Model Setup. We begin by describing a P4D deal from the US which serves as a motivating example of the stylized model described below. Shire Pharmaceuticals introduced an extended release version of its ADHD drug called Adderall XR in 2001. Under the ▇▇▇▇▇-▇▇▇▇▇▇ terms it had exclusivity until April 2005 (initial exclusivity was until October 2004, but then had received pediatric extensions). The underlying patents for the extended release version, unless invalidated, were effective until 2018. In November 2002, ▇▇▇▇ laboratories filed an abbreviated new drug application (ANDA) which was followed by a second filing by IMPAX in November 2003. Patent litigation ensued, but ▇▇▇▇▇ settled with both parties before any court outcome. Shire settled with IMPAX (the second filer) to enter the market no later than December 2010, but with a non-exclusive license. It also settled with ▇▇▇▇ laboratories (the first filer), which acknowledged that Shire’s patents were valid and to agreed to delay entry until April 1, 2009. At that point, ▇▇▇▇ would enter with a 180-day exclusive licence from Shire and pay royalties as a proportion of its profits from the sales of generic Adderall XR over the exclusivity period [▇▇▇▇ Laboratories, Inc., 2006]. Per the terms of the agreement, ▇▇▇▇ would also be allowed to enter earlier if another party were to launch a generic version of the drug. Similarly, Teva (which had acquired ▇▇▇▇ laboratories in the meantime) started marketing generic version of Adderall XR in the US on April 2, 2009, and six months later IMPAX also entered the market. For a discussion on side payments and additional examples, see ▇▇▇▇▇▇▇▇ [2007]. Further details of patent litigation and market entry rules in the US and EU are given in section 3 and our model is based on these institutional details. We propose a dynamic game Γ with J +1 players that illustrates the essential elements of interactions between a brand name firm B (player 0), which is protected by a patent, and J ≥ 1 potential generic challengers (G1, . . . , GJ ). As in ▇▇▇▇▇▇▇▇ and ▇▇▇▇▇▇▇ [2007], our game unfolds in the shadow of a trial. Our stylized game is designed to capture the market authorization rules and main features of P4D cases described earlier and stylized below. (1) There are two periods, period 1 which is pre-patent expiration, and period 2, which is post-patent expiration period. (2) In period one, the J potential entrants can sequentially contest entry. The branded firm can offer a payment to a challenger to stay out of the market during period one (a P4D deal), and guarantee the order of entry in the post-patent period, as long as the patent is not invalidated by another challenger (order of entry is not guaranteed if the patent is invalidated).7 (3) If at any stage a challenger (say the jth) does not accept a P4D deal and wins the court case (patent is invalidated), that challenger enters immediately in period one. However, the remaining J − j entrants can only enter in the next period. In a later section we relax the assumption of exclusivity for first successful challenger to no exclusivity for anyone, or exclusivity restricted to just the first filer. (4) Additionally, if the jth firm wins the court case, the brand can opt to launch an authorized generic (AG), either itself at an additional cost θ and earn two profits from the brand and its generic product, or via any of the previously paid-off firms, in which case it earns profits from the brand plus a licensing fee L. If the brand launches an AG, period one consists of a triopoly. In what follows we also assume that if the brand launches an AG externally, it is only via the first generic challenger.8 (5) Payoffs from the second period are discounted by factor δ ∈ [0, 1]. Further, in this period we assume a competitive oligopoly ensues among the J + 1 firms, and there are no licensing 7For instance, the branded firm can allow a generic to use its own production facilities to achieve all regulatory market approval requirements and enter just before other generic firms enter. 8This is a simplification but follows the example from ▇▇▇▇▇-▇▇▇▇ deal mentioned above. An alternative is to randomize. agreements, as the patent has expired. However, the profits and/or market shares are not equal as the order of entry matters, i.e., one of the generic products has a first mover advantage over the other generics. For the base case we assume that the second through the last generic entrants all earn the same profit (which is less than that of the first generic entrant).9 Based on the rules above, the game is as follows. The patent can be challenged in any of the Γj stages by a generic challenger ▇. In the first stage Γ1 of the game, generic G1 can choose to stay out of the market, in which case the monopoly continues and the game ends, or challenge entry. If it contests entry, the brand makes an offer of X1 to G1 to stay out of the market. If the offer is rejected, litigation ensues. If it is not rejected, the process is repeated with the second challenger. For exposition, the game is depicted in the Figures (1) and (2) below for the special case of J = 2 when there are only two potential challengers. The game and payoffs differ slightly for the first versus the second challenger, and hence we show these two cases explicitly, but the generalization to J > 2 challengers is similar to the second challenger case and we discuss that later. Continuing with the example of just two potential challengers (J = 2), we denote equilibrium profits due to the sales of the branded or generic drugs in any period by ΠM , ΠD# and ΠT# where M, D, and T stand for profits in monopoly, duopoly and triopoly market structures respectively, and the subscripts j ∈ {0, 1, 2} are for the brand and first and second generic entrants. The superscript ‘#’ is set to 1 or 0 to indicate when an authorized generic has been launched either by the branded firm itself (self-AG) or via one of the paid off generic firms in a P4D deal, and accounts for the possibility of price coordination and joint profit maximization. While we do not specify the exact values of the profits here, we assume that monopoly profits are greater than industry profits in a duopoly, which are in turn greater than industry profits in a triopoly. Further, profits are negatively correlated with entry order, and thus in a triopoly, the branded firm has the highest profits followed by those of the first and then the second generic entrant. Note that the jth generic challenger is not necessarily the same as jth entrant since a generic firm can choose to stay out of a market, and hence we denote the profits of the jth player by Vj. For example, consider the case where generic 1 has been paid off and hence agrees to stay out of the market and generic 2 enters the market and duopoly ensues between the brand and the second generic firm. Then, the equilibrium profits for the three players in the first period would be given by (V D, V D, V D) = (ΠD, 0, ΠD). Similarly, Lj is the adjustment to the final payoffs of the jth player due to any licensing agreements for an AG and we use the notation V˜T 1 = V T1 + δV T# + Lj to indicate sum of equilibrium profits from the two periods plus any licensing fee (note that we use the superscript T 1 on the sum of profits even if the second period is not necessarily T1, as long as the first period is T1). Also, since we assume that if an AG is launched it is only via the first challenger, we can simplify the notation to L1 = −L0 = L and L2 = 0. 9We later relax this assumption and allow the successful generic one to earn more than other generics if it enters in period one (i.e. to model an incumbency advantage). Note that only equilibrium profits from sales are shown in the nodes. The final payoffs include also litigation costs and AG costs as indicated along the branches. Figure 1. Game Tree (Γ1) If at any of the two stages the generic rejects the offer, litigation ensues and the involved parties incur the costs of c0 and cj (to be paid at the end of Γj). We assume c0 is sufficiently low for B to always prefer litigation over unopposed entry and the ensuing competition. The outcome of the litigation is modeled by the fictitious player (N, Nature), who decides randomly with probabilities 1 − πj and πj, respectively whether the brand B is successful with its lawsuit over patent infringement or not.10 As shown in Figures (1) and (2), the brand has the option of launching an AG at several of its decision nodes. For convenience, we will denote the subgames that start at these nodes as Γj,y, where j denotes the challenger and y = {B, G} denotes the relevant path of the game: y = B if either the brand wins the case or if the generic stays out, and y = G if the generic wins. Note also that in the first stage when G1 is the current challenger, the branded firm has the option to launch AG itself, 10As a special case we let πj = π for all j where π then represents the strength of the patent with π = 0 being a very strong patent and π = 1 being a very weak patent. Note that only equilibrium profits from sales, P4D payments and licensing fees are shown in the nodes. The final payoffs also include litigation costs and AG costs as indicated along the branches. Figure 2. Game Tree (Γ2) whereas in the later stages, the option to launch an AG is only via the first paid-off generic firm. Hence, the first P4D deal contains - unlike the successive P4D deals - an (implicit) option to become an AG producer.11 Alternatively, if a self-AG is not launched and the generic does not challenge (as in Γ1,B), the order of entry between the two generics for the second period in randomized (and hence the profits are depicted as expected generic profits). Further, if the branded firm launches a generic itself, the firm incurs a fixed entry cost θ of entering a generic market (or δ · θ if the generic is launched in the second period). If both generic challengers have accepted P4D payments, the game ends at the Γ3 node with payoffs given by (ΠM − X1 − X2, X1, X2) + δ(ΠT0, ΠT0, ΠT0) (which note is similar to Γ2,B with an adjustment of X2 payment to the second challenger). 11Note also that in Γ1,G if the brand launches its own AG in the first period, the first and second generics’ profits in post patent period are set equal to δΠT 1/2, i.e., they split the profits associated with a third product in a triopoly as there is no incumbency advantage for generic one, even though it enters in first period. The alternative extreme would be to assign δΠT 1 to generic one and zero economic profit to generic two with similar adjustments in Γj,G for j > 1 cases. We consider the outcomes from such incumbency advantage in a later section. The final payoff to a player along a path of the game Γ consists of the corresponding (continuation) profit in the ensuing market structure adjusted by the P4D payments and/or litigation costs received and/or paid along the path. Except for some specific values of the parameters, the finite game Γ has a unique subgame perfect equilibrium (SPE) that can be readily computed by backward induction. In particular, we can compute the minimum offer that Gj, j = 1, 2, will accept in the SPE from the condition, uj(Γj+1) + Xj = πjuj(Γj,G) + (1 − πj)uj(Γj,B) − cj, (1) where uj(·) is the expected payoff to player j in the unique SPE of the subgame. The condition (1) makes the (risk neutral) player Gj indifferent between accepting Xj - and getting the left hand side (lhs) of (1) - and rejecting it - and expecting the right hand side (rhs) of (1). The brand B (player 0) will make the offer Xj in equilibrium, whenever its expected SPE payoff u0(Γj+1) after paying Xj (receiving Xj if it is negative) exceeds its expected payoff from the litigation, i.e., when,
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