Course Hero - We put you ahead of the curve!
You have requested the below document.
- Title: pdf_CC32
- Type: Notes
- School: UCLA
- Course: ANDERSON 3243
- Term: Fall
competition Cournot under yield uncertainty: The case of the U.S. in uenza vaccine market The number of rms producing in uenza vaccine for the U.S. has been declining steadily in the recent past with essentially only two manufacturers supplying the vaccine since 2002. Numerous articles in the popular press have blamed low market price, insu cient incentives and uncertain demand as the key causes. However, economic theory predicts that an oligopolistic market with unregulated but costly entry will experience excess entry and oversupply, not the undersupply observed in the market for in uenza vaccine in recent years. In this paper we show how interaction between yield uncertainty in the production process and rms strategic behavior can contribute to a high degree of concentration in an industry and a reduction in the industry output and the expected consumer surplus in equilibrium. We analyze the social trade-o between risk pooling (by diversi cation of supply) and economies of scale (by avoiding duplication of xed costs). We conduct numerical analysis with realistic parameters to assess the impact of yield uncertainty on the U.S. in uenza vaccine market. We conclude tentatively that increased xed cost (such as regulatory cost) is likely to be a larger driver of concentration in this particular market than yield uncertainty. 1 Introduction The number of rms producing in uenza vaccine for the U.S. has been declining steadily in the recent past. Two manufacturers, Sano Pasteur and Chiron, had been supplying all injectible vaccine since 2002, down from around ve in the 1990s and more than a dozen in the 1970s (Brown, 2004), and a third manufacturer (Glaxo-SmithKline) entered the U.S. market after the 1 supply crisis in the 2004-2005 season. The non-injectible vaccine Flumist still only accounts for 2% of the total market. Articles in the popular press have blamed low market price, insu cient incentives and uncertain demand for this high degree of concentration and for the frequent vaccine shortages observed in the recent years (Forbes, 2004; Newsweek, 2004; Time, 2004). However, the existing evidence does not conclusively support these claims. The price for in uenza vaccine, unlike other vaccines, is not controlled by the government (Danzon et al., 2004) and has increased from $2 to around $8 per dose in the past ve years (Forbes, 2004). Also, the demand for in uenza vaccine has been increasing steadily over the past decade as can be seen from the immunization rates (O Mara, 2003). Other possible reasons for exit of rms include mergers and acquisitions, plant closures resulting from inability to meet stringent regulatory standards and the market for vaccines being less pro table and much smaller compared to that for other pharmaceutical products. Danzon et al. (2005) argue that high country-speci c regulatory cost is one of the key factors that would drive the long term equilibrium in the U.S. vaccine market to be characterized by one or u two suppliers. While these hypotheses might provide some explanation for the reduction in equilibrium number of rms over time, they do not address the question of whether this equilibrium is socially optimal. The American Antitrust Insititute has argued for more government involvement in order to build surge capacity, claiming that the free market process is not working satisfactorily (American Antitrust Institute, 2004). Economic theory, on the other hand, predicts that an oligopolistic market with unregulated but costly entry, such as the in uenza vaccine market, will experience excess entry and oversupply compared to the social optimum. One additional characteristic of the in uenza vaccine market that further complicates the situation has received considerable attention recently in the trade literature but not yet in the academic literature: the yield uncertainty in the production process. The manufacturing process for in uenza 2 vaccine involves growing the virus in chicken eggs and later extracting, purifying, inactivating and packaging the vaccine (Gerdil, 2002). Due to the inherent uncertainty regarding the growth characteristics of the viral strains, the quantity of vaccine that can be obtained per chicken egg is uncertain (National In uenza Vaccine Summit, 2006; National Vaccine Advisory Committee, 2003; Powermed, 2005; Gerson Lehrman Group, 2005; GAO, 2001). The magnitude of the challenge posed by the yield uncertainty in in uenza vaccine production is illustrated by quotes such as ...the yield of candidate strains sometimes is not as high as desired which results in fewer doses, or strains may take additional time to obtain optimal yields, resulting in delays in the availability of vaccine (National Vaccine Advisory Committee, 2003) and The rst [major factor contributing to the delay in vaccine availability in 2001] was that two manufacturers had unanticipated problems growing one of the two new in uenza strains introduced into the vaccine for 2000-01 (GAO, 2001). We model the e ect of yield uncertainty on the in uenza vaccine market using using a twostage game of oligopolistic competition. In the rst stage, rms simultaneously decide whether to enter the market by incurring a xed cost of entry. In the second stage, each entering rm selects the target production quantity. Then each rm yield is realized, actual quantity produced is s brought to the market and price emerges according to the traditional model of quantity (Cournot) competition. We employ this model to answer the following speci c questions: (i) What is the impact of yield uncertainty on the quantity produced by each rm, total output of the industry and total number of rms in the market under competitive equilibrium i.e., without any intervention by the social planner? (ii) What is the impact of yield uncertainty on consumer welfare? How should society trade o the risk pooling value of supply diversity against the e ciency of having a single source? (iii) What conditions result in less entry and lower production as compared to the social optimum? 3 (iv) Which regulatory interventions (supply-side or demand-side) are more e ective and under what conditions? Although originally inspired by the in uenza vaccine market, our model applies to several other industries with yield uncertainty including bio-pharmaceuticals, semiconductor manufacturing and agriculture. The rest of the paper is structured as follows. Section 2 contains background information on the in uenza vaccine market. Section 3 reviews related literature from operations management (OM), economics and public health economics on yield uncertainty, competition and vaccinations respectively. In section 4, we present the basic model. Section 5 outlines the main results concerning the competitive equilibrium and socially optimal solutions while section 6 discusses numerical experiments with data pertinent to the U.S. in uenza vaccine market. We provide some concluding remarks in section 7. 2 Background on in uenza The 20th century has seen several in uenza pandemics with the most severe in 1918 causing close to 20 million deaths worldwide (WHO, 2002). Every year, 10-20% of the population gets in uenza and nearly 36,000 people die of the resulting complications in the U.S. alone (Thomspon et al., 2003). In uenza and resulting complications are the sixth largest cause of death in the U.S. (Martone, 2002) with estimated annual costs of $11-18 billion (WHO, 2002). The most important reason for the persistence of in uenza epidemics is the uncanny ability of the virus to continuously adapt itself every season, a phenomenon called the antigenic drift. As a result, the composition of the vaccine has to be reviewed every year and can undergo frequent changes. The recommendations for vaccine composition are made every year by the WHO, in February for the northern hemisphere (for the u season lasting from October to February of next year) and in August for the southern hemisphere 4 (for the season lasting from May to August of next year). In addition, periodically, antigenic u shift, in which genetic material of di erent strains of virus are recombined results in pandemics. (The current public focus on avian stems from the anticipation of a similar pandemic due to u antigenic shift in the H5N1 strain.) The challenges faced by the in uenza vaccination system in the U.S. can be categorized broadly into demand-side challenges and supply-side challenges. Supply-side challenges in an in uenza supply chain arise primarily from the combination of long production lead-time, short immunization season and frequent changes in the vaccine composition. Production of in uenza vaccine involves a long and complex biological process. The virus is grown in chicken eggs and later inactivated, puri ed and processed to manufacture the vaccine (Gerdil, 2002). The entire process takes six to eight months. Hence the manufacturers have to decide on the production quantity long before complete information about demand is available. In addition, due to the continuous change in the constituent strains, unused vaccine from the previous season cannot be utilized this season. Williams (2005) and Yadav (2005) provide a detailed discussion of these distinguishing features of the in uenza vaccine supply chain and suggest various improvement opportunities. These challenges are compounded by the high yield uncertainty in the production process, discussed earlier. Due to the long lead-time it is impossible to take any recourse later if faced with a particularly virulent strain of the virus or higher than expected demand or lower than expected yield (Danzon et al., 2005). The e ects of uncertain yield on the supply of vaccine are clearly evident from the recent U.S. experience. In 2004-05, Chiron vaccine manufacturing plant in the s U.K. was shut down by regulators due to bacterial contamination resulting in a reduction of total supply to the US market by about 50%, causing unprecedented shortages. The U.S. had also faced considerable shortages in the 2003-04 and 2001-02 seasons due to an early onset of the epidemic u and unexpected delays in the production process respectively. 5 On the demand side, immunization rates are lower than is socially optimal. Immunization of elderly citizens has been shown to be cost bene cial (Nicol et al., 1998) and is recommended. However, in the U.S., as recently as 2002-03, the immunization rate was only around 60% among the elderly and even lower in other population groups (O Mara et al., 2003). Most other countries have even lower immunization rates. Key factors cited by elderly people for low immunization rates include perceived good health, lack of advice from medical personnel and negative views on e cacy and safety of the vaccine (Evans and Watson, 2003). Other probable factors include lack of health insurance and high cost of vaccination. More generally, public health economists have long argued that individuals fail to internalize the positive externalities arising from vaccination, resulting in lower rates of immunization than is socially optimal. See Philipson (2003) for more details. 3 Literature Review This work draws on and contributes to three distinct streams of literature. First, we extend the literature on the stochastically proportional yield model in operations management (OM) to a competitive setting. Second, we show how yield uncertainty a ects the existing results in the oligopoly literature that discusses various models of competition with endogenous entry. Third, we build on the public health economics literature to which we also contribute by simultaneously studying supply and demand side factors traditionally analyzed in isolation. The model of yield uncertainty employed here has been widely used in the OM literature and is referred to as the stochastically proportional yield model. However, most of this OM literature considers the impact of yield uncertainty on either the production planning decisions of a single rm or procurement decisions of a single rm buying from multiple non-competitive suppliers with uncertain yields (Yano and Lee, 1995). Henig and Gerchak (1990), in a single rm model, show using an approximation that a higher yield variance results in lower optimal target production 6 quantity. The rst part of our analysis shows that this result extends to a competitive setting. Anupindi and Akella (1993) and Gerchak and Parlar (1990) discuss the value of diversi cation in the case of a given number of unreliable suppliers. Recently Federgruen and Yang (2005) and Dada et al. (2007) consider the problem of procurement from multiple suppliers with di ering reliability and cost. In the second part of our analysis, the number of suppliers is endogenously determined through an entry game. Carr et al. (2005) consider a competitive model of demand and capacity uncertainty. They show that a reduction in yield uncertainty can reduce the rm pro t, if process improvement leads s to an e ective over-capacity in the industry resulting in sti er price competition. One of our results is consistent with this, but in our model the increase in quantity produced is a rational decision rather than a direct outcome as in Carr et al. (2005). Moreover, Carr et al. (2005) do not consider entry decisions and their main focus is on studying the interaction between process improvement and competitive forces. In short, we contribute to the OM literature by studying the impact of yield uncertainty on strategic decisions of the rm such as entry and production quantity. Other applications of OR/OM models to the in uenza context have focused on control (Finkelstein et al., 1981) and management (Longini et al., 1977) of epidemics and on strain selection (Wu et al., 2005). Williams (2005) and Yadav (2005) provide a detailed review of the structure of the in uenza vaccine supply chain and propose that an information hub and government buy-back scheme would improve its performance. Vives (1999) provides a detailed account of the vast literature related to the Cournot (1838) model of oligopolistic competition. Part of our paper focuses on the question of rm entry in the context of oligopoly. Mankiw and Whinston (1986) compare the number of rms in the freeentry equilibrium with the number of rms that a social planner would choose. They show that under a decreasing inverse demand function and convex cost structure entry of an additional rm 7 reduces the output of incumbents. Ignoring the integer constraint on the number of rms, this e ect is su cient to ensure that there will always be excess entry relative to the social optimum. Von Weizs cker (1980) reaches similar conclusions using a linear inverse demand function and numerical examples. We show that in contrast to Mankiw and Whinston (1986), adding yield uncertainty leads to less entry than optimal in a homogeneous goods market with business stealing e ect. Thus, we contribute to the literature on oligopoly by including yield uncertainty, so far not considered in the context of Cournot competition. Traditionally, the uncertainty studied in Cournot models has been related to demand or cost or that resulting from players private information (Leland, 1972; Klemperer and Meyer, 1986). However, yield uncertainty is fundamentally di erent from demand or cost uncertainty as it relates to a decision variable (production quantity) rather than an exogenous parameter; this has some important consequences. Our model of consumer demand for vaccines is closely related to Brito et al. (1991), where consumers di er in the cost of vaccination, and a consumer likelihood of contagion from the s unvaccinated population depends on the number of unvaccinated individuals. This interaction between individually rational decisions and epidemiological dynamics has recently received attention in the public health economics literature, reviewed by Philipson (2003). 4 4.1 Model Formulation Modeling the supply We assume that the industry consists of n manufacturing rms denoted by i 2 f1; 2; : : : ng possessing identical manufacturing process. If qi is the production quantity targeted by rm i (as re ected by the total number of chicken eggs chosen), then the actual quantity produced is given by qi = where i i qi , is a random variable re ecting the random yield per egg for rm i: Since the yield 8 uncertainty results in a random proportion of the target quantity being produced, this multiplicative model is also known as the stochastically proportional yield model. Yano and Lee (1995) mention that this model is appropriate when relatively large batch sizes are used, when the variation of the batch size from production run to production run tends to be small or when the yield losses might be relatively predictable for any particular set of conditions, but the conditions are not predictable. All these criteria are met in the case of in uenza vaccine production. Since all rms have the same production technology, i i is identically distributed for all rms. In addition, we assume that the = E[ i ] and 2 are independent (GAO, 2001). Let = Var[ i ] 8i. We include two marginal costs: (i) c1 per unit target quantity and (ii) c2 per unit actually produced. In our context, the rst cost is driven by the number of chicken eggs and the second cost corresponds to the cost of bottling and packaging the actual vaccine produced. 4.2 Modeling the demand bq, where e is a random variable We assume a linear inverse demand function given by p = a (e) ^ denoting the e cacy of the vaccine. The reservation price a depends on how e ective the vaccine is ^ against the circulating virus strain in the coming season and allows for the fact that the customers would be willing to pay more for a more e ective vaccine. While selecting the target production quantity, the rm does not know what the actual e cacy will be but does know the underlying distribution. We assume that e is independent of the yield uncertainty variable i, i.e., the e cacy of the viral strains selected for the vaccine is independent of the production characteristics for those strains. In some seasons due to the antigenic drift the strains in the vaccine will usually be less e ective, but that does not appear to be related to the production yield that will be obtained with these strains. 9 4.3 Modeling the market We model competition among rms as a two-stage game. First, the rms simultaneously decide whether to enter the industry. Each entering rm incurs a xed cost f . The manufacturers for in uenza vaccine decide on production quantities six to eight months before the onset of the u season. Hence the Cournot (1838) model is reasonable for the competition among the entering s rms in the second stage. Each rm sets its target production quantity, q i . After that, each rm yield i is realized, total production q = inverse demand function. X i iqi occurs and price p is set according to the above We solve this two-stage game using backward induction. We rst solve the second stage game for a given number of rms in the industry and derive the equilibrium target production quantities and pro ts as a function of this number. Then we analyze the rst stage game to nd the equilibrium number of rms in the market. 5 5.1 Equilibrium of the two-stage game Post entry competition under yield uncertainty In the second stage, given that there are n rms in the industry, each rm decides a target quantity q i at a cost of c1 q i . The uncertainty is resolved during the production process and qi = i qi is the actual quantity produced at a cost of c2 qi . The market price is given by the inverse demand Pn function p = a (e) b ^ and the expected pro t of the ith rm is given by i (qi ) = j=1 qj h i Pn Ee; i a (e) b ^ qj qi c1 q i c2 qi . Substituting for qi = i qi in this expression, de ning j=1 c= c1 + c2 and noting that e and i are independent, we obtain b@ 0 n X j=1 j q j AA i (q i ) = E 4@a 20 11 iqi cq i 5 3 (1) 10 where a = Ee [^ (e)]. Let a i denote the maximum pro t of the ith rm. Then, writing q i to denote the decisions of all rms other than i, the decision problem of the ith rm is = max qi 0 i (q i ; q i ) i (2) The equilibrium is found by solving the following set of equations: 2 X @ i (q i ) = (a c)E [ i ] 2bq i E 2 bE 4 i i @q i qi =q i i6=j Since E [ i ] = and Var[ i ] = 2 8i, the above system of equations has a unique solution = . Since we are primarily interested j qj 5 3 = 0 8i (3) that is symmetric. De ne the coe cient of variation in analyzing the impact of yield uncertainty on market and socially optimal solutions, we keep constant and only analyze the changes in rather than . Hence we can express all our results in terms of which simpli es the exposition considerably. The unique equilibrium of the Cournot game is straightforward: Lemma 1 The second stage Cournot game with yield uncertainty has a unique equilibrium in which: (i) The target quantity of each rm is given by qi = (a c) b[(n+1) 2 +2 2] 8i. ac b(n+1+2 2 (ii) The expected quantity produced by each rm is given by E[qi ] = qi = ) 8i. (iii) For given n, each rm target quantity and expected quantity is decreasing in the yield uns certainty as measured by . (iv) The expected pro t of each rm is given by i (n) = (a c)2 ( 2 +1) b(n+1+2 2 )2 8i. (v) For given n, each rm expected pro t is rst increasing and then decreasing in . s All proofs are provided in Appendix. Note that in the absence of any uncertainty, i.e., expected quantity produced reduces to qi = ac b(n+1) , = 0, the = (a c)2 : b(n+1)2 while expected pro t reduces to i 11 both familiar from Cournot competition without yield uncertainty. Moreover, for given n, higher yield uncertainty leads each rm to produce lower expected quantity. 5.2 Entry game Next, we focus on the rst stage of the game. We assume that there is a large population of identical potential entrants. Each of these potential entrants has a reservation pro t level of zero. All rms simultaneously decide whether to enter the market or not. We are not interested in which speci c rms out of the potential population enter, but only in the equilibrium number of entrants. For n 2 N to be the equilibrium number of rms in the industry, we must have and i (n i (n ) f + 1) f , as otherwise entering rms are losing money or earning su cient pro ts to attract additional entrants. Temporarily relaxing the integer constraint, the equilibrium number of entrants, x 2 R+ satis es i (x ) = f . Let nu 2 N denote the equilibrium number of rms under yield uncertainty and nd 2 N be the corresponding equilibrium number of rms for the deterministic case. Similarly, let xu , xd 2 R+ be the respective equilibrium numbers after relaxing the integer constraints. Let bnc denote the largest integer less than or equal to n. Lemma 2 The number of rms in the industry at equilibrium with and without yield uncertainty j k j k p a a pc pc is given by nu = 1+ 2 (1 + 2 2 ) and nd = 1 respectively. bf bf We now use these results to determine the impact of yield uncertainty on the equilibrium number of rms nu using the deterministic equilibrium number nd as benchmark. One might expect that uncertainty always (weakly) reduces the number of entrants in equilibrium, but the following proposition shows that that is not necessarily true. Proposition 1 The equilibrium number of rms under uncertainty (nu ) and in the determinn o ac ac istic case (nd ) satisfy (i) nu nd if pbf > 4 and or pbf 4 and (ii) nu nd if 1 12 n a pc bf > 4 and j o 1 , where a pc bf 1 , r ac p 2 bf 1 2 1. Recall that 1 equilibrium without uncertainty (nd k = nd . Thus, if the industry can support at most three rms at 3), then any amount of yield uncertainty (weakly) reduces the number of rms at equilibrium. However, if the industry can support three or more rms at equilibrium without uncertainty, then the equilibrium number of rms is (weakly) greater than nd if the uncertainty is lower or equal to a certain threshold. In other words, large uncertainty always results in exit of rms from the industry relative to the deterministic case, while limited uncertainty can actually cause more rms to enter in certain cases. To understand this, observe that yield uncertainty has two e ects on rms expected pro ts. First, yield uncertainty reduces the expected quantity produced by each rm. This has a negative e ect on the expected pro t. A secondary e ect is that yield uncertainty increases the market price by reducing output; this a ects all the units and not just the marginal units, which has a positive e ect on the expected pro t. For small levels of uncertainty, the positive e ect can dominate the negative e ect and hence cause a net increase in the expected pro t, thus attracting new entrants. However, for large uncertainty the net e ect is always negative and hence lowers the equilibrium number of rms. Moreover, from Proposition 1, it follows that the threshold level of uncertainty 1 is non-decreasing in the intercept of the inverse demand function a and non-increasing in the xed cost of entry f , marginal cost c, and price sensitivity b. In other words, if the industry has very high cost of entry, then relatively small uncertainty can reduce the number of entrants. 5.3 Total vaccine supply While clearly related to the number of entrants, we are ultimately interested in the impact of yield uncertainty on total expected quantity of vaccine produced in equilibrium, since in our model that is directly linked to the number of vaccinations and hence to the health care outcome for society. 13 Let qd be the total quantity produced at equilibrium in the absence of uncertainty and E [qu ] the total expected quantity produced at equilibrium under uncertainty. In Proposition 1, we already saw that limited levels of uncertainty can lead to increased entry. However, the next proposition shows that this is not true for total vaccine supply. Proposition 2 E[qu ] qd 8 0, i.e., the expected quantity produced by the market under yield uncertainty is lower than or equal to that in the deterministic case. The above result is true even for some levels of uncertainty where the number of rms at equilibrium is higher than in the base case, i.e., 1 > > 0. One might expect that higher quantities imply better societal outcomes, as production quantity equals number of vaccinations in our model. However, as seen below, we can prove this only in certain range of the yield uncertainty. 5.4 Social welfare To characterize the impact of these e ects on consumers, we compare the consumer welfare with and without yield uncertainty. At equilibrium, let E[CSu (qu )] denote the expected consumer welfare under uncertainty and let CSd (qd ) denote the consumer welfare in the absence of uncertainty. Formally, when total quantity produced is q, total expected consumer utility from vaccination is E Rq 0 (a bu)du and the expected amount paid by the consumers for vaccination is E [(a bq)q]. Hence, the expected consumer welfare in the case of yield uncertainty is given by E[CSu (qu )] = hR i q b E 0 u (a bu)du (a bqu )qu = 2 E (qu )2 . Similarly, in the deterministic case, the expected b consumer welfare is given by CSd (qd ) = 2 (qd )2 . Then; Proposition 3 De ne 2 , min the expected consumer welfare in equilibrium with and without uncertainty satisfy E[CSu (qu )] CSd (qd ) if > 2. n >0: jp 1+ 2 (a c) p bf k (1 + 2 2 ) + 2 j a pc bf 1 ko . Then, Also, 2 1 14 This result states that large uncertainty reduces the expected consumer welfare when compared to the deterministic case . The contrast with Proposition 2 (which holds for any level of uncertainty) arises because consumer welfare depends on E q 2 , not on E [q]. We are unable to compare CSd (qd ) and E[CSu (qu )] when < 2. 5.5 First best solution Having understood the impact of yield uncertainty on equilibrium entry and quantity supplied, we can now consider various interventions that could drive the market outcomes closer to the socially optimal ones. First, we formulate and solve the decision problem of a social planner who wants to maximize the total social welfare, then we compare the socially optimal solution to the equilibrium outcomes derived above. First-best denotes the solution to the social planner problem of maximizing the total social s welfare or the total surplus of society by choosing the number of rms (n) and the target production quantity of each rm (qi ) (Vives, 1999). This presupposes the existence of an omnipotent and omniscient benevolent agency, possibly government, that can costlessly and perfectly control both the structure of the industry and the conduct of the rms in the industry. Then the social planner s problem can be formulated as: Z q max E [W (q; n)] = max E qi ;n qi ;n n X i=1 n X i=1 (a bq)dq E [cq] nf 0 where q = qi = i qi denotes the total quantity produced by n rms. The rst term is the total expected consumer utility, i.e., the area under the demand curve for consumers who do purchase, and the second term is the expected variable cost of production if q(n) is the total quantity produced. The third term is the total cost of entry incurred by society if n rms enter the 15 industry. Simplifying we obtain the following social planner problem: s max E [W (q(n); n)] = max (a qi ;n qi ;n c)E(q) b E(q 2 ) 2 nf (4) This problem can be solved optimally by rst xing n and optimizing over qi , which we call the quantity problem. In the second step, we substitute the optimal qi in the original problem and optimize over n. We call this the structural problem. f f Lemma 3 Let qi b denote the rst-best planned production quantity of the ith rm and let qi b denote hi 2(a f f the corresponding actual quantity produced. Then (i) qi b = b[(n+1) c) 2 ] = 2qi and (ii) E qi b = 2 +2 2(a c) ; b(n+1+2 2 ) where = as de ned earlier. For a given number of rms, the socially optimal target quantity and expected production quantity for each rm is twice that under competition. The next step is to characterize the socially f optimal number of rms. Substituting the expression for qi b in (4) and simplifying, the structural problem is: max E [W (n)] = max n n 2 (a c)2 n(1 + 2 ) b n+1+2 22 nf (5) and the result is characterized in the following proposition. Proposition 4 Let nf b denote the number of rms in the rst best solution. Then 1 1 + 2 2 . Also nf b = 1 if 2(1+ 2 nf b < ) (a c) p . bf In the deterministic case, where = 0, it is easy to see that nf b = 1. This is in accordance with the existing intuition that the rst-best solution in a deterministic setting involves having a benevolent monopoly which produces the socially optimal quantity since society then incurs the j k c) p xed cost of entry only once. nd = (a bf 1 3 implies that nf b = 1 8 > 0. In other words, if the xed costs are high enough that the market can support at most three entrants in the deterministic case, then the rst-best solution is to have benevolent monopoly regardless of the level 16 of uncertainty. However, when nd > 3, there exist levels uncertainty of for which society prefers multiple suppliers since the value of diversity of supply is greater than the additional xed cost of entry. This risk pooling e ect occurs due to the concavity of consumer welfare. Each incremental unit of vaccine produced less leads to higher social costs. 5.6 Seond-best solution Now, we turn to the case where the social planner can regulate the number of rms in the industry, but cannot regulate their conduct, so that the entering rms engage in Cournot competition in the post-entry game. This solution is referred to as the second-best structural regulation or simply second-best (Vives, 1999). We shall focus on this case in greater detail due to the restrictive assumptions required for the rst-best. The social planner problem in this case is given by s b E(q 2 ) 2 max E [W (q(n); n)] = max (a n n n(a c) b(n+1+2 2 ) c)E(q) nf (6) Substituting E(q) = from Lemma (1) and E(q 2 ) = (a " c)2 1 2b (a c)2 n(n+ 2 ) b2 (n+1+2 2 )2 22 in (6), we obtain: # max E [W (q(n); n)] = max n n 1+2 +n 2 (n + 1 + 2 2 )2 nf (7) Relaxing n to x 2 R+ , it can be veri ed that E [W (q(x); x)] is strictly concave. Hence by restricting n 2 N+ , E [W (q(n); n)] can have at most two maximizers. Let nsb denote the element u of this set of two maximizers and let nsb denote the deterministic optimum. We begin by analyzing d the deterministic case and then extend the analysis to the case with uncertainty. Proposition 5 In the absence of yield uncertainty, the second-best number of rms is given by nsb 2 d a pc bf 2 3 1 1; a pc bf 2 3 1 + 1 . Also, nsb d 1 nd . In other words, in the absence of yield uncertainty, the equilibrium number of rms can be less than the second-best number of rms but not by more than one. This is because in the second-best 17 outcome, the rms are making positive pro ts causing more rms to enter. This result is identical to that of Mankiw and Whinston (1986). However, we show that including yield uncertainty can change the relationship between the second-best number of rms nsb and the equilibrium number u of rms under uncertainty nu . The result is summarized in the following proposition: Proposition 6 The number of rms in equilibrium nu and in the second-best solution nsb satisfy u p2 2 ac sb (i) nsb > nu if 1 nu if < 3 , where 3 > 0 solves pbf = 2(1+22+) 2 1+ . u 3 and (ii) nu This shows that if the yield uncertainty is larger than a certain threshold, then the number of rms at unregulated equilibrium will be less than in the second-best case. Since the total expected quantity produced n(a c) b(n+1+2 2 ) is increasing in n, the industry undersupplies at equilibrium whenever uncertainty is higher than that threshold. In contrast, for a low level of uncertainty, the outcome is the same as that in the deterministic case. 6 Application to the U.S. in uenza vaccine market In this section, we apply our model to the U.S. in uenza vaccine market. Our objective is to determine whether and how much yield uncertainty might have contributed to the observed exit from the U.S. in uenza vaccine market. We start with calibrating our model, then analyze the market equilibrium, evaluate the impact of demand-side and supply-side public policies on social welfare, and conduct sensitivity analyses. 6.1 Model calibration In this section, we derive the demand function for the U.S. market by assuming uniformly distributed consumer valuations which leads to a linear inverse demand function. Consider M individuals, who each demand zero or one unit of vaccine in a single-period context. The individuals di er only 18 in the expected cost incurred if they do not get vaccinated, denoted by v. This cost re ects the likelihood of getting infected and the resulting costs of health care, lost income etc. We assume perfect vaccination, i.e., after vaccination consumers stay perfectly healthy and do not incur any health care or other costs; relaxing this would not change our results. Following Brito et al. (1991), we assume that v follows a uniform distribution F (v) on the range [v; v]. Let p be the price of one dose of vaccine and let v be the valuation of the threshold consumer who is indi erent between getting vaccinated and not getting vaccinated, given price p. Then for a rational consumer, who does not account for the positive externality of vaccination mentioned earlier, p = v and q(p) = (1 F (v ) = vv vv F (v )) M is the total demand at that price. Substituting vp vv and p = v , we get q(p) = M . De ning a , v and b , (v v) M, the following inverse demand function is obtained: p=a bq (8) In our subsequent analysis we assume that the individuals are rational, but including a positive externality does not change our results. While we chose parameter values (summarized in Table 1) that represent the U.S. situation to the degree possible, some of the values are inevitably, at best, very rough estimates. The population (M ) was chosen to the be the U.S. population, which is Table 1: Parameter values for the U.S. in uenza vaccine market Parameter Value v($) 0 v($) 8 c ($) 3 f ($ million) 40 M (million) 300 0:64 approximately 300 million (U.S. Census Bureau, 2004). The lower limit of the customer valuation (v) can be normalized to zero. The upper limit of the customer valuation ( ) of $8 was chosen based on anecdotal evidence (Nichol, 2001) of the direct cost of vaccination and the fact that vaccination is a covered bene t under insurance for many customers. The true value of higher. But, the underlying distribution is also unlikely to be uniform. If 19 is likely to be much is interpreted as the upper limit of the mass market willingness to pay, then $8 seems to be reasonable. The value of s u = 0 was based on the fact that individuals behave in a self-interested manner with respect to vaccination. The variable cost (c) of $3 per dose was based on the costs of procurement from the manufacturers (O Mara et al., 2003) and assuming around 50% gross margin. The value of f is also not directly available. Gottlieb (2004) reports that an investment of around $300 million is required to build a new in uenza vaccine plant. A 10-20% cost of capital on this investment translates into an annual xed cost of $30 - $60 million dollars. During the 2000-01 season, Parkedale announced its departure from the in uenza vaccine market, writing o $45 million (Danzon et al., 2004). Based on these data, we chose an annual xed cost of $40 million, but let it vary from $20 million to $100 million in our sensitivity analysis. The yield uncertainty ( ) is even less observable. We rst estimated industry-wide yield uncertainty using the data in Table 2 (Strikas, 2005). We used the time-variation of quantity produced and supplied as a proxy for the underlying yield uncertainty. Since vaccination begins in October, manufacturers aim to supply all the vaccine by that time. The year-to-year variability in the degree to which supply is late is an indicator of the yield uncertainty. However, in order to control for idiosyncratic variations such as that in 2004 due to the Chiron failure, we normalized the quantity supplied until October (A) by the total supply for that year (B). We then calculated the standard deviation and mean of this normalized quantity (D) to estimate these values of for the industry. We corroborated using simply the standard deviations for total annual quantity produced (C) and 0:38. total annual quantity supplied (B) during 1999-2004. All these values were in the range 0:33 This would be an underestimate for the true coe cient of variation , since it does not take into account the variation between the targeted quantity and the nal quantity produced in each year. We assumed the value of = 0:45 based on this exercise. This value is the for the industry 20 Table 2: Quantity of in uenza vaccine produced and distributed (million doses) Year 1999 2000 2001 2002 2003 2004 Supplied by Oct. (A) 75.8 26.6 43.0 82.7 80.0 51.0 Total supplied, entire season (B) 76.8 70.4 77.7 83.0 83.1 57.1 Total produced (C) 77.2 77.9 87.7 95.0 86.9 61.0 D= 0.987 0.378 0.553 0.996 0.963 0.893 A B comprising two rms for the period under consideration. Assuming that the yields for the two existing rms are independent of one another, we calculated the but let of . for each rm as p 2 0:45 = 0:64, vary from 0 to 3 in the sensitivity analysis in the light of uncertainty about the true value 6.2 Analysis of market equilibrium Here, we calculate the equilibrium for the U.S. in uenza vaccine market as predicted by our model. Table 3 summarizes the di erence between the deterministic ( = 0) and the stochastic yield case ( = 0:64) for the equilibrium and optimal (second-best) solutions. The results show that even a relatively low level of yield uncertainty ( = 0:64) can eliminate the excess number of entrants at equilibrium predicted by the deterministic model, in this case because the number of entrants in the optimal (second-best) solution increases. More importantly, yield uncertainty results in a substantial reduction (17%) of expected total quantity produced and a corresponding reduction in social welfare (27%) and consumer welfare (19%). The equilibrium prediction from our model under uncertainty matches fairly closely with the observations from the U.S. market. While two rms had been in the market from 1999 to 2004, a 21 Table 3: Equilibrium and second-best solutions for the in uenza vaccine case Solution Socially optimal (second-best) number of rms Equilibrium number of rms Equilibrium industry output (million doses) Equilibrium price ($/dose) Equilibrium pro t per rm ($ million) Equilibrium consumer surplus ($ million) Equilibrium social welfare ($ million) =0 2 3 141 4:25 58:6 180 319 = 0:64 3 3 117 4:88 57:0 146 258 Di erence 50% 0% 17% 15% 3% 19% 24% third rm has entered in 2005. The real equilibrium industry output is around 100 million doses, in the same ballpark as the predicted 117 million. While this in itself certainly does not imply that our model and parameter values are correct, it seems to indicate some face validity. 6.3 Evaluation of demand-side and supply-side policies Having calculated the equilibrium outcomes, we now compare the performance of various policy interventions intended to improve the social welfare. These policies can be classi ed into demandside and supply-side measures. An example of a demand-side policy is to improve awareness about the bene ts of vaccination and thus implicitly shift the demand curve upward through an increase in a in (8). A supply-side intervention discussed in this paper is structural regulation, where the government regulates market entry through instruments such as entry taxes, subsidies or regulatory costs. Technological interventions such as a new production process to reduce the yield uncertainty are not considered here. We vary a (or correspondingly v) and from Table 1, keeping other parameters xed, to compare the demand- and supply-side interventions under di erent levels of yield uncertainty. We only compare the outcomes of these policies, not the costs, since that is not 22 our focus and we are not aware of any reasonable data to estimate these costs. Clearly any actual policy decision would require analysis of the costs as well as bene ts. First, we study the impact of demand-side policies on the structure of the industry (nu and nsb ) for di erent values of a and . In Figure 1, u correspondingly represents a $ increase in the value of a (or ) above the base case considered in section 6.2, where v = $8. We consider = 0:4; 0:8; 1:2 and 1:6. This corresponds to a 10% to 40% increase in the valuation of the average consumer given our intial range of [v; v] = [0; 8]. Figure 1 veri es the result proved in Proposition 6: yield uncertainty beyond a particular threshold can cause less entry than is socially optimal. This threshold increases as the demand curve shifts up, i.e., as a increases. Figure 1: Equilibrium and socially optimal number of rms as a function of policies. 10 for di erent demand-side 10 = $0.4 8 Number 8 Number = $0.8 6 4 2 0 0 nu* nus 6 4 2 0 0.5 1 1.5 2 0 nu* nus 0.5 1 1.5 2 Coefficient of variation Coefficient of variation 10 = $1.2 8 Number 10 8 Number = $1.6 6 4 2 0 0 nu* nus 0.5 1 1.5 2 6 4 2 0 0 nu* nus 0.5 1 1.5 2 Coefficient of variation Coefficient of variation Next, we study the impact of supply- and demand-side interventions on social welfare under various conditions depending on the value of . Let E [W (n; )] denote the social welfare when n rms enter the market and is as de ned above. This can be interpreted as the e ect of demand-side interventions such as improved awareness or better insurance coverage. The base case is given by 23 E [W (nu ; 0)], i.e., no supply- or demand-side intervention. We measure the performance of demandside intervention by calculating rd , calculating rs , E [W (nsb ;0)] u E[W (nu ;0)] . E[W (nu ; )] E[W (nu ;0)] and the performance of supply-side intervention by in Figure 2. The results show We plot rd and rs as a function of Figure 2: Impact of demand-side and supply side policies on social welfare as a function of . 7.00 rd, rs 6.00 rd ( =$1.2) rd ( =$1.6) 5.00 4.00 rd ( =$0.8) 3.00 rd ( =$0.4) 2.00 rs 1.00 0 0.5 1 1.5 2 Coefficient of Variation that in the given parameter range, a demand-side policy causing a 10% increase in the valuation of an average consumer already results in a higher increase in social welfare than a supply-side policy of regulated entry; higher increases in valuation result in even higher social welfare. Recall, though, that the costs of implementing these policies are not included in our model and we do not know whether achieving a 10% increase in the valuation of an average consumer may be much more complex and costly than implementing a supply-side policy of regulated entry. 6.4 Sensitivity analysis Since we are most uncertain about our estimates of f and , we conducted a sensitivity analysis to ascertain the impact of these parameters on our results. Recall that we have assumed constant and hence the change in to be is a ected through change in . The results are summarized in 24 Table 4. Each cell contains a pair (nu ; nsb ); i.e., the equilibrium number followed by the second-best number of rms for each combination of f and . The cells with X indicate that the industry is Table 4: Equilibrium and socially optimal number of rms for di erent values of f and f ($ millions) 20 30 40 50 60 80 100 =0 5; 5 4; 3 3; 2 3; 2 2; 2 2; 1 2; 1 = 0:5 6; 3 4; 3 3; 2 3; 2 2; 2 2; 2 1; 1 =1 6; 5 4; 4 3; 3 3; 3 2; 2 1; 2 1; 1 = 1:5 6; 6 4; 4 3; 3 2; 2 1; 2 1; 1 1; 1 =2 6; 6 3; 4 1; 3 1; 2 1; 1 X X = 2:5 4; 6 1; 3 1; 2 X X X X =3 X X X X X X X not viable for those combinations of f and : even one rm producing the vaccine would result in negative social welfare. We pay particular attention to cases where nu < nsb , highlighted in bold, as those are the cases where yield uncertainty can help explain lower entry than is socially optimal. The sensitivity analysis demonstrates that at higher levels of the xed cost of entry, f , even relatively low levels of yield uncertainty can result in less than optimal entry at equilibrium. Table 4 also shows that for a given level of yield uncertainty, an increase in xed cost f reduces the equilibrium number of rms. This provides an indirect support for the hypothesis that increased regualatory cost could explain the exit of rms from the U.S. market. Conversely, for relatively low levels of yield uncertainty, including those in the range we estimate and for most values of xed cost, the presence of yield uncertainty does not change the equilibrium number of rms. To summarize, the predictions of our model match reasonably closely with the observations from the U.S. market as seen from Table 3. Without accounting for the cost of implementation, we found that demand-side policies had higher impact on social welfare than the supply-side policies 25 over the range of parameters valid for the U.S. market, although our analysis only addresses a part of this issue and su ers from signi cant limitations. 7 Concluding remarks In this paper we analyze the e ect of yield uncertainty in Cournot competition. The model is based on the context of the market for in uenza vaccine, but applies to other settings with yield uncertainty, xed cost of entry and Cournot competition. We show that if yield uncertainty is su ciently large, less rms will enter in equilibrium than at the social optimum with regulated entry. This is in contrast to the traditional result (on Cournot competition without yield uncertainty) that excess entry will occur at equilibrium relative to the second-best social optimum. We also show that this uncertainty can reduce the expected total industry output and the expected consumer surplus in equilibrium. These results continue to hold even in the presence of positive externalities of vaccination. We report numerical analyses with parameter values pertinent to the U.S. in uenza vaccine market. The predictions for number of rms and quantity supplied are broadly comparable to what we actually observe in practice. In the relevant range of parameters, we nd that including yield uncertainty eliminates the excess entry compared to the socially optimal number of rms predicted by the traditional oligopoly model. However, in explaining the exit of rms from the market over years, increasing xed cost appears to be a more signi cant factor than the yield uncertainty. We also compare the performance of demand-side and supply-side policies aimed at improving social welfare. We nd that demand-side policies, though possibly much more di cult and costly to implement, are likely to be signi cantly more e ective than supply-side policies for the range of parameters characterizing the U.S. market. Acknowledgements 26 The authors would like to thank the Director and Product Manager of a distributor of in uenza vaccine, the V.P. (Specialty clinical services), Pharmacist Specialist and Pharmacy Buyer from a community hospital, and William Comanor, UCLA School of Public Health, for helpful discussions. The authors would also like to thank Sushil Bikhchandani, Scott Carr, the associate editor and three anonymous reviewers for their insightful comments on an earlier version of this paper. References American Antitrust Institute. 2004. Could the federal government have prevented the vaccine u shortage? An industrial organization perspective. October 29. Anupindi, R., R. Akella. 1993. Diversi cation under supply uncertainty. Management Sci. 39. 944-963. Billeter, M. 2004. Meeting the quality standards for in uenza, pneumococcal vaccination. Infectious Disease News. Brito, D.L., E. Sheshinski, M.D. Intriligator. 1991. Externalities and compulsory vaccinations. Journal of Public Economics. 45. 69-90. Brown, D., 2004. How U.S. got down to two makers of vaccine. Washington Post. October 17, u 2004. A01-04. Chick, S., H. Mamani, D. Simchi-Levi. 2006. Supply chain coordination and in uenza vaccination. Working paper. Cournot A., 1838. Recherches sur les principes mathematiques de la theorie des richesses. English edition (ed. N. Bacon): Researches into the Mathematical Principles of the Theory of Wealth. Macmillan, New York. 1987. Danzon, P.M., N.S. Periera, S.S. Tejwani. 2005. Vaccine supply: A cross national perspective. Health A airs. 24. 706-717. 27 Dada, M., N. C. Petruzzi, L. B. Schwarz. 2007. A newsvendor procurement problem when s suppliers are unreliable. Manufacturing & Service Operations Management. 9. 9-32. Evans, M.R., P.A. Watson. 2003. Why do older people not get immunized against in uenza? A community survey. Vaccine. 21. 2421-2427. Federgruen, A., N. Yang. 2005. Optimal supply diversi cation under general supply risks. Working paper. Finkelstein S., C.N. Smart, A.M. Gralla, C.R. d Oliviera. 1981. A two-stage model for the control of epidemic in uenza. Management Sci. 27. 834-846. Forbes. 2004. Broken Eggs. November 1. GAO. 2001. Flu vaccine: Supply problems heighten need to ensure access for high-risk people. Report to congressional requesters. Gerchak, Y., M. Parlar. 1990. Yield variability, cost trade-o s and diversi cation in the EOQ model. Naval Res. Logist. 37. 341-354. Gerdil, C. 2002. The annual production cycle for in uenza vaccine. 21. 1776-1779. Gottlieb, S. 2004. Vaccine makers get a shot in the arm. Forbes.com. October 11. Henig M., Y. Gerchak. 1990. The structure of periodic review policies in the presence of random yield. Oper. Res. 38. 634-643. Klemperer P., M. Meyer. 1986. Price-competition vs. quantity competition - The role of uncertainty. The RAND Journal of Economics. 17. 618-638. Kurian, S., D.S. Blog, K.M. Sherin. 2004. Optimizing vaccine availability and utilization: Position statement of the American College of Preventive Medicine. American Journal of Preventive Medicine. 26. 372-374. Leland, H.E. 1972. Theory of the rm facing uncertain demand. Amer. Econ. Rev. 62. 278-291. Longini, I.M., E. Ackerman, L.R. Elveback. 1977. An optimization model for in uenza epidemics. 28 Math. Biosci. 38. 141-157. Mankiw, N.G., M.D. Whinston. 1986. Free entry and social ine ciency. The RAND Journal of Economics. 17. 48-58. Martone. 2000. In uenza: The virus, the disease and how to protect yourself. National Foundation of Infectious diseases. National In uenza Vaccine Summit. 2006. Comments on the status of prebooked in uenza vaccine for 2006-2007. http://www.ama-assn.org/ama/pub/category/13732.html National Vaccine Advisory Committee. 2003. Strengthening the Supply of Routinely Recommended Vaccines in the United States. http://www.hhs.gov/nvpo/bulletins/nvac-vsr.htm#intro Newsweek. 2004. The shot asco. November 1. u Nichol, K.L. 2001. Cost-bene t analysis of a strategy to vaccinate healthy working adults against in uenza. Arch. Int. Med. 161. 749-759. Nichol, K.L., J. Wuorenma, T. von Sternberg. 1998. Bene ts of in uenza vaccination for low-, medium- and high-risk senior citizens. Arch. Int. Med. 158. 1769-1776. O Mara, D., K. Fukuda, J.A. Singleton. 2003. In uenza vaccine: ensuring timely and adequate supply. Infect. Med. 20. 548-554. Philipson, T. 2003. Economic epidemiology and infectious diseases. Handbook of Health Economics. Vol 1B. (eds. Culyer A.J., J.P. Newhouse) Powermed. 2005. Pandemic in uenza and biothreat preparedness: Role of PMEDT M DNA vaccines. www.powermed.com Suzumura K., K. Kiyono. 1987. Entry barriers and economic welfare. Rev. Econ. Studies. 54. 157-167. Strikas, R.A. 2005. Driving increased in uenza vaccine uptake. The 2005 National In uenza Summit, Chicago, IL. 29 Thompson W.W., D.K. Shay, E. Weintraub, L. Brammer, N. Cox, L.J. Anderson, K. Fukuda. 2003. Mortality associated with in uenza and respiratory syncytial virus in the United States. JAMA. 179-186. Time. 2004. The Snafu. November 1. u U.S. Census Bureau. 2004. Nation adds 3 million people in last year; Nevada again fastest-growing state. http://www.census.gov/Press-Release/www/releases/archives/population/003153.html. Vives, X. 1999. Oligopoly Pricing: Old Ideas and New Tools. The MIT Press. Cambridge, Massachusetts. von Weizs cker, C.C. 1980. A welfare analysis of barriers to entry. Bell Journal of Economics. 11. 399-420. WHO. 2002. In uenza vaccines: WHO position paper. Weekly epidemiological record. 28. 239. Williams, D. G. 2005. The in uenza vaccine supply chain: structure, risk and coordination. Unpublished thesis. Massachusetts Institute of Technology - Zaragoza International Logistics Program. Wu, J.T., L.M. Wein, A.S. Perelson. 2005. Optimization of in uenza vaccine selection. Oper. Res. 53. 456-476. Yadav, P. 2005. Value of Creating a Redistribution Network for In uenza Vaccine in the United States. Fifth Workshop on Business Aspects of Closed Loop Supply Chains. Nashville TN. Yano C.A, H.L. Lee. 1995. Lot sizing with random yields: A review. Oper. Res. 43. 311-334. 230- Appendix: Proofs Proof of Lemma 1: The proof is analogous to that for the deterministic case. The ith rm solves the concave maximization problem in qi given by (2). Hence the rst order condition in (3) is necessary and su cient to obtain the equilibrium target quantities. Using E [ i j] = E [ i] E [ j] 30 (random variables i and j are independent), V ar ( i ) = E 2 i (E [ i ])2 = 2, E [ i] = 8i (each rm has the same yield distribution), and simplifying (3), we obtain a unique solution to the above set of equations given by: qi = (a c) b [(n + 1) 2 + 2 8i (9) 2] The expected production quantity of the ith rm is given by E [qi ] = E [ i ] qi = (a c) 2 b [(n + 1) 2 + 2 = (a c) b n+1+2 8i (10) 2] 2 which is decreasing in . This completes the proof for parts (i), (ii) and (iii). Substituting (9) in (1) and simplifying yields i (n) = (a c)2 ( 2 + 1) b(n + 1 + 2 2 )2 @ i (qi ) @ (11) = (a c)2 2 (n 3 2 b(n+1+2 2 ) 2 Proof of Lemma 2: We rst ignore the integer constraint on the number of rms and solve for xu by using the condition i (xu ) which proves part (iv). Di erentiating (11) w.r.t. , we obtain q @ i (qi ) @ i (q ) < 0 for > n 2 3 if n > 3 and @ i < 0 for 8 > 0 if n @ ) . Hence, 3, which proves part (v). = f . Using (11) and rearranging the terms we get: (12) Since Proof of Proposition 1: Since the derivatives w.r.t. (a c) p xu = p 1 + 2 (1 + 2 2 ) bf p c) p (x) is decreasing in x, nu = bxu c = (a bf 1 + 2 (1 + 2 2 ), using (12). i 0 and 2 2 have the same signs, we , we get dxu d2 2 focus on the latter due to ease of analysis. Di erentiating (12) w.r.t. (a c) p 2 bf =0 = p1 1+ 2 2 2 = =0 (a c) p 2 bf 2. For (a c) p bf 4, xd =) nu nd 8 > 0 (a c) p bf dxu d2 Next, consider the case (a c) p bf 2 0 =) xu =0 > 4. Substituting = 0 in (12), we get xd = xd () 1 1; this with (12) and simplifying, we obtain xu Since bxu c = nu and bxd c = nd , we obtain where 1 1 = r 1. Comparing ac p 2 bf 1 2 1. =) nu nd and =) nu nd . 31 Proof of Proposition 2: With yield uncertainty, the total quantity produced at equilibrium is E [qu ] = qd = nu X E [qi ] = i=1 nd (a c) b(nd +1) . using (10). Without yield uncertainty ( = 0 and nu = nd ), we get k j j k p c) ac p Comparing the two and using nd = pbf 1 and nu = (a bf 1 + 2 (1 + 2 2 ) : E [qu ] () (1 + 2 2 ) ac p bf 1 (a c) p p 1+ bf 2 nu (a c) , b(nu +1+2 2 ) qd (1 + 2 2 ) (13) Consider the following inequality: (1 + 2 2 ) ac p bf (a c) p p 1+ bf 2 1 (1 + 2 2 ) 2 (14) Write LHS and RHS for left and right hand side of (14) respectively. RHS = LHS at since we are only interested in the sign of the derivative, we can di erentiate w.r.t. j k @LHS ac ac = 2 pbf 1 and @RHS = pbf p 1 2 2. So, @RHS 2 < @RHS 2 < @2 @2 @2 @2 2 1+ >0 =0 @LHS @2 2 = 0. Again, . Note that 2 2 @LHS @2 =0 = >0 . Hence (14) and consequently (13) holds 8 > 0. Hence, we have qd E [qu ] 8 0. Proof of Proposition 3: Begin with calculating E (qu )2 and E[CSu (qu )]. Using (9) and simplifying we obtain: E[CSu (qu )] = (a c)2 nu (nu + 2 ) nu (nu + 22 = 2 2b(nu + 1 + 2 ) 2 )f (15) Hence, for the deterministic case ( = 0 and nu = nd ), CSd (qd ) = Next, de ne h( ) , (a c) p bf (nd )2 f 2 2 (16) j a pc bf expressions for nd and nu and (15) and (16), proves the rst part of the result. Also, nd nu + 2 k 1 , which is unimodal in for j k ac > 0 and h(0) = 0. So, 2 = min f > 0 : h( ) 0g exists. Thus, > 2 =) pbf 1 > jp k p (a c) c) p p 1 + 2 (1 + 2 2 ) + 2 1 + 2 (a bf (1 + 2 2 ) + 2 . This, along with the bf p 1+ 2 (1 + 2 2 )+ > 2 =) =) nd nu () > 1. Hence 2 > 1. 32 Proof of Lemma 3: Note that E(q) = E 2 2 i n X i=1 E(q 2 ) = E 4 =E qi !2 3 " n X i=1 2 qi + E [ i ] E [ 5=E " n X i=1 qi = E # # n X i=1 2 qi j] X i6=j 2 3 X +E4 qi qj 5 i6=j " n X i=1 i qi = # n X i=1 qi . Similarly, qi qj = ( 2 + 2 ) n X i=1 2 qi + 2 X i6=j qi qj Substituting the expressions for E(q) and E(q 2 ) in (4) and simplifying, we obtain 8 2 3 n n < X X X b4 2 2 qi ( + 2) qi + 2 qi qj 5 max E [W (q(n); n)] = (a c) qi ;n : 2 i=1 i=1 i6=j nf 9 = ; We rst maximize over qi , keeping n xed. The resulting objective function is jointly concave # " n X b in qi with rst order condition (a c) = 2 2 2 + 2 qi + 2 8i. This condition and qk k=1 f we obtain qi b = 2(a c) b[(n+1) 2 +2 consequently the optimal solution qi is symmetric in i. Summing over i and utilizing symmetry 2] . Similarly, the expected quantity produced by each rm is given by E [qi ] = 2(a c) . b(n+1+2 2 ) Proof of Proposition 4: First, consider the continuous relaxation of (5) to x 2 R+ . The rst order condition w.r.t. x gives 2(a c)2 (1+ 2 )(1+2 2 x) bf = x+1+2 23 . Note that the left hand side is decreasing in x and positive only for x < 1 + 2 2 . The right hand side is increasing in x and always positive. Since we require that the expected quantity produced is non-negative for any n, it is required that n range 1 2(a c)2 (1+ 1. Thus, if a solution exists to this equation, it is unique and lies in the n < 1 + 2 2 . Also, a necessary and su cient condition for a solution to exist is given by 2 )(1+2 2 x) x=1 bf > x+1+2 23 x=1 or alternately 2(1+ 2 ) < (a c) p . bf If this condition is 0. not satis ed, then n = x = 1, since otherwise would imply that dE[W (x)] dx x=1 Proof of Proposition 5: Consider the continuous relaxation of (7) in x 2 R+ instead of n 2 N. Clearly, E [W (q(x); x)] is also a concave function of x and has a unique optimum given by the following rst-order condition, @E [W (q(x); x)] (a c)2 2(1 + 2 2 )2 + x 2 (1 + 2 2 ) = @x 2b (x + 1 + 2 2 )3 33 2 f =0 (17) For j Clearly, nsb d = 0; this yields k a pc 1 and xd = bf 1 xsb d = a pc bf 2 3 1 and hence nsb 2 d xsb ; xsb + 1 . Recall that nd = d d xd . Now nsb d 1 xsb d xsb d xd . a pc bf 1. First check that xsb d 1 nd . xd =) nsb d Proof of Proposition 6: We de ne xu and xsb corresponding to nu and nsb respectively. Thus u u xsb = u x: @EW (x) @x x=xsb u = 0 . First, in order to compare xu and xsb , we calculate u @EW (x) . @x x=xu Using (17) we get: @E [W (q(x); x)] @x Also, since f = i (xu ) = x=xu (a c)2 2bf " 2 1+2 22 + xu 2 1+2 2 2 (xu + 1 + 2 2 )3 # f = (a c)2 (1+ 2 ) b(xu +1+2 2 )2 at equilibrium: 2 1+2 22 @E [W (q(x); x)] @x = x=xu (a c)2 2bf " + xu 2 1+2 2 2 (xu + 1 + 2 2 )3 22 # xu (a c)2 (1 + 2 ) b(xu + 1 + 2 2 )2 2 = h (a c)2 2 1+2 2bf (xu + 1 + 2 2 )3 xsb is u a p c. bf 3 1+2 2 2 i 2(xu + 1 + 2 2 ) xsb ; xsb + 1 . Thus, u u p2 2 ac nsb . Combining these two conditions we obtain 2(1+22+) 2 1+ > pbf =) xsb =) nu xu u u p2 2 sb . It is easy to check that 2(1+2 ) 1+ is an increasing function of . De ning 3 = nu nu 2+ 2 p2 2 ac > 0 : 2(1+22+) 2 1+ = pbf , the above condition is equivalent to nu nsb if u 3 . This proves the rst part of the proposition. For the second part of the proposition, combining the fact p2 p2 2 2 a p c =) x that 2(1+22+) 2 1+ xsb and xu xs =) nu nsb , we obtain 2(1+22+) 2 1+ u u u u bf a pc bf A su cient condition for xu p2 2 xu xsb if 2(1+22+) 2 1+ u @E[W (q(x);x)] @x x=xu 0. Substituting xu and simplifying gives In addition,nu = bxu c and nsb 2 u =) nu nsb . Note that the left hand side of this expression is increasing in u nsb if u 3. and using the same argument as above we conclude that nu 34
Find millions of documents here - Study Guides, Homework Solutions, Papers, Exam Answer Keys and more.
Course Hero has millions of course related materials that will enable you to learn better, faster and get an A in all your courses.
Below is a small sample set of documents:
pdf_CC32.pdf
Path: UCLA >> ANDERSON >> 994 Fall, 2008
Path: UCLA >> ANDERSON >> 16436 Fall, 2008
Path: UCLA >> ANDERSON >> 994 Fall, 2008
Path: UCLA >> ANDERSON >> 119 Fall, 2008
Path: UCLA >> ANDERSON >> 119 Fall, 2008
Path: UCLA >> ANDERSON >> 646 Fall, 2008
Path: UCLA >> ANDERSON >> 997 Fall, 2008
Path: UCLA >> ANDERSON >> 997 Fall, 2008
Path: UCLA >> ANDERSON >> 997 Fall, 2008
Path: UCLA >> ANDERSON >> 997 Fall, 2008
Path: UCLA >> ANDERSON >> 990 Fall, 2008
Path: UCLA >> ANDERSON >> 990 Fall, 2008
Path: UCLA >> ANDERSON >> 19419 Fall, 2008
Path: UCLA >> ANDERSON >> 985 Fall, 2008
Path: UCLA >> ANDERSON >> 985 Fall, 2008
Path: UCLA >> ANDERSON >> 459 Fall, 2008
Path: UCLA >> ANDERSON >> 459 Fall, 2008
Path: UCLA >> ANDERSON >> 772 Fall, 2008
Path: UCLA >> ANDERSON >> 744 Fall, 2008
Path: UCLA >> ANDERSON >> 894 Fall, 2008
Path: UCLA >> ANDERSON >> 659 Fall, 2008
Path: UCLA >> ANDERSON >> 659 Fall, 2008
Path: UCLA >> ANDERSON >> 659 Fall, 2008
Path: UCLA >> ANDERSON >> 812 Fall, 2008
Path: UCLA >> ANDERSON >> 998 Fall, 2008
Path: UCLA >> ANDERSON >> 998 Fall, 2008
Path: UCLA >> ANDERSON >> 682 Fall, 2008
Path: UCLA >> ANDERSON >> 682 Fall, 2008
Path: UCLA >> ANDERSON >> 1210 Fall, 2008
Path: UCLA >> ANDERSON >> 1960 Fall, 2008
Path: UCLA >> ANDERSON >> 1960 Fall, 2008
Path: UCLA >> ANDERSON >> 1960 Fall, 2008
Path: UCLA >> ANDERSON >> 1960 Fall, 2008
Path: UCLA >> ANDERSON >> 1960 Fall, 2008
Path: UCLA >> ANDERSON >> 1960 Fall, 2008
Path: UCLA >> ANDERSON >> 1960 Fall, 2008
Path: UCLA >> ANDERSON >> 1960 Fall, 2008
Path: UCLA >> ANDERSON >> 1960 Fall, 2008
Path: UCLA >> ANDERSON >> 1960 Fall, 2008
Path: UCLA >> ANDERSON >> 1340 Fall, 2008
Path: UCLA >> ANDERSON >> 1340 Fall, 2008
Path: UCLA >> ANDERSON >> 1569 Fall, 2008
Path: UCLA >> ANDERSON >> 1569 Fall, 2008
Path: UCLA >> ANDERSON >> 1218 Fall, 2008
Path: UCLA >> ANDERSON >> 841 Fall, 2008
Path: UCLA >> ANDERSON >> 842 Fall, 2008
Path: UCLA >> ANDERSON >> 841 Fall, 2008
Path: UCLA >> ANDERSON >> 841 Fall, 2008
Path: UCLA >> ANDERSON >> 841 Fall, 2008
Path: UCLA >> ANDERSON >> 558 Fall, 2008
Path: UCLA >> ANDERSON >> 583 Fall, 2008
Path: UCLA >> ANDERSON >> 583 Fall, 2008
Path: UCLA >> ANDERSON >> 583 Fall, 2008
Path: UCLA >> ANDERSON >> 583 Fall, 2008
Path: UCLA >> ANDERSON >> 583 Fall, 2008
Path: UCLA >> ANDERSON >> 583 Fall, 2008
Path: UCLA >> ANDERSON >> 583 Fall, 2008
Path: UCLA >> ANDERSON >> 583 Fall, 2008
Path: UCLA >> ANDERSON >> 583 Fall, 2008
Path: UCLA >> ANDERSON >> 583 Fall, 2008
Path: UCLA >> ANDERSON >> 583 Fall, 2008
Path: UCLA >> ANDERSON >> 583 Fall, 2008
Path: UCLA >> ANDERSON >> 583 Fall, 2008
Path: UCLA >> ANDERSON >> 583 Fall, 2008
Path: UCLA >> ANDERSON >> 583 Fall, 2008
Path: UCLA >> ANDERSON >> 583 Fall, 2008
Path: UCLA >> ANDERSON >> 583 Fall, 2008
Path: UCLA >> ANDERSON >> 583 Fall, 2008
Path: UCLA >> ANDERSON >> 583 Fall, 2008
Path: UCLA >> ANDERSON >> 583 Fall, 2008