(α, c)) 4(1+k∗(α, c)−α)2=ck∗(α, c)(A.6) and ks(α, c)satisfies: ∂W ∂k=0⇔3 8−α(1−α) (1−2ks(α, c)) 8(1+ks(α, c)−α)3=cks(α, c)(A.7) 123 94 SERIEs (2011) 2:75–95 It suffices to show that ∂W ∂kk=k∗(α,c)>0. From (A.5) and (A.6), the derivative of the social welfare function evaluated at k=k∗(α, c)is: ∂W(k,α)|k≥α ∂kk=k∗(α,c) =3 8−(1+k∗(α, c))2−α(1+2k∗(α, c)) 4(1+k∗(α, c)−α)2−α(1−α) (1−2k∗(α, c)) 8(1+k∗(α, c)−α)3 =3 8−1 4−α(1−α) 4(1+k∗(α, c)−α)2−α(1−α) (1−2k∗(α, c)) 8(1+k∗(α, c)−α)3 =1 8−α(1−α)(3−2α) 8(1+k∗(α, c)−α)3 As 1 +k∗(α, c)−α>1, it follows that α(1−α)(3−2α) (1+k∗(α, c)−α)3<1 and therefore ∂W ∂kk=k∗(α,c)>0.  Proof of Proposition 5The effect of αin social welfare is given by: dW dα=∂W ∂k dk∗(α, c) dα+∂W ∂α (A.8) From Proposition 4,ask∗(α, c)<ks(α, c), it follows that ∂W ∂kis positive. Let α∈α∗(c), 1. From Proposition 3,dk∗(α) dα<0 and social welfare will decrease with αdW dα<0if ∂W ∂α <0. From (A.4), the effect of αin social welfare, keeping the level of innovation constant, is given by: ∂W ∂α =− k∗(α, c) 8(1+k∗(α, c)−α)31+k∗(α, c)−α(1−2α) +2α(1−α) =− k∗(α, c) 8(1+k∗(α, c)−α)31+k∗(α, c)−α1+2k∗(α, c) (A.9) When 1 +k∗(α, c)−α(1+2k∗(α, c)) ≥0, it follows from (A.8) that dW dα<0. If 1+k∗(α, c)−α(1+2k∗(α, c)) < 0, the effect of αin social welfare is ambiguous. Let α∈[0,α∗(c)]. From Proposition 3,dk∗(α) dα>0 and social welfare will increase with αdW dα>0if ∂W ∂α >0. From (A.9), it is required that 1 +k∗(α, c)−α(1+ 2k∗(α, c)) ≤0. If 1+k∗(α, c)−α(1+2k∗(α, c)) > 0, the effect of αin social welfare is ambiguous.  123 SERIEs (2011) 2:75–95 95 References Bardey D, Bommier A, Jullien B (2010) Retail price regulation and innovation: reference pricing in the pharmaceutical industry. J Health Econ 29(2):303–316 Brekke KR, Königbauer I, Straume OR (2007) Reference pricing of pharmaceuticals. 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