A Model of Generic Drug Shortages: Supply Disruptions, Demand Substitution, and Price Control


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1 A Model of Generic Drug Sortages: Supply Disruptions, Demand Substitution, and Price Control SangHyun Kim, Fiona Scott Morton Yale Scool of Management, Yale University, 165 Witney Avenue, New Haven, CT January 015 In recent years, te U.S. ealt care system as been plagued by sortages of generic sterile injectable drugs. Te sortage problem arose suddenly witin a sort period of time, and remained persistent afterwards. To gain insigts into wat may ave led to tis situation, we develop a model tat incorporates key caracteristics of te generic drug manufacturing industry, including random supply disruptions, demand substitution across di erent brands, and regulatory control of prices. In our model, rms competitively coose teir production capacities anticipating future supply disruptions. Tese capacity coices in turn determine product availability. Our equilibrium analysis reveals a number of counterintuitive results. First, drug availability increases as productions become more prone to disruptions. Second, allowing temporary price increases during sortages may or may not increase drug availability, depending on ow severe te sortage is. Based on tese insigts, we discuss ow external factors may ave interacted wit rm decisions to result in te current drug sortage situation. 1 Introduction Starting in te late 000s, tere ave been signi cant and persistent sortages of generic drugs in te United States. Number of sortage instances tripled between 005 and 010, increasing from 61 to 178 in ve years. Te majority of te sortages occur in te category of sterile injectable drugs, including cancer drugs, anestetics, and antibiotics, wic accounted for 80% of te sortages tat occurred in despite te fact tat tey represented only 9% of te entire generics market in volume (U.S. Food and Drug Administration 011). Because many of tese drugs are medically necessary, sortages can create a serious and even lifetreatening situation. Altoug te situation as sown signs of improvement recently, te number of sortages remains very large (U.S. Government Accountability O ce 014). Tese sortages occur mainly because of disruptions in te manufacturing process. It was reported tat about alf of te sortages in were due to quality and oter production issues (FDA 011). Notable examples include fungal or bacterial contaminations, introduction of particulate matters in vials, and equipment failures (GAO 014). Discovery of suc issues requires cleanup and restoration, slowing down productions or forcing temporary closure of te entire facility in some cases. 1
2 In teory, tese disruptions need not result in product sortages; if tere are enoug excess capacities or inventories tat can be utilized as backups, ten sortages can be avoided. Te fact tat sortages ave become commonplace indicates tat te generics industry s management of capacities as been a ected by some external factors. Responding to te serious nature of te problem, te FDA as instituted a number of remedial procedures tat are designed to minimize te impact of sortages. Tey include: expediting regulatory approval processes, facilitating information saring among stakeolders, and easing importation of substitutes from overseas (FDA 011). Tese measures ave been successful to some degree, as te FDA credits tem for preventing 38 sortages in 010 (in te year wen 178 sortages were reported). Wile tese e orts elp alleviate te impact of sortages, tey are stopgap measures tat do not address systematic aw. A permanent solution as proven to be elusive, in spite of numerous publised expert opinions, congressional earings, and a presidential executive order tat led to te FDA actions (U.S. House of Representatives 011, Harris 011). Te continued sortage situation as prompted many to speculate on te structural causes rooted in market dynamics. Some fault te government s price control policies, caracterizing tem as arbitrary mecanisms tat dampen te rms incentives to invest in capacity and modernize facilities (see 6 for more discussions). Some blame te FDA, believing tat te agency s aggressive regulatory interventions often result in unnecessary production sutdowns and delays (Graam 01). Oters are suspicious about te role of Group Purcasing Organizations (GPO), wic act as intermediaries on bealf of ospitals but introduce ine ciencies in te parmaceutical supply cain. It as also been suggested tat an increased number of patent expirations ave led te generics manufacturers to sift teir limited resources to producing new drugs at te expense of existing ones (U.S. Department of Healt and Human Services 011). Despite many conjectures, none of tem as been singled out as te de nitive cause of te problem. Instead, it is likely tat te problem arises from a combination of factors economic, operational, and regulatory tat are closely intertwined (FDA 011). Among many features of tis multifaceted problem, a number of distinctive ones stand out. First, demands for sterile injectable drugs are relatively stable and unresponsive to price canges, because te drugs are medically necessary and insured consumers do not directly bear te price di erences. Second, supply disruptions for a given drug occur sporadically, caused by unforeseen events tat are often beyond te rms control. Tird, di erent brands of a given drug are perfect substitutes. Even toug some level of di erentiation is acieved troug packaging and advertising, te underlying products are cemically equivalent and terefore tey are clinically indistinguisable. Fourt, te industry is igly concentrated, in most cases wit tree or fewer competing manufacturers for a
3 given drug (HHS 011, FDA 011). Finally, regulations limit te extent to wic drug prices can vary over time. Te Medicare Modernization Act of 003 required tat annual price increases be capped at a small percentage, tus creating price stickiness (U.S. House of Representatives 011). In tis paper we develop a stylized model tat captures tese unique features and analyze ow tey interact, generating insigts into te dynamics of sortages and te conditions tat may exacerbate tem. Speci cally, we focus on researc questions tat ave important practical implications to te generic drug industry but ave not yet been rigorously studied. Tey are: (1) Knowing tat future supply disruptions will reduce production outputs, ow do manufacturing rms optimally coose teir capacities? () How does te sticky price policy in uence te rms incentives? (3) Wat is te net impact of external factors and rm decisions on product availability? (4) Wy did te sortage problem arise suddenly during a sort period of time, and wy as it been so persistent? Te analysis of our model reveals tat rms optimal capacity coices result in an unexpected equilibrium outcome: iger probability of supply disruptions leads to iger product availability. In oter words, as productions become more prone to disruptions (i.e., less reliable production process), consumers ave a better cance of obtaining te products. Tis reversal of te usual relationsip between reliability and availability also in uences ow availability canges wit oter environmental variables. Based on tese observations, we propose a ypotesis tat explains te current drug sortage situation. We also nd tat te policy of keeping te price xed may or may not ave a positive e ect on product availability; altoug temporary price increases lower availability in general, tere are situations were te opposite is true. We identify te conditions under wic tis appens, and discuss policy implications. Te rest of te paper is organized as follows. After a brief review of te related literature in, in 3 we present model assumptions and derive matematical expressions tat form a basis of te analysis. In 4 we analyze te base model in wic product price is assumed to be xed. Tis is followed by 5, were we consider an alternative model assumption under wic price increases temporarily during sortages. Based on te insigts gained from 45, in 6 we propose a ypotesis tat may explain te current sortage crisis, concluding in 7. Note tat B. and B.3 in te Appendix contain te results of additional model extensions ( Centralized Firm s Decisions and Capacity and Reliability Coices Under Fixed Price ) tat we refer to trougout te paper. Related Literature Te generic drug industry as been te subject of many studies in economics, as its unique market dynamics provide a fertile ground for testing te teories from te industrial organization literature. Generic versions of a particular drug are introduced to te market after te patent protection enjoyed 3
4 by te original drug developer expires, creating opportunities for oter parmaceutical rms to o er cemically equivalent substitutes. Tese entries usually result in a bifurcated market in wic te market sare of te branded drug srinks substantially as pricesensitive buyers opt for its generic counterparts. At te same time, competition among te entrants lowers te price of generics. From an economic perspective, tese penomena can be described by te models of monopoly, market entry and deterrence, vertical di erentiation, and oligopolistic competition (Caves et al. 1991, Scerer 1993, Frank and Salkever 1997). Tese features are furter enriced by te fact tat te parmaceutical industry is igly regulated, a ecting te rms incentives and overall e ciency of te market. Over te years researcers ave sed ligt on tese issues empirically, suc as rms market entry decisions (Scott Morton 1999), relationsip between drug prices and te degree of competition (Rei en and Ward 005, Olson and Wendling 013), and impact of regulations on competition (Danzon and Cao 000). Despite te ric body of literature in economics tat investigates various aspects of te generic drug industry, very few discuss sortages. Tis is not surprising, given tat te issue as not caugt te attention of researcers until recently; altoug generic drug sortages in te U.S. ave always existed to some degree, only in te past few years as te problem reaced te status of a national crisis. 1 Tis development prompted many commentaries from ealt care researcers and professionals, wo diagnose te problem based on conceptual economic arguments (for example, see Jensen and Rappaport (010), Scweitzer (013), and Woodcock and Wosinska (013)). To date, owever, tere as been a deart of studies tat examine te sortage problem using rigorous analytical frameworks. An exception is an article by Yurukoglu (01), wo develops an econometric model and argues tat te current sortage situation is attributed to te implementation of Medicare Modernization Act of 003, wic led to a signi cant drop in drug prices tat likely reduced capacity and maintenance investments. Our paper lls tis void in te literature. Borrowing te ideas from operations management (OM), we develop a stylized gameteoretic model tat captures key caracteristics of te generic drug industry and analyze te incentive dynamics under supply disruptions. Tis approac makes sense especially because supply disruption management as been one of te most active areas of researc in OM; for a recent survey of tis literature, see Snyder et al. (014). Te majority of works in tis area focus on procurement settings in wic a buyer as a contractual relationsip wit one or more suppliers wose production outputs are random due to supply socks. Representative articles include 1 A series of New York Times articles since 011 illustrates te persistent and serious nature of te problem: U.S. Scrambling to Ease Sortage of Vital Medicine (Harris 011); Drug Sortages Persist in U.S., Harming Care (Tomas 01); Drug Sortages Continue to Vex Doctors (Tavernise 014). 4
5 Tomlin (006), Dada et al. (007), Wang et al. (010), and Yang et al. (009). Aydin et al. (01) survey te models of supply disruptions under decentralized decisionmaking, te category tat our paper belongs to. Of tese, Deo and Corbett (009), Tang and Kouvelis (009), and Kim and Tomlin (013) are particularly relevant since tey consider orizontal competition among rms, similar to wat we assume in our paper. Tese autors, owever, focus on oter aspects of competition (market entry decisions, correlated disruption risks, etc.) tat we do not take into account. In our model, competition is driven by inter rm demand substitution tat occurs after supply disruptions. Not many researcers ave studied a similar dynamic, wit notable exceptions by Tang and Kouvelis (009) and Tomlin (009) wo also incorporate demand substitutability but in problem settings quite distinct from ours (centralized decision, sole vs. dual sourcing). Te way we represent demand substitution resembles te newsvendor competition model by Netessine and Rudi (003). Unlike in Netessine and Rudi (003), owever, demand substitution in our model is triggered by supply uncertainty, not by demand uncertainty. In sum, tis paper represents one of te rst rigorous studies on te current generic drug sortage situation. We obtain novel insigts from te model tat combines te ideas from OM and economics tat tus far ave existed in separate domains, tus contributing to te OM literature on supply disruptions as well as te literature on economics of te parmaceutical industry. 3 Model As we previewed in 1, te generic sterile injectable drug industry as a few idiosyncratic caracteristics wic we igligt in our model. Tey are: (i) te industry for a given drug typically consists of a small number competing manufacturers; (ii) di erent brands of a drug are perfectly substitutable; (iii) te demands for a drug are stable over time and exibit relatively small variability; (iv) production output drops wen a random and exogenous sock reduces capacity. To capture tese succinctly, we develop a model of duopoly in wic two manufacturers facing constant demands and random supply disruptions compete on capacities. Te capacity decisions determine product availability as well as te amount of demands tat spill over from one rm to te oter during disruptions. Building on tis framework, we consider two model variations under di erent assumptions on price. In 4 we study a base model in wic te price is assumed to be xed. In 5, we modify te base model by assuming tat te price varies wit total industry capacity wen sortages occur. Te base model re ects te current government policy tat limits sortterm price increases. Despite te policy, owever, price increases during sortages ave been reported, including tose from gray markets (Link et al. 01). Wit te modi ed model we aim to discover ow price variations a ect rm decisions and product availability. For expositional clarity, in te remaining space of tis section we 5
6 describe te model based on xedprice assumption. Variableprice modi cation is introduced in 5. We make some simplifying assumptions around tis setup in order to construct a parsimonious model tat allows tractable analysis. 3.1 Supply Disruptions and Economics Te industry consists of two rms, denoted by te subscript i f1; g, tat manufacture a omogeneous product wose units are perfectly substitutable (e.g., cemically equivalent cancer drugs). Firm i owns and operates manufacturing facility i. Te rms and teir facilities ave symmetric caracteristics, as we elaborate below. Te unit price of te product is p. For simplicity, we normalize te unit cost of production to zero and assume tat production lead time is negligible. All rms are risk neutral. Consumer demands for te product arrive at te market over an in nite orizon, starting at time zero. In order to focus on te impact of supplyside uncertainty, we make a simplifying assumption tat te demands arrive deterministically at a constant rate of per unit time. At eac moment, te arriving demands are equally divided and directed to te two rms. Eac rm dedicates production capacity at its facility wose rate of output perfectly matces tat of te demand stream it receives. We refer to tis fullyutilized production capacity as regular capacity. Terefore, eac facility receives its allotted demands at te rate of = per unit time and produces exactly at tat rate using te regular capacity. Witout loss of generality, we assume tat te costs of setting up capacities are sunk at time zero. In addition to regular capacity, eac rm maintains spare capacity wic is normally idle but may be utilized wen production is disrupted. (Tese distinct capacity types are analogous to running stocks and safety stocks found in inventory models, e.g., Groenevelt et al. 199b.) We present te assumptions on disruptions next, followed by spare capacity. Clearly, all demands are lled wen productions on regular capacities run continuously. However, some demands may be lost wen a sock arrives at one of te facilities and disables a portion of regular capacity, creating a supply sortage by reducing production output. Examples of suc socks in te drug manufacturing environment include bacterial or particulate contamination of a production line and equipment failure (GAO 014). We assume tat regular capacities at bot facilities are subject to independent and repeated socks. Once a sock arrives, it takes a random amount of time to recover full capacity at eac facility. Tus, at any given moment eac facility is eiter in te undisrupted state (wit full capacity) or in te disrupted state (wit partial capacity), alternating between te two states as time progresses. Let i be te fraction of time in steady state during wic facility i is in te disrupted state; in te remaining 1 i fraction of time, te facility is in te undisrupted state. If te states evolve as an alternating renewal process wit mean durations 1= i and 1= i, ten i = i=( i + i ). 6
7 Hence, iger i means less reliability of facility i. We refer to i as disruption probability at facility i, since it represents te cance tat te facility is in te disrupted state at a random point in time. 1 We assume tat te disruption probabilities are su ciently small so tat te approximation 0 applies, i.e., te probability tat bot facilities are simultaneously disrupted is negligible. Tis assumption captures te fact tat disruptions are relatively rare events, and it also enables tractable analysis. (A similar assumption is commonly adopted in te spare parts inventory management literature tat considers lowprobability, igconsequence events; see Serbrooke (1994) and Muckstadt (005).) Wit tis approximation, te industry is in one of tree states at any given moment: (i) neiter facility is disrupted, wit probability (1 1 )(1 ) 1 1 ; (ii) facility 1 is disrupted and facility is undisrupted, wit probability 1 (1 and facility is disrupted, wit probability (1 1 ). ) 1 ; (iii) facility 1 is undisrupted We assume tat, upon arrival, a sock instantaneously reduces te regular capacity at facility i from = to = i, were i [0; 1] is a random variable tat represents te percentage of regular capacity tat survives te sock s impact and can be utilized for production. 3 For parsimony, we refer to i simply as capacity yield. (Te proportional random yield assumption is widely adopted in te OM literature; for example, see Federgruen and Yang (009), Deo and Corbett (009), and Tang and Kouvelis (011).) We assume tat 1 and are independent and identically distributed wit mean E [ i ], i = 1;, saring te same pdf f and cdf F satisfying f (0) > 0, F (0) = 0, and F (1) = 1. Moreover, we assume tat F is logconcave, a mild condition satis ed by many commonly used probability distributions (Bagnoli and Bergstrom 005) and employed by oter OM researcers (e.g., Cacon and Zang 006). For expositional ease, we use te notations F (x) 1 G (x) R x 0 F (y) dy = E[(x i) + ]. F (x) and To make up for reduced capacity, eac rm maintains spare capacity tat is utilized only during disruptions. (GAO (014) reports te use of spare capacities by generic drug manufacturers.) To simplify te analysis we assume tat spare capacities are una ected by socks, unlike te regular capacities. Tis distinction is reasonable in a situation were spare capacities are maintained in a secure location and utilized only during disruptions as backups, in contrast to fullyutilized regular capacities tat are constantly exposed to te sources of socks tat may a ect daily production operations. Wile tis dicotomy is an idealized assumption, it is instrumental in deriving key analytical results; note tat a similar assumption is commonly made in te spare inventory management models. Firm i sets s i units of spare capacity at time zero and maintains tem afterwards, incurring te olding cost per 3 Te assumption tat capacity is reduced by a fraction is consistent wit a report by Woodcock and Wosinska (013): In most cases, rms ave been able to continue production wile improvements were being made, altoug often at a reduced rate. 7
8 unit of capacity per unit time. Te olding cost may include te overead cost of maintenance as well as te opportunity cost of capital; ence, is similar to its counterpart in inventory models. (Note tat te olding cost of regular capacity is normalized to zero, since it does not play a meaningful role in our analysis.) To rule out trivial equilibrium outcomes in wic supply sortages never occur, we restrict attention to parameter combinations tat satisfy te following relationsips: 0 s i = and p i < for i = 1;. Te rst condition states tat te amount of spare capacity at eac facility (s i ) does not exceed te maximum amount of regular capacity (=). Te second condition states tat te maximum pro t tat eac unit of spare capacity can generate during disruptions at a single facility (p i ) is outweiged by a rm s cost of olding te capacity (). Notice tat spare capacity in our model plays a role similar to tat of safety stocks in inventory models. We do not explicitly take inventory into account for two reasons. First, sterile injectables do not ave long self lives, and terefore manufacturers typically carry only a small amount of safety stocks (HHS 011). Tis implies tat a volume increase is more likely to be acieved by expanding production capacity, rater tan carrying more inventories. Second, an analysis of inventory competition under random supply disruptions and demand substitution is intractable, due to te di culty of identifying parsimonious matematical expressions for interdependent inventory levels at te two facilities tat evolve according to a random process. In fact, even a model of noncompetitive inventory control in te same setting presents analytical callenges, requiring somewat arbitrary assumptions to gain tractability (see, for example, Groenevelt et al. 199a,b). Altoug capacity and inventory are not mapped precisely to one anoter, teir roles are analogous in tat spare capacity captures te main tradeo s of carrying safety stocks. Finally, we assume tat all variables tat we introduced tus far, including te distribution functions, are common knowledge. 3. Demand Spillovers and Performance Measures In line wit te assumption tat product units are perfectly substitutable, we assume tat all demands tat are allotted to facility i but cannot be lled tere are immediately redirected to facility j. If te redirected demand is un lled again, it is lost. (In teir examination of te FDA data, Woodcock and Wosinska (013) report similar spillover e ects.) In te presence of suc demand spillovers, ten, eac rm s sales rate (measured in quantity per unit time) depends on te states of te two facilities at any given moment. Clearly, te sales rate is equal to te allotted demand rate wen neiter facility is disrupted; since eac as ample capacity to ll its allotted demand rate of =, no demand spillover occurs. Wen one of te facilities is disrupted, owever, te sales rates di er from te allotted demand rates as capacity reduction may lead to demand spillover. 8
9 To quantify tis, suppose tat facility 1 is disrupted wit percentage yield 1 < 1 wile facility is undisrupted at a random point in time. Wit regular and spare capacities combined, te realized total capacities tat rm 1 and rm possess are Q 1 = 1 = + s 1 and Q = = + s, respectively. Demand spillover from facility 1 to facility occurs if Q 1 is smaller tan te allotted demand rate =, or equivalently, if 1 < 1 s 1 =. Te amount of spillover is equal to (= Q 1 ) +, were () + max f0; g. Hence, rm 1 s sales rate is equal to min f=; Q 1 g, wereas rm s sales rate is equal to min = + (= Q 1 ) + ; Q, taking into account te additional demands tat are spilled over. Summarizing te outcomes in all tree facility states, te sales rate at facility i is equal to 8 >< R i >: if neiter facility is disrupted, min ; i + s i if facility i is disrupted but facility j is not, n min + + o j s j ; + s i if facility j is disrupted but facility i is not. (1) Tus, rm i s longrun average sales rate is E [R i ] = (1 i j ) + E min ; n i + s i i + E min + + oi j s j ; + s i j ; () wic is obtained by taking expectations of eac outcome in (1) and weiging tem by teir respective steadystate probabilities. (Recall tat te above expression is an approximation under te assumption i 1 for i = 1; suc tat 1 0.) Te measure of consumer welfare in our model is availability, denoted as A, wic is de ned as te longrun average ll rate at te industry level tat accounts for all tree states of facilities. Wen neiter facility is disrupted, ll rate is equal to one since tere is ample capacity in te industry to satisfy all demands. Wen one of te facilities is disrupted, owever, ll rate (te ratio between expected industry sales rate and total demand rate ) may fall below one because of capacity reduction; it is equal to E [min f; Q 1 + Q g] =. Enumerating all tree states as above, we can write availability as A = (1 i j ) + E min ; i + s i + + s j i + E min ; + s i + j + s j j : (3) As expected, te following relationsip can be veri ed using te expressions in () and (3): A = E [R 1 + R ] =. Tat is, availability is equal to te ratio between te longrun average industry sales rate and te demand rate. As we prove in Lemma B.1 found in te Appendix, E [R i ] and A de ned in () and (3) can be expressed in terms of te distribution function F via te notation G (x) = R x 0 F (y) dy. First, rm i s 9
10 longrun average sales rate E [R i ] can be written as E [R i ] = ig 1 s i + jg 1 s j jg (1 s i+s j ) + : (4) In te context of our problem, te function G() represents te expected fraction of demands tat cannot be ful lled because of insu cient capacity (i.e., fractional excess demand). 4 Tus, te expression in (4) can be interpreted as: rm i s expected sales rate in te absence of supply disruptions (te rst term on te rigtand side) adjusted downward by te demands tat cannot be ful lled at disrupted facility i (te second term), adjusted upward by te demands tat cannot be ful lled at disrupted facility j and spilled over to undisrupted facility i (te tird term), and nally adjusted downward by te demands tat cannot be ful lled eiter at disrupted facility j or at undisrupted facility i (te last term). Similarly, availability A can be written as A = 1 i + j G (1 s i+s j ) + : (5) Tis is interpreted as: maximum availability of one (te rst term) adjusted downward by te fraction of demands tat cannot be ful lled at eiter facility wen disruptions occur (te second term). Te expressions (4) and (5) are used extensively in our analysis. 3.3 Objectives and Decisions As we outlined at te beginning of tis section, we consider two scenarios based on weter or not te price varies wit total industry capacity. In eac scenario, we study a simultaneousmove game between rm 1 and rm wo set teir spare capacity levels s 1 and s competitively at time zero. Eac rm s payo is equal to its longrun average pro t, denoted by i, i = 1;. Te exact expression for i is presented in 4 and 5. Firm i sets s i in order to maximize its payo i. We identify te Nas equilibrium of tis capacity game, denoted by te superscript, and study its properties. To deliver clean insigts, in several places of our analysis we pay special attention to te symmetric equilibrium tat emerges under te assumption 1 =, i.e., te two facilities ave identical disruption probabilities. 4 For example, te second term on te rigtand side of (4) is derived as follows. Conditional on facility i being disrupted (wit probability i) and left wit realized total capacity of + si, te amount of excess demand at eac moment is + si since te facility receives te demands at te rate of. Ten te expected excess demand is E i + R si = 1 si = 1 s i 0 x f (x) dx = G 1 s i. Terefore, te expected fraction of total demands tat cannot be ful lled by reduced supply is equal to G 1 s i. 10
11 4 Capacity Coices Under Fixed Price 4.1 Equilibrium of Capacity Game In tis section we assume tat unit price of te product is set to a constant value p at time zero and remains uncanged afterwards. Applied to te generic drug industry, tis assumption simulates an idealized version of te price control sceme mentioned in 1 wic creates price stickiness. Wit te price xed at p, rm i s payo is equal to i = s i + pe [R i ], i.e., te cost of olding s i units of spare capacity plus te expected revenue in longrun averages. (Recall tat te cost of installing capacity is assumed to be sunk at time zero.) Using te expression of E [R i ] derived in (4), te payo function can be written as i = s i + p 1 i G 1 s i + j G 1 s j j G (1 s i+s j ) +i : (6) As we outlined in 3.3, at time zero rm 1 and rm set teir spare capacity levels s 1 and s in a simultaneousmove game. Te Nas equilibrium of tis game is speci ed in te following result. (Note tat, as we assumed in 3.1, in all results below we restrict attention to parameter combinations tat satisfy p i < for i = 1;.) Proposition 1 (Equilibrium under xed price) A unique Nas equilibrium (s 1 ; s ) of te capacity game exists and is identi ed as follows. (a) If 1 + p, ten (s 1 ; s ) = (0; 0). (b) If max f 1 ; g < p < 1 +, ten (s 1 ; s ) = (bs 1; bs ) were bs 1 > 0 and bs > 0 satisfying bs 1 +bs < is a solution to te system of equations i F 1 s i + j F 1 s i+s j = p for i; j f1; g wit i 6= j. All proofs are found in A of te Appendix. Te proposition identi es te condition under wic rms ave incentives to old nonzero spare capacities: p ( 1 + ) >. If tis condition is violated, no spare capacity exists in te industry, tus maximally exposing consumers to supply disruptions. Te condition states tat investing in spare capacity is economically justi ed if te pro t tat a unit of spare capacity can generate during disruptions at any of te two facilities exceeds te cost of olding it. In oter words, not only sould te unit price be su ciently ig but also te disruptions in te entire industry occur relatively frequently (i.e., relatively large 1 + ). Tis implies tat eac rm s capacity investment decision sould factor in reliability of te oter rm s facility, in addition to tat of te rm s own facility. Tis externality arises because of demand spillovers; if rm j as insu cient amount of capacity during disruptions at its facility, te resulting excess demands are directed to rm 11
12 i, creating additional revenue opportunities for te latter and increasing te value of eac unit of spare capacity it olds. From tis reasoning it is clear tat te two rms spare capacities act as substitutes, wic can be matematically veri ed using te results in Proposition 1. As we observe from Proposition 1, te equilibrium decisions (s 1 ; s ) are implicitly speci ed as a system of equations tat is not readily interpretable. For tis reason, we rst examine a simpli ed setting. Corollary 1 Suppose tat i is uniformly distributed for i = 1; and 1 = =. Ten: (a) s 1 = s = 0 and A = 1 if p ; (b) s 1 = s = 3 1 p > 0 and A 1 = 1 18 p if < p <. In tis special case were capacity yield i is uniformly distributed and te rms ave identical disruption probabilities 1 = =, we obtain closedform solutions for te equilibrium spare capacities s 1 = s and te corresponding availability A. As it turns out, te most important properties of te equilibrium are captured in tese simple equations. We discuss tem next. 4. Properties of Equilibrium First, te equations for s 1 and s in Corollary 1 con rms te intuition tat rms increase teir spare capacity investments in response to iger price, provided tat spare capacities are pro table; from te corollary, we see tat s i = 0 if price p is small (part (a)) but s i > 0 increases in p if p is su ciently large (part (b)). As a direct consequence, availability A is independent of te price p for small p but it increases in p oterwise. In oter words, iger pro t margin for spare capacity raises availability by providing te rms wit more investment incentives; see Figure 1(a) for illustrations. Wile tis is te expected beavior, an interesting pattern emerges once we examine ow it interacts wit te disruption probability. Consider varying te value of. If p is small, it is clear from te expression for A in Corollary 1(a) tat a iger cance of disruptions (iger ) leads to lower availability; tis is in line wit intuition. However, if p is large, te opposite is true. From te expression for A in Corollary 1(b), we see tat iger leads to iger availability. In oter words, as facilities are expected to encounter more frequent and prolonged disruptions, product availability goes up. Te contrast between tese two regimes implies tat te availability ranking for di erent values of is reversed as te price p increases. (For example, te tree examples in Figure 1(a) sow tat availability is ranked in te order of = 0:08, = 0:10, and = 0:1 wen p is small, wereas te order canges to = 0:1, = 0:10, and = 0:08 wen p is large.) Figure 1(b) presents a di erent view of te same observation, clearly sowing tat availability increases in provided tat rms ave incentives to invest in spare capacities (s i > 0). 1
13 p = 1. A θ = 0.1 θ = 0.10 θ = 0.08 A p = 1.0 p = p (a) Availability as a function of price p θ (b) Availability as a function of disruption probability θ Figure 1: Availability A at te symmetric equilibrium under xed price. Te left panel sows A as a function of price p for tree values of symmetric disruption probability 1 = =. Te rigt panel sows A as a function of for tree values of p. In tese examples, te capacity yield random variable i is assumed to be uniformly distributed wit = 0: and = 1. In fact, tis is a general result it is not a byproduct of restrictive assumptions in Corollary 1. Even in cases were te capacity yield is nonuniformly distributed and/or te disruption probabilities are asymmetric ( 1 6= ), te same reversal olds. Tis is proved in te next proposition. Proposition At te nonzero equilibrium (s 1 ; s ) = (bs 1; bs ) speci ed in Proposition 1+bs i > i > 0 for i = 1;. Tus, availability A evaluated at te equilibrium increases in disruption probability at eiter facility ( 1 or ), given tat te rms nd it pro table to invest in spare capacities. Tis is a more general result tan te one inferred from Corollary 1, since it is no longer assumed tat disruption probabilities at te two facilities vary in tandem and te only condition imposed on capacity yield random variable is tat its cdf F is logconcave. Summarizing, te equilibrium of te capacity game is establised in suc a way tat, as te disruption probabilities 1 and gradually increase, te corresponding canges in availability A exibit a sarp reversal. If disruptions are rare and sort (small 1 + ), ten availability decreases in 1 and. By contrast, if disruptions are frequent and long (large 1 + ), ten availability increases in 1 and. Te necessary and su cient condition for tis reversal is tat spare capacities are pro table for te rms, i.e., te condition p ( 1 + ) > from Proposition 1 is satis ed. As Proposition reveals, availability increases in disruption probabilities because te rms respond to te latter cange by increasing spare capacity levels (s 1 + s increases in i); te greater te capacities, te iger te availability. Tis rm response is intuitive, since te economic value of spare 13
14 capacities increases if disruptions occur more often and last longer. However, it is not immediately clear wy tis endogenous capacity coice determines te direction of availability cange, since it represents only an indirect e ect of increased i. Te direct e ect of increased i puses availability in te opposite direction, because witout te rms interventions, more frequent and prolonged disruptions (i.e., less reliability) lower availability. Proposition states tat te net outcome of tese two competing forces is suc tat te indirect e ect dominates te direct e ect. Wat is signi cant from tis result is tat no middle ground is reaced, resulting in an unambiguous reversal of te usual relationsip between reliability and availability: iger cance of disruptions leads to iger availability. Interestingly, a key driver of tis counterintuitive result is te assumption tat capacity yield random variable i as a logconcave distribution function. Given tat logconcave distributions are ubiquitous (most wellknown distributions used in probability modeling belong to tis category; see Bagnoli and Bergstrom 005), we see tat te seemingly contradictory relationsip between reliability and availability is quite robust. In fact, it can be proved tat te same result olds even in alternative scenarios were tere is no competition; see Proposition B. in te Appendix. To gain intuition, let us examine a simpler variation of te model in wic a single rm managing one facility sets te spare capacity level s given tat it faces te disruption probability and demands arriving at rate. An analysis similar to te one above reveals tat, for su ciently large p and, te rm cooses bs > 0 tat satis es te optimality condition F (1 bs=) = =p, wic maps to te equilibrium condition found in Proposition 1. Te leftand side of tis equation represents te longrun average cance of sortage occurrences; conditional on te facility being disrupted wit probability, sortage occurs if te te total supply + bs is less tan te total demand, i.e., wit probability Pr( + bs < ) = F (1 bs=). Rearranging te above equation, we get: 1 F 1 bs = p : (7) p Tis equation is analogous to te newsvendor formula, as te leftand side can be interpreted as te instock probability (in longrun average) and te rigtand side as te critical ratio. At te optimal coice bs, availability is A = 1 G 1 bs : (8) Notice te similarity between te expressions for instock probability and availability appearing in (7) and (8); te only di erence between tese two service measures is tat te distribution function F in (7) is replaced wit G in (8), te two linked via te relationsip G (x) = R x 0 F (y) dy. Now, consider wat appens wen disruption probability is increased by an in nitesimal amount. 14
15 Di erentiating (7) wit respect to and rearranging te = F (1 bs=) f (1 bs=) : (9) Te leftand side of tis equation denotes te rate of cange in te normalized spare capacity level bs= in response to a percentage increase in disruption probability. Hence, (9) speci es ow te rm adjusts its spare capacity to maintain te optimality condition (7), or equivalently, to preserve payo maximizing instock probability. To see ow tis capacity adjustment impacts availability, we di erentiate (8) wit respect to and combine te result wit (9) = G 1 bs + F = f (1 bs=) G (1 bs=) + F (1 bs=) f (1 bs=) > 0; (10) were te inequality follows from logconcavity of F since it implies F (x) =f (x) > G (x) =F (x) (Bagnoli and Bergstrom 005). Tus, we arrive at te same conclusion as above: even in te single rm, singlefacility case, availability increases in disruption probability provided tat te rm as an incentive to invest in spare capacity. 5 From tese discussions we conclude tat te availability increase originates from te rm s optimal adjustment of capacity tat preserves instock probability, wic turns out to overcompensate for te loss of availability. A reasoning based on te newsvendor model analysis elps understand tis result better. Recall tat instock probability quanti es a binary service outcome, namely te cance tat all demands are satis ed by supply, 6 wereas availability (or ll rate) quanti es an incremental service outcome, namely te fraction of demands satis ed by supply. As tese de nitions convey, instock probability is a less exible service measure tan availability is, and as suc, it is more sensitive to a cange in environmental variables; a drop in instock probability due to increased is greater tan a corresponding drop in availability. Consequently, compensating for te loss in instock probability requires a greater amount of capacity tan wat is needed to compensate for a similar loss in availability. Logconcavity of te distribution function F ensures tat te rm s extra compensation 5 Alternatively, one can sow te same result using anoter property of logconcave distribution functions. Combining (7) wit (8), we can write availability as A c G(1 bs=) G(z) R z = 1, were te ratio can be rewritten as z x f(x) dx p F (1 bs=) F (z) 0 F (z) (z) via integration by parts. Bagnoli and Bergstrom (005) call (z) meanadvantageoverinferiors function, and prove tat logconcavity of F implies tat it is monotone increasing. In te context of our problem, (z) can be interpreted as follows. Suppose tat customers arrive at te facility and form a queue, obtaining te product on a rstcome, rstserve basis. Te customer at te z t percentile is at te position z in te queue. Ten te expected sales conditional on a stockout occurrence for te z t percentilecustomers is R z x f(x) R z dx, and terefore (z) = z x f(x) dx measures te 0 F (z) 0 F (z) gap between te queue position of an un lled customer and te expected number of lled demands. Hence, tat (z) is monotone increasing captures te notion tat te marginal contribution to expected sales by an additional customer in te queue is diminising. Tis monotonicity gives rise to our > 0 in (10). Te binary de nition of instock probability enters into a newsvendor rm s optimality condition because te rm weigs te cance of selling a marginal unit of capacity versus te cance of not selling it. 15
16 to preserve te payo maximizing instock probability results in iger availability tan were it started, as illustrated in (10). In essence, overcompensation arises because te sortage risk as a greater impact on te rm s pro tability tan it does on consumers cance of obtaining te product. Returning to te duopoly setting of our model, we see from Proposition tat te overcompensation e ect survives under competition. In fact, tis e ect is ampli ed by competition. Tis is because decentralization provides eac rm wit an extra motive to increase its capacity: demandstealing. As selfinterested rms do not fully internalize te negative externality tat demand substitution creates, tey attempt to pro t from eac oter s spillover demands by in ating teir capacities. As a result, competition raises availability (tis is proved in Proposition B. found in te Appendix). Note tat an analogous result is wellknown in inventory competition models (e.g., Netessine and Rudi 003), and it is replicated in our setting were demand substitution occurs because of supply uncertainty, not because of demand uncertainty as assumed by most in te literature. Terefore, te availability increase we observe in Figure 1 is caused by a combination of two e ects: rms inerent tendency to overcompensate for te loss of availability, plus teir competitive capacity overinvestment to steal eac oter s demands. Tis implies tat te reversal of te usual relationsip between reliability and availability, wic we ave discussed tus far, is likely to be prevalent in te generic drug industry were competition is common. In 6 we discuss te implications of tis nding in te context of generic sterile injectable drug sortages. 5 Capacity Coices Under SortageInduced Price Increase Building on te insigts from te last section, we now relax te xed price assumption and study ow rms capacity decisions and availability are in uenced by price variation. Speci cally, we modify te base model by assuming tat unit price of te product increases in te amount of capacity sortfall. As we brie y mentioned in 1, suc sortageinduced price increases are observed in practice despite te government s e ort to contain tem. Evidently te main reason for instituting a price control policy is to minimize te ealt care expenditure and protect consumer welfare. However, te impact of price variation on product availability is not clearly understood; our analysis in tis section seds ligt on tis issue. We model price variation as follows. Suppose tat unit price of te product adjusts instantly to te total industry capacity Q 1 + Q present at a given moment, were Q i = i = + s i is a random variable denoting te capacity tat facility i possesses. Te industry undergoes a sortage if and only if a disruption reduces te total capacity to Q 1 + Q <, i.e., te demand rate exceeds available production capacity. We assume tat te price is xed at a minimum value p 0 if tere is ample capacity (Q 1 +Q ), wereas te price increases in proportion to te amount of sortfall if a sortage occurs 16
17 and all capacities are utilized (Q 1 + Q < ). Note tat te xed minimum price assumption re ects te idea tat unutilized capacities bring no economic bene ts. Given tis speci cation, te price is a random variable since its value depends on realization of Q 1 + Q. Tis price, denoted by P, is equal to P = p 0 + b ( Q 1 Q ) + were te coe cient b represents te rate at wic te price increases in proportion to te amount of sortfall ( Q 1 Q ) +. We refer to b simply as price sensitivity. (Note tat we recover te xed price model of te last section by setting b = 0 and p = p 0.) As before, we assume tat at most one facility is disrupted at a given moment via te assumption i 1 and apply te approximation (1 1 ) (1 ) 1 1. Since Q 1 + Q = + i + s 1 + s wen facility i is disrupted wit capacity yield i and facility j is undisrupted, te price P de ned above can be written as 8 < p 0 P = : p 0 + b if neiter facility is disrupted, + (11) i s 1 s if facility i is disrupted but facility j is not. Firm i s payo is ten equal to i = s i + E [P R i ], were te sales rate R i is de ned in (1). Combining (1) and (11) and taking expectations yield i = s i + p 0 1 i G 1 s i + j G 1 + b 4 i 1 s i+s j G + b 4 i 1 s i+s j F (1 s i+s j s j j G (1 s i+s j (1 s i+s j ) + + j 1 + s i G ) + s R (1 i +s j i ) +i (1 s i+s j ) +i ) + 0 x f (x) dx : (1) Similar to wat we assumed in te last section, we restrict attention to cases wit p 0 i < for i = 1;. Te expression for availability A is uncanged from (5), since its de nition does not explicitly include price. However, te availability is indirectly a ected by price variation, since te rms optimal coices for capacity levels s 1 and s are in uenced by te price. As before, we assume tat te rms set s 1 and s at time zero in a simultaneousmove game, coosing te values tat maximize teir payo s 1 and given in (1). In order to isolate te impact of price variation, in te following discussions we focus on te symmetric Nas equilibrium of tis game by assuming 1 = =, i.e., disruption probabilities at te two facilities are identical. Te equilibrium is speci ed in te next proposition. Proposition 3 (Symmetric equilibrium under sortageinduced price increase) Suppose 1 = = and b p 0 3, wic implies p 0 of te capacity game exists and is identi ed as follows. (a) If p 0 b 4 (3 1) 1, ten s 1 = s = 0. b 4 (3 1) 1 < p 0. A unique symmetric equilibrium (s 1 ; s ) 17
18 (b) If p b 0 4 (3 1) 1 < < p 0, ten s 1 = s = bs 0; 4 is a solution to te equation p 0 F 1 s + F 1 4 s b F 1 4 s 3G 1 4 s =. Te proposition demonstrates tat te basic structure of te equilibrium is uncanged from tat speci ed in Proposition 1, were price was assumed to be xed. Namely, as in te xed price case, tere exists a tresold value for under wic rms do not invest in spare capacities (s i over wic tey do (s i incentives to invest in spare capacities. = 0) and > 0); again, a su ciently large cance of disruptions provides te rms wit A departure from Proposition 1, owever, is tat tis tresold value, = p 0 b 4 (3 1) 1, now depends on two additional parameters: price sensitivity b and expected capacity yield = E [ i ]. Tis raises te question: ow does te equilibrium adjust in response to canges in tese parameters? In particular, do te rms invest more or less in spare capacities if tey expect to see a greater price increase following a sortage (i.e., larger b)? Te answer is found in te next result. Proposition 4 At te nonzero equilibrium (s 1 ; s ) = (bs; bs) speci ed in Proposition 3(b), te following @ > 0 > 0. (b) 3, < 0 @b > 0 < 0. If < 1 3, on te oter and, tere exists a unique cuto = c > 0 for < @b < 0 < 0 for > c. Part (a) of te proposition parallels Proposition of te xed price case. Te result sows tat, even wen te price is allowed to vary according to (11), te equilibrium is establised in suc a way tat availability A increases in disruption probability if te combination of price coe cients p 0 and b makes capacity investment pro table for te rms. Tis result is analogous to Proposition ; ence, te main insigt from te last section tat te usual relationsip between reliability and availability is reversed is robust to price variation. Part (b) answers te question above, namely, ow price sensitivity b in uences te (symmetric) capacity coices s 1 = s = bs and te corresponding availability A. As te result sows, tese relationsips are not straigtforward; bs and A may or may not increase in b. Moreover, te answer depends on expected capacity yield, or equivalently, expected capacity loss percentage 1. Consider te rst case 1=3, i.e., wen capacity loss is expected to be relatively small. Te proposition states tat bot bs and A decrease in b in tis case. (Compare four curves for A in Figure (a) tat correspond to di erent values of b.) Tis is intuitive; if te rms anticipate tat a sortage will lead to a precipitous price increase (large b), ten tey ave incentives to restrict spare capacity levels so 18
19 0.98 b = b = 0 A b = 0.6 A b = θ (a) ρ = θ (b) ρ = 0.1 Figure : Availability A at te symmetric equilibrium under te variable price speci ed in (11), as a function of disruption probability 1 = =. Te four curves in eac panel correspond to b = 0, b = 0:, b = 0:4; and b = 0:6 (only te rst and last are labeled in te gure). Te examples in panel (a) ave = 0:5 wit uniform distribution for capacity yield i wile te examples in panel (b) ave = 0:1 wit power distribution F (x) = x 1=9. As in Figure 1, = 0: and = 1 are assumed. tat tey can take advantage of te price jump. 7 As te condition 1=3 indicates, owever, tis intuition is valid insofar as disruptions do not cause severe capacity loss. Indeed, if capacity loss is expected to be signi cant ( < 1=3), te opposite can appen: equilibrium spare capacity level bs and availability A increase in b. Tat is, rms anticipating a sarp price increase reserve more spare capacities, reversing te earlier observation. Figure (b) identi es te region in wic tis reversal appens. From tis discussion, we see tat a sortageinduced price increase creates subtle dynamics. An examination of te conditions under wic te reversal arises elps us understand tis result. According to Proposition 4(b), te reversal appens wen disruptions occur relatively infrequently ( smaller tan te cuto c but large enoug to ensure bs > 0) and te expected capacity loss is severe (small ). Te rst condition discourages rms from olding a large amount of spare capacities, since tey will be mostly idle. Because of restricted capacity, ten, supply sortfall during sortages is expected to be signi cant and so is te price jump. Te second condition, on te oter and, magni es te supply sortfall at eac facility since a disruption causes severe capacity loss. As a direct consequence, spillover demands increase; more demands originally allotted to a given facility are diverted to te oter facility. In sum, relatively infrequent disruptions and severe capacity loss togeter lead to a situation were rms are likely to face ig price and ig volume of spillover demands during disruptions. Tis creates an opportunity for eac rm to increase its pro t, acieved by expanding its spare capacity 7 Yurukoglu (01) o ers te same insigt: If payments are iger in sortage periods, ten manufacturers ave incentives to create arti cial sortages. 19
20 and capturing more spillover demands tat contribute a ig pro t margin. It ten follows tat greater price sensitivity b ampli es tis e ect, i.e., rms capacityexpansion incentives become stronger. Tis explains te reversal observed in Proposition 4(b) and Figure (b). Tis reasoning reveals tat te key to te observed reversal is eac rm s demandstealing motive. Tis is a purely competitive e ect; as we sow in Proposition B.4 found in te Appendix, tis countervailing e ect disappears in an alternative setting were a single rm makes capacity decisions for te entire industry. Hence, we conclude tat competition introduces a tradeo tat establises a nuanced relationsip between price variation and availability. Intuition suggests tat availability will be lowered if te price increases during sortages, since rms can exploit tis market response by creating arti cial sortages wit restricted capacities to in ate te price. As our analysis reveals, owever, rms incentives to acieve tis outcome are muted wen capacities are too restricted, causing a signi cant amount of spillover demands. Tese secondary demands may be attractive to te rms enoug to reverse teir capacity restrictions, tus resulting in iger availability. 6 Implications to Generic Sterile Injectable Drug Sortages Troug te analysis in 4 and 5, we uncovered new dynamics tat arise from interactions among supply disruptions, demand substitution, competition, and price control. All of tem are unique features of te generic drug industry, and terefore te insigts we gained from our analysis ave important implications to te sortage situation tat te industry is currently undergoing. In tis section we discuss ow our model predictions may explain some of te observed penomena. Two facts about te current sortage situation are particularly striking: a sudden, dramatic rise of sortages and teir persistency. Togeter, tey indicate tat te problem was caused by a structural sift in te industry. Healt care experts, policy analysts, and academics ave proposed several ypoteses to identify te cause. Some ave suggested tat industrywide mergers and consolidations ad been te direct cause of te problem (U.S. House of Representatives 01, Scweitzer 013), but te study by HHS (011) found little evidence to support tis view. Tere are also tose wo point to te industry s adoption of justintime (JIT) manufacturing practices as te culprit, since it advocates keeping minimum amount of inventory (Gerett 01, Gordon 013). However, tis does not explain te sudden rise of sortages, since JIT ad been around for many years prior to te onset of te problem. Peraps te most visible and controversial ypotesis is tat a cange in Medicare Part B reimbursement policy, part of te Medicare Modernization Act (MMA) of 003, was directly responsible for causing te sortages. Tis policy cange introduced a new basis for calculating drug purcase reimbursement rates, and it ad an e ect of signi cantly reducing average payments to service providers 0
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