Chain Supply Coordination and Inuenza Vaccination Stephen E. Chick Hamed Mamani March 13, 2006 David Simchi-Levi
Abstract Billions of dollars are being allocated for inuenza pandemic preparedness, and vaccination is a primary weapon for ghting inuenza outbreaks. The inuenza vaccine supply chain has characteristics that resemble the news vendor problem, but possesses several characteristics that distinguish it from typical supply chains. Differences include a nonlinear value of sales (caused by the nonlinear health benets of vaccination due to infection dynamics) and vaccine production yield issues. We show that production risks, taken currently by the vaccine manufacturer, lead to insufcient supply of vaccine. Unfortunately, several supply contracts that coordinate buyer (governmental public health service) and supplier (vaccine manufacturer) incentives in industrial supply chains can not fully coordinate the inuenza vaccine supply chain. We design a variant of the cost sharing contract and show that it provides incentives to both parties so that the supply chain achieves global optimization and hence gurantees sufcient supply of vaccine.
INSEAD; Technology and Operations Management Area; Boulevard de Constance; 77300 Fontainebleau France. stephen. [e-mail] MIT Operations Research Center; 77 Massachusetts Ave. Bldg E40-130; Cambridge, MA 02139, USA. [e-mail] MIT Department of Civil and Environmental Engineering, and The Engineering System Division; 77 Mass. Ave. Rm. 1-171; Cambridge, MA 02139, USA. [e-mail]
1 Inuenza: Overview, Control and Operational Challenges
Inuenza is an acute respiratory illness that spreads rapidly in seasonal epidemics. Globally, annual inuenza outbreaks result in 250,000 to 500,000 deaths. The World Health Organization reports that costs in terms of health care, lost days of work and education, and social disruption have been estimated to vary between $1 million and $6 million per 100,000 inhabitants yearly in industrialized countries. A moderate, new inuenza pandemic could increase those losses by an order of magnitude (WHO, 2005). This paper provides background about inuenza and vaccination, a key tool for controlling inuenza outbreaks, then highlights some operational challenges for delivering those vaccines. One challenge is the design of contracts to coordinate the incentives of actors in a supply chain that crosses the boundary between the public sector (health care service systems) and private sector (vaccine manufacturers). Some experts suggest the U.S. government should promise to purchase a xed amount of u vaccinedespite the cost and the likelihood that some of the money would end up being wasted. Canada, for instance, has contracts with vaccine makers to cover most of its population. That takes much of the risk out of the companys business, but still lets it manufacture additional doses for the private market(WSJ, Wysocki and Lueck, 2006) I recently met with leaders of the vaccine industry. They assured me that they will work with the federal government to expand the vaccine industry, so that our country is better prepared for any pandemic. Im requesting a total of $7.1 billion in emergency funding from the United States Congress(George W. Bush, 2005)
We then present a model of a governments decision of purchase quantities of vaccines, which balances the public health benets of vaccination and the cost of procuring and administering those vaccines, and a manufacturers choice of production volume. We characterize the optimal decisions of each in both selsh and system-oriented play, then assess whether several contracts can align their incentives. Due to special features of the inuenza value chain, wholesale price and pay back contracts are shown to be unable to fully coordinate decisions. We conclude by demonstrating a variation of a cost sharing contract that can coordinate concerns for both public health outcomes and production economics.
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1.1 Inuenza and Inuenza Transmission Inuenza is characterized by fever, chills, cough, sore throat, headache, muscle aches and loss of appetite. It is most often a mild viral infection transmitted by respiratory secretions through sneezing or coughing. Complications of inuenza include pneumonia due to secondary bacterial infection, which is more common in children and the elderly (e.g., see http://www.cdc.gov/flu, or Janeway et al. 2001). The various strains of inuenza experience slight mutations in their genome through time (antigenic drift). This allows for annual outbreaks, as previously acquired adaptive immunity may not cover emerging strains. Every few decades, a highly virulent strain may emerge that causes a global pandemic with high mortality rates. This may be caused by a larger genomic mutation (antigenic shift). The three pandemics that occurred in the twentieth century came from strains of avian u. The Spanish u (H1N1) of 1918 killed 2040 million people worldwide (WHO, 2005), far more than died in World War I. Milder pandemics occurred in 1957 (H2N2) and 1968 (H3N2). The H5N1 virus is the most likely potential culprit for a future pandemic (http://www.who.int/csr/disease/influenza/).
1.2 Vaccination as a Control Tool Vaccines can reduce the risk of infection to exposed individuals that are susceptible to infection, and can reduce the probability of transmission from a vaccinated individual that is infected with inuenza (Longini et al., 1978; Smith et al., 1984; Longini et al., 2000; Chick et al., 2001). Vaccines therefore act on the basic reproduction number, R0 , the mean number of new infections from a single infected in an otherwise susceptible population (Dietz, 1993). If R0 can be reduced below 1, then the dynamics of a large outbreak can be averted. Let f 0 be the so-called critical vaccination fraction, the minimum fraction of the population to vaccinate to reduce the reproduction number to 1 (Hill and Longini, 2003). Vaccination is seen as a principal means of preventing inuenza. Although vaccination policies may vary from country to country, particular attention is typically those those aged 65 or more, health care workers, and
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those that may have certain risk factors (Bridges et al., 2002; WHO, 2005). Vaccination can be complemented with antiviral therapy. Vaccination is cost effective. Nichol et al. (1994) found that immunization in the elderly saved $117 per person in medical costs. Weycker et al. (2005) argue for the systematic vaccination of children, not only the elderly, as a means to obtain a signicant population-wide benet for vaccination.
1.3 Operational Challenges in the Inuenza Vaccine Supply Chain Gerdil (2003) overviews the highly challenging and time-constrained vaccine production and delivery process. We focus on the predominant method, inactivated virus vaccine production. For the northern hemisphere, the WHO analyzes global surveillance data and in February announces the selection of three virus strains for the fall vaccination program. Samples of the strains are provided to manufacturers. High-volume production of vaccine for each of the three strains then proceeds separately. Production takes place in eleven day old embryonated eggs, so the number of eggs needed must be anticipated well in advance of the production cycle. Blending and clinical trials begin in May-June. Filling and packaging occur in July and August. Governmental certication may be required at various steps for different countries. Shipping occurs in September for vaccination in October-November. Immunity is conferred two weeks after vaccination. The southern hemisphere uses a separate 6-month cycle. Within two 6-month production cycles, almost 250 million doses are delivered to over 100 countries per year. Figure 1 provides a graphic summary. Saluzzo and Lacroix-Gerdil (2006) provide additional information, particularly with respect to avian u preparedness. There are several key operational challenges that are presented by the inuenza vaccine value chain. A challenge at the start of the value chain is antigenic drift, which requires that inuenza vaccines be reformulated each year. Inuenza vaccines are one-time news-vendor products, as opposed to all other vaccines, which closely resemble (perishable) EOQ-type products. Not only are production volumes hard to predict, but the selection of the target strains is a challenge. Wu et al. (2005) develop an optimization model
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Figure 1: Inuenza vaccine time line. of antigenic changes. Their results suggest that the current selection policy is reasonably effective. They also identify heuristic policies that may improve the selection process. Another challenge occurs toward the end of the value chain, after vaccines are produced. That involves the allocation of vaccines to various subpopulations, and the logistics of transhipment to insure appropriate delivery. Hill and Longini (2003) describe a mathematical model to optimally allocate vaccines to several supbopulations with potentially heterogeneously mixing individuals. Weycker et al. (2005) use a different, stochastic simulation model to illustrate the benets of vaccinating certain subpopulations (children). Those articles do not discuss the logistics of delivery. Yadav and Williams (2005) propose an information clearinghouse for vaccine supply and demand to provide a market overview and help to eliminate order gaming and price gouging, as well as demand forecasting tools, and regional vaccine redistribution pools to shift supplies from areas with surpluses to areas experiencing shortages. This paper is concerned with a challenge in the middle of the value chain: the design of contracts that align manufacturer choices for production volume and the need for protability, and governmental choices that balance the costs and public health benets of vaccination programs. Special characteristics of the inuenza vaccine supply chain that differentiate it from many other supply chains include a nonlinear value of a sale (the value of averting an infection by vaccination depends upon nonlinear infection dynamics), and p. 4
a dependence of production yields on the virus strains selected for the vaccine. Current production technology for inactivated virus vaccines, market forces, and business practices also combine to limit the ability to stockpile vaccines, limit production capacity, and slow the ability to respond to outbreaks. Governmental and industry partnerships may help to improve responsiveness (Pien, 2004; Bush, 2005; Wysocki and Lueck, 2006). The ideal way to structure those partnerships is an open question. This paper addresses one dimension of that multi-faceted question. Section 2 presents a model to assess contractual mechanisms that align manufacturer risks and incentives with governmental health care policy objectives for inuenza vaccination. Section 3 and Section 4 analyze the model. A variant of the cost sharing contract, which we show can align incentives for public health benets and production costs, also increases production volumes. Increased production volumes for annual vaccination are consistent with the recommendations of the Pandemic Inuenza Plan of the U.S. Dept. of Health and Human Services (2005). Section 5 discusses implications and limitations of the analysis.
2 Joint Epidemic and Supply Chain Model
The work here unites two previously separate streams of literature. The epidemic literature provides epidemic models and cost benet analysis for interventions such as vaccination (Jacquez, 1996; Diekmann and Heesterbeek, 2000; Hill and Longini, 2003), but does not address logistical and manufacturing concerns. The supply chain literature addresses logistical and manufacturing concerns in general, but does not address the special characteristics of the inuenza vaccine supply chain highlighted above. We use simplied epidemic and supply chain models to focus on contractual issues between a single government and a single manufacturer. The single government is intended to represent centralized aggregate planning decisions for vaccination policy. The government initially announces a fraction f of a population of N individuals to vaccinate. Given the demand by the government, the manufacturer then decides how much to produce. Production volume decisions are indexed by the number of eggs, nE , a critical factor in inuenza
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vaccine production. Production costs are c per egg. The actual amount produced, nE U , is a random variable that is indexed by a yield, U . We assume that the yield U has a continuous probability density function fU (u) with mean and variance 2 . This assumption means that the yield is affected by the specic strain of the virus, and may vary from year to year, more so than from one statistically independent batch to the next within a given production campaign. The manufacturer then sells whatever vaccine is produced, up to the amount initially requested by the government (a maximum of N f d doses, where N is the population size, and d is the number of doses per individual). Unmet demand is lost, and excess vaccines are discarded (due to antigenic shift). When acting separately, the government seeks to minimize the variable cost of procuring, pr , and administering, pa , each dose, plus the total social cost of the outbreak, bT (f ), where T (f ) is the total number of infected individuals by the end of the outbreak, and b is the average direct and indirect cost of inuenza infection per outbreak (Weycker et al., 2005, provides estimates of such costs). Dene f to be the maximum fraction of the population for which the net benet of administering more vaccine is positive, and dene f similarly with respect to both vaccine procurement and administration costs, f = sup{f : bT (f ) + pa N d < 0, for f such that T (f ) exists} f = sup{f : bT (f ) + (pa + pr )N d < 0, for f such that T (f ) exists}.
(1) (2)
The epidemic model determines the number of individuals, T (f ), that are infected by the end of the outbreak. While vaccine effects and health outcomes may vary by subpopulation, and vaccination programs can take advantage of that fact (Weycker et al., 2005), we simplify the model in order to focus on contract issues for production volume, rather than including details about optimal allocation of a given volume. We use a deterministic compartmental model of N homogeneous and randomly mixing individuals (Diekmann and Heesterbeek, 2000), of which a fraction S0 of the population is initially Susceptible. A fraction I0 is Infected and infectious (an initial seeding due to exposure from exogenous sources). After recovery, individuals are
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Table 1: Summary of Notation. Supply Chain nE Number of eggs input into vaccine production by the manufacturer U Random variable for the yield per egg, with pdf of fU (u), mean , and variance 2 d Doses of vaccine needed per person c Unit cost of production for manufacturer, per egg input pr Revenue to the manufacturer from government, per dose of vaccine pa Cost per dose for government to administer vaccine b Average total social cost per infected individual (direct + indirect costs) Z Number of doses sold from manufacturer to government W Number of doses administered by government to susceptible population Outbreak N Total number of people in the population R0 Basic reproduction number, or expected number of secondary infections caused by one infected in an otherwise susceptible, unvaccinated population f fraction of the population to vaccinate announced by government to manufacturer T (f ) Total number infected during the infection season, a function of the fraction vaccinated I0 The initial fraction of infected people introduced to the population S0 The initial fraction of susceptible people in the population Vaccine effects on transmission, including susceptibility and infectiousness effects Linear approximation to number of direct and indirect infections averted by a vaccination f0 The critical vaccination fraction (fraction of population to vaccinate to halt outbreak) f The maximum fraction for which (free) vaccine can be cost-effectively administered f The maximum fraction for which vaccine can be cost-effectively procured and administered N k Relates vaccination fractions and vaccine production inputs, k = fnEd Removed and no longer infectious. This so-called SIR epidemic model is consistent with the natural history of infection of inuenza. Table 1 summarizes the notation. We assume that vaccination removes some fraction of individuals from the pool of susceptibles, where is interpreted as a combination of vaccine effects. If S0 = 1 I0 f , then T (f ) = N p, where the so-called attack rate p (Longini et al., 1978) satises p = S0 (1 + I0 eR0 p ). S0
(3)
The critical vaccination fraction is f 0 = (R0 1)/(R0 ) when R0 > 1 (Hill and Longini, 2003). Rather than deriving results via such an implicit characterization from the epidemic model, we derive results for a nonincreasing T (f ) 0 with specic general characteristics. Appendix B describes why it is reasonable to consider two functional forms: a piecewise linear T (f ), or a strictly convex T (f ). This removes p. 7
the details of an implicit solution for an epidemic model from the supply chain analysis. Section 3 handles the piecewise linear case. Section 4 handles the convex case. Any further characteristics of the epidemic model that are needed below are compatible with Longini et al. (1978), specialized to one subpopulation.
2.1 Game setting The epidemic and supply chain models above dene a sequential game. The government announces a fraction f of the population for which it will purchase vaccines. The manufacturer then decides on a production quantity, indexed by nE , in order to maximize expected prots (minimize expected costs), subject to potential yield losses and market capacity constraints. The manufacturer problem is: min M F = E [cnE pr Z] (net manufacturer costs)
nE
s.t. Z = min{nE U, f N d} (doses sold yield and demand) nE 0 (nonnegative production volume)
(4)
So that the optimal production level is not zero, n > 0, we assume: E Assumption 1 The expected revenue exceeds the cost per egg, pr > c, so vaccines can be protable. Given that assumption, we characterize the optimal production quantity. Proposition 1 For any random egg yield, U , with pdf fU (u), and given the order quantity D = f N d by government, the optimal production level for the manufacturer is
fNd n E
0
ufU (u)du =
c . pr
(5)
Claims that are not justied in the main text are proven in Appendix A. A useful corollary follows directly. Corollary 1 If c, pr , fU (u), N and d are held constant, then the relationship between the fraction of people to be vaccinated, f , and optimum production level, nE , is linear. That is, there is a xed constant, k G , such that k G nE = f N d. p. 8
The government problem is to select a fraction f that indexes demand, knowing that the manufacturer will behave optimally, as in (5), and may deliver less, in expectation, than what is ordered due to yield losses. The government may order some excess (even f > f ), in order to account for potential yield losses. In this base model, we assume that the government purchases up to the amount it announced, but will administer only those doses that have a nonnegative cost-health benet.
W min GF = E bT ( N d ) + pa W + pr Z f
(net government costs) (doses bought yield and demand) (doses given doses bought, cost effective level) (6) (manufacturer acts optimally) (fraction of population) (nonnegative production volume)
s.t. Z = min{nE U, f N d} W = min{nE U, f N d, f N d}
fNd nE
0
c ufU (u)du = pr
0f 1 nE 0
Such a two-actor game has a Nash equilibrium (Nash, 1951), which we identify below.
2.2 System setting The system setting assesses whether the manufacturer and government can collaborate via procurement contracts to reduce the sum of their expected nancial and health costs, to a level that is below the sum of those costs if each player acts individually as in Section 2.1. System costs do not include monetary transfers from government to manufacturer. Formally, the system problem is
W min SF = E bT ( N d ) + pa W + cnE f,nE
(total system costs) (doses given yield, demand, cost effective level) (7) (fraction of population) (nonnegative production volume).
s.t. W = min{nE U, f N d, f N d} 0f 1 nE 0
This formulation does not explicitly link f and nE together, since we seek system optimal behavior rather than local prot-maximizing behavior. p. 9
3 Piecewise Linear Number of Infected Figure 2 plots the attack rate, p, which is directly proportional to the total number infected, T (f ), as a function of the fraction of initially exposed individuals, I0 and reasonable values of R0 for inuenza transmission (Gani et al., 2005). If there are few that are initially infected due to exogenous exposure (small I0 /S0 ), then Appendix B justies the following piecewise linear approximation for T (f ). M N f, 0 f f 0 T (f ) = 0, f 0 f 1, where is interpreted here as the marginal number of infections averted per additional vaccination.
0.8 0.9
(8)
0.7
Io = 0 Io = 0.005 Io = 0.01 Io = 0.05 Io = 0.1
0.8
0.7
Io = 0 Io = 0.005 Io = 0.01 Io = 0.05 Io = 0.1
0.6 0.6 0.5
attack rate
attack rate
0 0.1 0.2 0.3 0.4 0.5 0.6 fraction vaccinated 0.7 0.8 0.9 1
0.5
0.4
0.4
0.3 0.3 0.2 0.2 0.1
0.1
0
0
0
0.1
0.2
0.3
0.4 0.5 0.6 fraction vaccinated
0.7
0.8
0.9
1
(a) R0 = 1.67
(b) R0 = 2.0
Figure 2: Attack rate versus f for different values of I0 and R0
We seek structural results to compare the values of the game equilibrium and system optimum. With this approximation for T (f ), the maximum cost-effective number of individuals to vaccinate equals the critical vaccination fraction, f = f 0 . The governments objective function from Problem (6) is GF = E b max{M W , 0} + pa W + pr Z . d
(9)
The manufacturer problem is the same. p. 10
The systems objective function from Problem (7) is SF = E b max{M W , 0} + pa W + cnE . d (10)
3.1 Optimal solutions for game and system settings This section describes the equilibria of the game setting and the optimal system solution for the manufacturer and government. It assumes that the parameters of the model in Section 2 are given. A series of assumptions and results are developed to show that the optional system solution requires a higher vaccine production level than in the game setting. Section 3.2 uses those results to design contracts that create a new game, to get individual actors to behave in a system optimal way. If the following assumption were not valid, then even free vaccines would not be cost effective. Assumption 2 The expected health benet of vaccination exceeds the administration cost, b pa d > 0. Proposition 2 Let f S , nS be optima for the system setting with objective function in (10). If Assumption 2 E holds, then (1) f S could be any value between f 0 and 1; and (2) nS satises E
f 0Nd nS E
0
ufU (u)du =
c
b d
pa
.
(11)
The next assumption implies that vaccination is cost effective from the governments point of view. Assumption 3 The expected health benet of vaccination exceeds the cost of administering and procuring the doses, b (pa + pr )d > 0. Observe that if Assumption 3 does not hold, then vaccines at market costs are not cost effective. To see this, set f = min{f, f 0 }. Then for all 0 f 1, GF (f, nE ) b
fNd nE
(M
0
nE u )fU (u)du + b(M N f ) d
fNd nE
fNd nE
fU (u)du
fNd nE
+(pa + pr )nE
0
ufU (u)du + (pa + pr )(f N d)
fNd nE
fU (u)du
fNd nE
1 = bM + nE (pa + pr )d b d
0
ufU (u)du + f N (pa + pr )d b
fU (u)du.
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If b (pa + pr )d < 0, then GF (f, nE ) > bM for all f, nE > 0, and f G = nG = 0 would be optimal. E Given Assumption 3 and Proposition 2, we can compare the values of (5) and (11) to obtain Corollary 2. Corollary 2 Let f S , nS be optimal values of the system problem and dene k S = E optimal values of the game setting and dene k G = The concept k =
fNd nE f GN d . nG E f 0N d . nS E
Let f G , nG denote E
If Assumption 3 holds, then k S < k G .
that relates vaccination fractions to vaccine production volumes is useful below.
Proposition 2 characterized the optimal vaccine fraction and production level for the system setting. We now assess optimal behavior in the game setting. (5) indicates that it sufces to characterize the optimal vaccine fraction, which then determines the optimal production level in the game setting. Proposition 3 Let f G , nG be optimal solutions for the game setting, and set k G = E holds, then f G f 0 . Furthermore, f G = f 0 if and only if b ( + pa + pr ) d
kG 0 f GN d . nG E
If Assumption 3
ufU (u)du + pr k G
kG
fU (u)du 0.
(12)
Although it may seem, at rst glance, that Condition (12) depends on f G through k G , this is not true. Given the problem data, the value of k G is determined by (5), independently of the values of f G and nG . The E condition in this claim is therefore veriable by having the initial data of the problem. Intuitively, the inequality in the second part of Proposition 3, Condition (12), shows that if b is sufciently higher than the other costs, then the game pushes the government to order a higher amount of vaccine than the amount specied by the critical vaccine fraction, f 0 . Theorem 1 uses our results on the optimal production level in the system setting, Proposition 2, and the game setting, Proposition 3, to prove the main result of this section: optimal production volumes are higher in the system setting than in the game setting. Theorem 1 Given Assumption 3 and the setup above, nS > nG . E E The intuition behind Theorem 1 is that the manufacturer bears all the risk of uncertain production yields in the game setting and hence is not willing to produce enough. p. 12
3.2 Coordinating Contracts The objective of this section is to design contracts that will align governmental and manufacturer incentives. We show that wholesale or pay back contracts can not coordinate this supply chain. We then demonstrate a cost sharing contract that is able to do so.
3.2.1 Wholesale price contracts In wholesale price contract, the supplier and government negotiate a price pr . Unfortunately, the system optimum can not be fully achieved just by adjusting the value of pr . Proposition 4 There does not exist a wholesale price contract which satises the condition in Assumption 3 and coordinates the supply chain.
3.2.2 Pay back contracts In a pay back contract, the government agrees to buy any excess production, beyond the desired volume, for a discounted price pc (with 0 < pc < pr ) from the manufacturer. This shifts some risk of excess production from the manufacturer to the government, and would typically increase production. We show that the pay back contract does not provide sufcient incentive to coordinate the inuenza supply chain, unlike typical supply chains, for any reasonable value of pc . Assumption 4 denes a reasonable pc as one that precludes the manufacturer from producing an innite volume for an innite prot. Assumption 4 The average revenue per egg at the discounted price is less than its cost, pc < c. The pay back contract increases the manufacturers prot by adding the revenue associated with nE U min{nE U, f N d} doses of excess production. This changes the manufacturer problem from Problem (4) to min M F = E cnE pr Z pc (nE U Z)
nE
s.t. Z = min{nE U, f N d} nE 0. p. 13
By adapting the argument of Proposition 1, the optimal production level n can be shown to satisfy E
fNd n E
0
ufU (u)du =
c pc . pr pc
(13)
The effect of this contract on the government problem in Problem (6) is to change the objective to GF = E b max{M W , 0} + pa W + pr Z + pc (nE U Z) , d
and to change the manufacturer acts optimally constraint, which determines the optimal production input quantity nE as a function of f , from (5) to (13). Denote the optimal values of this pay back contract problem by f N , nN . Set k N = E
fN Nd . nN E
Proposition 5 If Assumptions 1, 2 and 4 hold, then there does not exist a pay back contract which could coordinate this supply chain. In fact, under any pay back contract, the resulting production level is less than the optimal system production level, nN < nS . E E Proposition 5 suggests that compensating the manufacturer for having excess inventory is not enough to achieve global optimization. Indeed, a pay back contract does not compensate the manufacturer when the production volume, nE , is high while the yield, nE U is low. The cost sharing agreement described below is designed to address this issue.
3.2.3 Cost sharing contracts In a cost sharing contract, the government pays proportional to the production volume nE at a rate of pe per each egg. Such an agreement decreases the manufacturers risk of excess production, and provides an incentive to increase production. Here, we describe a contract that increases production to the system optimum, f 0 , nS . E
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With the cost sharing contract, the manufacturer problem is: min M F = E (c pe )nE pr Z
nE
s.t. Z = min{nE U, f N d} nE 0 The optimality condition for nE given f follows immediately, as for the original problem,
fNd n E
0
ufU (u)du =
c pe . pr
(14)
Cost sharing increases the governments costs, changing its objective function to: GF = E b max{M W , 0} + pa W + pr Z + pe nE , d
(15)
and resulting in the following optimization problem. min GF = E b max{M W , 0} + pa W + pr Z + pe nE d
f
s.t.
Z = min{nE U, f N d} W = min{nE U, f N d, f 0 N d}
fNd nE
0
ufU (u)du =
c pe pr
0f 1 nE 0 Denote the optimal solutions of this problem by f e , ne , and set k e = E For any given pr , choose pe > 0 so that
cpe pr f eN d ne . E b d
=
b pa d
c
. Such a pe exists since pr <
pa . If pe is
chosen this way, then k e = k S . Further, if pr satises Assumption 3, such a pe not only moves k e to k S , but it aligns the vaccination fractions and production volumes, as in Theorem 2.
cpe pr c
Theorem 2 If Assumption 3 holds and pe is chosen so that
=
b pa d
, then the optimal values (f e , ne ) E
for Problem (15) equal (f 0 , nS ), so this cost sharing contract will coordinate the supply chain. E
p. 15
The cost sharing contract can coordinate incentives, unlike the pay back contract, because the manufacturers risk of both excess and insufcient yield can be handled by the contracts balance between paying for outputs (via pr ) and for effort (via pe ).
4 Strictly Convex Number of Infected This section presumes that T (f ) is strictly convex. While T (f ) may not be convex for all choices of the parameters of the infection model, it is strictly convex for sufciently large I0 and values of R0 that are representative of inuenza (see Appendix B). This corresponds to a larger initial exposure to members of the population, such as may occur in an initial pandemic wave. Below we explore the game equilibrium and the optimal system solution; we then show that a variation of the cost sharing contract can coordinate the supply chain.
4.1 Optimal solutions for game and system settings The solution to the manufacturer problem in Problem (4) with convex T (f ) remains the same as above, as the manufacturers objective function does not depend upon T (f ). The analysis of the government problem in Problem (6) and the system problem in Problem (7) is somewhat more complicated when T (f ) is strictly convex, but the general ideas are similar to those in the linear model. For the system setting, the following analog of Proposition 2 holds. Proposition 6 If T (f ) is strictly convex, f is the solution of (1), and the optimum values of the system problem in Problem (7) are denoted by f S , nS , then (a) f S could be any value between f and 1; and (b) nS E E is the solution of the following equation:
0
fNd nS E
nS u b T ( E ) + pa ufU (u)du + c = 0. Nd Nd
The following analog of Proposition 3 for convex T (f ) characterizes the set of the game equilibria.
p. 16
Proposition 7 Let f G , nG denote the game solution, let k G = E
kG
f GN d nG E
and set nE =
fNd . kG
If T (f ) is strictly
convex, then (a)
0
ufU (u)du =
kG 0
c ; and (b) f G f if and only if pr
kG
b nE u T( ) + pa ufU (u)du + c + pr k G Nd Nd
fU (u)du 0.
(16)
Theorem 3, the main result of this section, shows that, as in the linear case, the system optimal production level exceeds that of the game equilibrium. The proof requires the following three lemmas. Lemma 1 If nG nS , then f G f . E E Lemma 2 Let f be the solution of bT (f ) + (pa + pr )N d = 0. Then f G > f . Lemma 3 Let k S =
kS 0 fNd . nS E
Then for all k > 0,
k 0
nS u b T ( E ) + pa ufU (u)du Nd Nd
nS u b T ( E ) + pa ufU (u)du. Nd Nd
Theorem 3 Let nS and nG denote the production level under the system optimum and game equilibrium, E E respectively. For all nonincreasing strictly convex T (f ), we have nS > nG . E E Thus, the theorem suggests that the production level set by the manufacturer, nE , is below the amount required by the system. Hence, the need for effective contracts.
4.2 Coordinating Contracts This section constructs a contract which can coordinate this supply chain. Unfortunately, the cost sharing contract of Section 3.2.3, dened by the pair pr , pe , does not coordinate the supply chain. Observe that in the piecewise linear case, the government orders enough, i.e., f G f 0 , even without the contract. This is not the case for the convex case, where without the contract, f G maybe smaller than f f S , see Proposition 6 and Proposition 7. Thus, the contract should provide incentive for the government to vaccinate a higher fraction of the population, and provide a manufacturer incentive to produce enough. Section 4.2.1 shows that this goal p. 17
can be achieved using a whole-unit discount for the vaccine purchased by the government. In return, the government will pay the manufacturer a portion of the production cost. The relation between the whole-unit discount and the cost sharing portion is such that the more people the government plans to vaccinate, the greater the discount they get and the higher its participation in the production cost.
4.2.1 Whole-unit discount/cost sharing contract Consider a contract where the vaccine price depends on the fraction of the population the government plans to vaccinate, that is, the government pays the manufacturer pr (f ) per dose. The cost sharing component of the contract is such that the government pays proportional to the production level, nE . The per unit price paid by the government, pe (f ) depends on f . This section rst constructs a specic class of pricing policies. It then shows how the original game is modied by the pricing policy. The section concludes with a proof that the given pricing policies indeed align incentives. The following two assumptions constrain the set of pricing policies of interest. Assumption 5 The price pr (f ) 0 has the following characteristics: 1. There is a whole-unit discount, i.e., pr (f ) 0. 2. The total vaccine cost (pr (f )f N d) is nondecreasing in f , (a) (pr (f )f N d) = pr (f )f N d + pr (f )N d 0 for all 0 f f . (b) pr (f )f N d + pr (f )N d = 0. 3. The total cost to the government excluding the cost sharing component is convex in f , (a) bT (f ) + pr (f )f N d + 2pr (f )N d 0 for all 0 f f . 4. There are no further volume discounts beyond a certain threshold, pr (f ) = pr (f ) for all f f 1. p. 18
If the derivative pr (f ) does not exist at f = f , then use the left derivative in Assumption 5. c pe (f ) Assumption 6 Given pr (f ), let pe (f ) 0 satisfy = pr (f ) In Assumption 6, k S =
fNd nS E kS 0
ufU (u)du for all f [0, 1].
is the same as before, where f , nS are the solutions for the system setting. E
Before proceeding, we show rst that the set of the conditions in Assumptions 5 and 6 results in a feasible set. We give an example that satises the conditions in Assumption 5, then modify it to obtain functions that satisfy all of the conditions in both assumptions. Consider the following pricing strategy, b T (f ) + T (f )f + T (0) , 0 f f fNd pr (f ) = pr (f ), f < f 1.
(17)
Claim 1 If 0 < < 1, then the pricing strategy introduced in (17) gives a nonnegative price for any f and satises all the conditions in Assumption 5.
Now we show that for some , (17) satises Assumption 6. It sufces to show that pe (f ) 0 for all f , so the goal is to choose a pricing strategy such that pr (f ) in f , it sufces to show that pr (0)
kS 0 kS 0
ufU (u)du c. Since pr (f ) is nonincreasing
ufU (u)du c. For any pr (f ) that satises (17), b T (f ) + T (f )f + T (0) f 0 f N d b T (f ) T (0) = T (f ) lim f 0 Nd f b T (f ) T (0) = Nd
pr (0) = lim pr (f ) = lim
f 0
Observe that
kS 0
b ufU (u)du . It sufces to have N d T (f ) T (0) c in order to insure that
Assumption 6 holds. This justies Claim 2: pricing strategies exist that satisfy both assumptions.
c }, (f )T (0)]
Claim 2 If 0 < < min{1, [T and 6.
then the pricing strategy pr (f ) in (17) satises Assumptions 5
p. 19
All the ingredients are in place to build a coordinating contract. The key idea is to keep the relationship between the optimal production level and order quantity linear. Assumption 6 accomplishes this. To see this, observe that this contract changes the manufacturer objective, for a given f , to: M F (nE ) = (c pe (f ))nE pr (f )nE
fNd nE
0
ufU (u)du pr (f )f N d
fNd nE
fU (u)du
By taking the derivatives, we have: M F (nE ) nE 2 M F (n ) E 2 nE = (c pe (f )) pr (f ) = pr (f )
fNd nE
0
ufU (u)du
fNd fNd fNd )fU ( )0 2 (n nE nE E
0
fNd n E
Therefore, this M F is convex in nE , and the optimal nE satises with Assumption 6, this implies that
0
fNd n E
ufU (u)du =
cpe (f ) pr (f ) .
Together
ufU (u)du =
kS 0
ufU (u)du. So for any given f , the optimal
production level for the manufacturer is linear in f , with n = E fNd . kS
(18)
Therefore this contract changes the government objective to
W min GF = E bT ( N d ) + pa W + pr (f )Z + pe (f )nE , f fNd nE
(19)
and changes the manufacturing constraint to
= k S . This restatement of the game setting for the
whole-unit discout/cost sharing contract permits the statement of the main result of this section. Theorem 4 For any pe (f ), pr (f ) that satisfy Assumptions 5 and 6, the optimal values of Problem (19), denoted by (f c , nc ), are equal to (f , nS ). That is, this cost sharing contract coordinates the supply chain. E E Proof: In order to analyze Problem (19), we again split it into two separate subproblems.
p. 20
Case 1 (0 f f ): In this case the optimization problem would be: min GF1 = b
f
fNd nE
0
nE u T( )fU (u)du + bT (f ) Nd
fNd nE
fU (u)du + pa nE
fNd nE
fNd nE
0
ufU (u)du
+pa f N d
fNd nE
fU (u)du + pe (f )nE + pr (f )nE
0
ufU (u)du
= cnE (by Assumption 6) +pr (f )(f N d)
fNd nE
fU (u)du
fNd kS
subject to the constraints f N d = k S nE ; 0 f f ; and nE 0. Substituting the constraint nE = into the objective function gives min GF1 = b
f kS 0
T ( kfS u)fU (u)du + bT (f )
kS fU (u)du kS
N + pa fkSd
kS 0
ufU (u)du
+pa f N d s.t. 0f f
kS
fU (u)du + c
fNd + pr (f )f N d kS
fU (u)du
We show that in this case the optimum value is at f . For this purpose, it is enough to analyze the rst derivative of GF1 . GF1 f
f Nd k = T ( S u)ufU (u)du + bT (f ) ufU (u)du fU (u)du + pa S k k 0 0 kS Nd +pa N d fU (u)du + c S + pr (f )N d fU (u)du + pr (f )f N d fU (u)du k kS kS kS
b kS
kS
S
=
Nd kS
kS
[
0
b f T ( S u) + pa ]ufU (u)du + c Nd k
kS
(20) fU (u)du
+ bT (f ) + pa N d + pr (f )N d + pr (f )f N d
We show that each of the two components in (20) is negative, making the derivative of GF1 negative for all 0 f f . To see this, rst note that the function J(f ) = function of f , as J (f ) =
kS b 0 [ N dkS T kS b 0 [ Nd T
( kfS u) + pa ]ufU (u)du is an increasing
( kfS u)]u2 fU (u) 0. Hence J(f ) J(f ), f f . However, (
nS u E Nd )
using f N d = nS k S , we get J(f ) = E J(f ) + c 0, so
kS
kS b 0 [ Nd T
+ pa ]ufU (u) = c (by Proposition 6). As a result
[
0
f b T ( S u) + pa ]ufU (u)du + c 0; Nd k p. 21
0f f
This shows that the rst parenthesis in (20) is negative. To show that the second term of the derivative of GF1 is also negative, we consider the term bT (f ) + pa N d + pr (f )N d + pr (f )f N d. The derivative of this expression is bT (f ) + pr (f )f N d + 2pr (f )N d, which is positive using the third part of Assumption 5. This means that bT (f ) + pa N d + pr (f )N d + pr (f )f N d bT (f ) + pa N d + pr (f )N d + pr (f )f N d for all 0 f f . Note that bT (f ) + pa N d = 0 by the denition of f , and that pr (f )N d + pr (f )f N d = 0 by the second part of Assumption 5. This suggests bT (f ) + pa N d + pr (f )N d + pr (f )f N d 0, 0f f
which shows the second term of the derivative of GF1 is also negative. By the strict convexity of T (f ), equality occurs only at f . Hence (20) implies that GF1 (f ) 0 for all 0 f f meaning that the minimum of GF1 is attained at f . The corresponding production value to f is nS (using 18). So in this case, the only E candidate for optimality is the system optimal solution. Case 2 (f f 1): In this case, using the denition of pr (f ), pr (f ) = pr (f ), and hence pe (f ) = pe (f ) for all f f . As a result, the government objective becomes: GF2 = b
0
fNd nE
nE u )fU (u)du + bT (f ) T( Nd
fNd nE
fNd nE
fU (u)du + pa nE
fNd nE
fNd nE
0
ufU (u)du
+pa f N d
fU (u)du + pe (f )nE + pr (f )nE
fNd nE
0
ufU (u)du
= cnE (by Assumption 5) +pr (f )f N d fU (u)du
subject to the constraints f N d = k S nE ; f f 1; and nE 0. Substituting the constraint f N d = k S nE to remove f from the objective gives: GF2 = b +pa f N d
fNd nE
0
fNd nE
nE u T( )fU (u)du + bT (f ) Nd
fNd nE
fU (u)du + pa nE fU (u)du
fNd nE
0
ufU (u)du
fU (u)du + cnE + pr (f )nE k S
kS
with the constraint f f replaced by the constraint nE nS . E p. 22
We show the derivative of the objective function in this case is positive and hence GF2 is minimized when that constraint is tight, nE i.e., = nS . Consider, E GF2 nE =
0
fNd nE
[
b nE u T( ) + pa ]ufU (u)du + c + pr (f )k S Nd Nd
kS
fU (u)du
(21)
The rst term above is exactly the function H(nE ) introduced in the proof of Proposition 7, and by using its nondecreasing property, we get H(nE ) H(nS ) for all nE nS . However, Proposition 6 suggests E E H(nS ) = E
0
fNd nS E
b [ Nd T (
nS u E Nd )
fNd nE
+ pa ]ufU (u)du = c. This implies that nE nS . E
[
0
b nE u T( ) + pa ]ufU (u)du + c 0; Nd Nd
By using this result with (21), we obtain the desired result, GF2 nE =
0
fNd nE
[
b nE u T( ) + pa ]ufU (u)du + c + pr (f )k S Nd Nd
kS
kS
fU (u)du
pr (f )k S
fU (u)du 0
In both case 1 and case 2, the optimum values for the game setting are f , nS . E
4.2.2 Coordinating Contract: Numerical Application This section uses the idea behind Theorem 4 together with estimates of parameters from the inuenza literature in order to develop a contract that can coordinate the supply chain empirically, even though the actual T (f ) may make a slight deviation from strict convexity. Hill and Longini (2003) suggest R0 = 1.87 and Weycker et al. (2005) argue that = 0.90 is a reasonable value for vaccine effects. We use the data from Weycker et al. (2005) to estimate the direct costs (not indirect) of each infected individual, with to b = $95 on average over the different groups. The vaccine price is set to pr = $12 (CDC, 2005). For vaccine administration costs, there are no explicit data, so we used pa = $40 as a base case, as did Weniger et al. (1998) for pediatric vaccines, being roughly the cost of doctor visit. We used d = 1 dose of vaccine, the usual value, per adult vaccinated. We are not aware of literature to dene the variance of vaccine production yields, so we assumed that U has a gamma distribution with mean p. 23
= 1 (Palese, 2006) and standard deviation = 1/5 = 0.2, so that U Gamma(25, 1/25). We assumed a population of N = 107 individuals and a production cost of c = $6 (not necessarily the actual number). Figure 3 depicts the optimal contract, governmental costs and manufacturer prots, for the special case of T (f ) that is based upon the above parameters and a large initial epidemic wave (using I0 = 0.1). While T (f ) in this case is not precisely convex (slight nonconvexity near f = 0.08), a strict application of the prices implied by (17) and Assumption 6 leads to a whole-unit discount price, pr (f ) (scale on left-hand of y-axis of 3(a)), and cost sharing price pe (f ) (scale on right-hand side of y-axis), that empirically coordinates incentives. The particular choice of = 0.215 for Figure 3 insures that the government overall vaccine procurement and health benet costs are reduced by the contract from $528M to $527M ; that the government orders more (up from f G = 0.65 to f = 0.68); that the manufacturer is willing to produce more (n increases from 6.3M E to 7M ); and that the manufacturers prots increase (from $32.8M to $33.7M ). For sensitivity analysis, we ran the contract under different administration costs [pa = $20, approximately the value in Pisano 2006 for Medicare reimbursement, and pa = $60]; and different values for health benets [b = $275, a value from Gessner 2000 converted into 2000 dollars, and b = $450, the combined direct and indirect costs calculated using data from Weycker et al. 2005]. For values of b $250, we found f = 1 due to the high benet of vaccination compared with the cost of administraction. In general, a smaller pa or higher b will increase f , and increasing pa or decreasing b will decrease f .
5 Discussion and Model Limitations
This work derived the equilibrium state of an interaction between a government and a manufacturer, with the realistic feature of a manufacturer that bears the risk of uncertain production yields. The model shows that a rational manufacturer will always underproduce inuenza vaccines in that setting, relative to the levels that provide an optimal system-wide cost-benet tradeoff. When the levels of exogenous introduction of inuenza into a population are small, leading to the
p. 24
Contract prices 16 2 5.45
x 10
8
Government Cost before contract after contract
pr
14 1.5 5.4
12
1
GF
5.35
10
0.5
5.3
pe
8 0 0.2 0.4 f 0.6 0.8 1 0 5.25 0.5 0.55 0.6 0.65 0.7 f 0.75 0.8 0.85 0.9
(a) pr (f ), pe (f )
x 10
7
(b) GF (f )
Manufacturer Profit before contract after contract
3.5
3
MF
2.5
2
4
5
6
7
8 nE
9
10
11
12 x 10
6
(c) M F (nE )
Figure 3: Cost sharing/whole-unit discount contract.
p. 25
piecewise linear approximation for T (f ) in Section 3, a relatively simple cost sharing contract can coordinate the incentives of the actors to obtain a system optimal solution. When the levels of exogenous introduction of inuenza into a population are somewhat large, as in a large-wave pandemic situation, the analysis of Section 4 may be appropriate. The simple cost sharing contract must be modied to account for the nonlinear population-level health benets that are provided by inuenza vaccination programs. It is therefore not surprising that the whole-unit discount/cost sharing contracts that can align incentives depend on the expected number of infections averted by a given magnitude of the vaccination program effort. There are several limitations of this model. Some of the limitations can be handled with existing methods. Other limitations could lead to interesting future work, but do not limit the value of insights above regarding contract design for governmental/industry collaboration for inuenza outbreak preparedness. One, an epidemic model with homogeneous and homogeneously mixing populations ignores the potential to target specic critical subpopulations, such as children or the elderly. In the short run, the contractual designs here that determine production volumes could be accompanied in a second stage analysis with other work (e.g., Hill and Longini, 2003) that can optimally allocate vaccines to different subpopulations. The generality of the analysis for piecewise linear or convex T (f ) allows some exibility in adapting the incentive alignment results here to more complex epidemic models that account for the prioritization of certain subgroups. Two, the analysis above assumes that the per person benet b and the cost to administer pa are constant. The results may generalize nicely to the case of variable marginal benets of vaccination, b(f ), as long as b(f )T (f ) is convex. Terms like bT (f ) in the denition of f , for example, would be replaced with (b(f )T (f )) = b (f )T (f ) + b(f )T (f ). Similarly, a convex increasing administration cost, pa (f ), might be appropriate too. The net effect of these two changes is expected to decrease the optimal vaccination fraction. Three, the model assumes that health consequences can be quantied by direct and indirect monetary
p. 26
costs, but a multi-attribute approach might be desired to more fully examine issues like the number of deaths or hospitalizations. These features can be modeled indirectly with the present work by assessing the number infected and applying the relevant morbidity and mortality rates. Four, the model assumes that the government can precisely specify the number of individuals to vaccinate. This is potential drawback of the other epidemic models mentioned in this paper, too. The inclusion of individuals choice to become vaccinated would also require much additional complexity. Five, the model currently examines a single manufacturer and a single government, and assumes that all parameters are known to all parties. The cost per dose and yield distributions are not likely to be public information, and there are several providers and many purchasers. Nevertheless the equilibrium still might still be modeled as an outcome of interactions between two rational actors of the model. Multiple buyers and suppliers would be an interesting extension. Contracts in the presence of multiple manufacturers and/or suppliers could be complicated, to avoid collusion on the part of a subset of the players.
6 Conclusion
This work developed the rst integrated supply-chain/health economics model of two key players in the inuenza vaccine supply chain: a government that purchase and administer vaccines in order to achieve an efcient cost-benet tradeoff, and a manufacturer that optimizes production input levels to achieve costeffective delivery of vaccines in the presence of yield uncertainty. The model indicates a lack of coordination for contracts that leave the manufacturer with the production yield risks. That lack of coordination results in vaccine production shortfalls. We show that a global social optimum cannot be fully attained by changing the vaccine price alone, or by reducing the risk of production yields by having the government contractually pay a reduced rate for doses that are produced in excess of the original demand. A variation of the cost sharing contract is one option that can align incentives to achieve a social optimum.
p. 27
Acknowledgments: We are grateful for discussions with Philippe Laurent, Industrial Product Leader for Inuenza, and Catherine Chevat, Health Economics Deputy Director, of Sano-Pasteur; and Michel Galy, former director of the Institut Merieux vaccine facility near Lyon, France (now part of Sano-Pasteur). The content of this paper is ours, and does not necessarily represent the positions of those individuals or of Sano-Pasteur.
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p. 30
Online Appendix A Appendix: Proofs of mathematical results
Proposition 1.
Proof: The expected cost function for the manufacturer is M F (nE ) = cnE pr E min{nE U, f N d} = cnE pr nE E min{U, = cnE pr nE ( = cnE pr nE
fNd nE
fNd } nE
fNd nE
0
fNd nE
ufU (u)du +
fNd fU (u)du) nE
fNd nE
0
ufU (u)du pr f N d
fU (u)du
So to get the minimum of M F we need to see the behavior of its derivative: M F nE = c pr = c pr = c pr Note that
2M F n2 E
fNd nE
0
fNd nE
0
fNd nE
fNd fNd fNd fNd fNd )fU ( )( 2 ) pr f N d fU ( )( 2 ) nE nE nE nE nE 2 2 (f N d) fNd (f N d) fNd ufU (u)du + pr fU ( ) pr fU ( ) 2 2 nE nE nE nE ufU (u)du pr nE ( ufU (u)du
2
0
N N = pr [( (f n3d) )fU ( fnEd )] 0 so the rst order optimality condition is sufcient. Hence the
E
optimum production quantity n is solution of the following equation: E
fNd n E
0
ufU (u)du =
c pr
Corollary 1. Proof: Immediate upon inspection of the values of the parameters.
Proposition 2. Proof: To show these results, we analyze SF in two different regions, f f 0 and f f 0 . Let SF1 (f, nE ) denotes the value of SF when f f 0 , and likewise SF2 (f, nE ) is the value of SF where f f 0 . Note that if f f 0 then W = Z = min{nE U, f N d}, and the value of SF1 is SF1 (f, nE ) =b
fNd nE
(M
fNd nE
0
nE u )fU (u)du + b(M N f ) d
fNd nE
fU (u)du (22) (f f 0 ).
+ pa nE
0
ufU (u)du + pa (f N d)
fNd nE
fU (u)du + cnE
p. 31
Online Appendix For f > f 0 , given that M N f = M N f 0 = 0, the value of SF is SF2 (f, nE ) =b
f 0Nd nE
0 0
nE u (M )fU (u)du + pa nE d
f 0N d nE
f 0Nd nE
0
ufU (u)du (23)
0
+ pa (f N d)
fU (u)du + cnE
(f f ).
The limits of integration in the right hand side of (23) use f 0 , not f . In order to get the overall optimal values for f S , nS , we solve the following two subproblems. E SF 1 = min SF1 s.t. 0 f f0 nE 0 SF 2 = min SF2 s.t. f0 f 1 nE 0
Optimality conditions for subproblem SF 1: The KKT conditions, if f f 0 , are,
N b
fNd nE
fU (u)du + pa N d
fNd nE
fU (u)du + 0 = 0
b d
fNd nE
0
ufU (u)du + pa
fNd nE
0
ufU (u)du + c = 0
(f f 0 ) = 0 f = nE = 0 ;
, 0 , 0,
where the rst equation is obtained by taking the derivative with respect to f and the second equation is obtained by taking the derivative with respect to the nE . Moreover , 0 , are KKT multipliers of constraints f f 0 , f 0, nE 0, respectively. Note that if Assumption 2 were not valid, then the second equation of KKT conditions would require > 0, and the third equation would imply that n = 0. E We are interested in the case where nE > 0, f > 0 which is a conclusion of Assumption 2. This implies that 0 = = 0, and the KKT conditions simplify:
N b + pa N d b + pa d
fNd nE fNd nE
fU (u)du + = 0
0
ufU (u)du + c = 0 0
(f f 0 ) = 0 ; p. 32
Online Appendix In the rst equation above, Assumption 2 suggests that > 0. If > 0, the last of the KKT conditions would give rise to f = f 0 . So SF1 will always get its minimum at the extreme f 0 . The optimal nE in this case can be obtained from the second equation of the KKT conditions and using the fact that f = f 0 , and
f 0Nd n E
0
ufU (u)du =
c
b d
pa
.
(24)
Optimality conditions for the problem SF 2: If f f 0 , then SF2 does not depend on f (the vaccine fraction declared by the government does not change the value of objective function). It follows that all values f 0 f 1 are optimum and so the rst part of the claim is proved. Now SF2 is a function of nE only and the derivative of GF with respect to nE is b SF2 = ( + pa ) nE d Note that
2 SF2 n2 E
0 0 0 f 0N d nE
0
ufU (u)du + c.
= ( b pa )( f nN d )( f nN d )fU ( f nN d ), which is nonnegative by Assumption 2, hence 2 d E E
E
SF2 (nE ) is a convex function on nE and the rst order optimality condition is sufcient. By getting the root of the derivative of SF2 above, we can see that the optimum nE for SF2 is the same as the solution of (24). So the optimum value for nS satises the same equation in both cases. E
Proposition 3.
Proof: If we dene GF1 , GF 1 like SF1 , SF 1 for the case where f f 0 , then GF1 is:
kG
GF1 (f, nE ) = b
(M
0
nE u )fU (u)du + b(M N f ) d
kG 0 kG 0 kG 0 kG 0 kG 0
kG
fU (u)du
kG
+(pa + pr )nE = bM b nE d
ufU (u)du + (pa + pr )(f N d)
fU (u)du
ufU (u)du N bf
kG
fU (u)du
kG
+(pa + pr )nE = bM b nE d
ufU (u)du + (pa + pr )f N d ufU (u)du b nE k G d
kG
fU (u)du
(
0 fU (u)du
= 1)
fU (u)du
kG
+(pa + pr )nE
ufU (u)du + (pa + pr )nE k G
kG 0
fU (u)du
(f N d = nE k G )
b + pa + pr ) = bM + nE ( d
ufU (u)du + k G
kG
fU (u)du
p. 33
Online Appendix By Assumption 3, the coefcient of nE in the last equality is negative, so the optimum value for nE in GF1 lies on the upper boundary, where f = f 0 . This proves the rst part of the claim. For the second part, similarly dene GF2 , GF 2 to represent the government objective functions for the cases f f 0 and f f 0 , respectively. Using the fact that T (f ) = 0 for all f f 0 , and the constraint In the second equation above, f = GF2 (f, nE ) = b
f 0Nd nE
nE k G Nd ,
to obtain
f 0Nd nE
0
nE u )fU (u)du + pa nE (M d
f 0Nd nE
kG
0
ufU (u)du + pr nE
0
ufU (u)du
+pa (f 0 N d)
f 0Nd nE
fU (u)du + pr (f N d)
kG
fU (u)du ufU (u)du + pa (f 0 N d)
f 0Nd nE
=b
0
nE u (M )fU (u)du + pa nE d
kG
f 0Nd nE
0
fU (u)du
+pr nE GF2 nE 2 GF 2 n2 E
0
ufU (u)du + k G
f 0Nd
kG
fU (u)du
kG 0
nE b = ( + pa ) ufU (u)du + pr d 0 f 0N d b f 0N d pa ) 2 fU ( ) =( d nE nE 0
ufU (u)du + pr k G
kG
fU (u)du
for f f 0 . Note that f nN d k G . By Assumption 2, E
2 GF2 n2 E
0, so GF2 is a convex function of nE . To nd
the minimum it sufces to look at the sign of its rst derivative. If Condition (12) holds, then Assumption 2 implies that
GF2 nE
0 on f f 0 , so that the minimum of GF2 for f [f 0 , 1] is obtained at f 0 . The
optimum for both GF1 and GF2 lead to the claimed optimum, namely f G = f 0 . If Condition (12) does not hold (i.e. ( b + pa + pr ) d
kG 0
ufU (u)du + pr k G
kG fU (u)du
< 0); then
because of the convexity of function GF2 on nE (non-decreasing derivative), there are two cases: Case 1: E ; n
GF2 nG E
= 0. In this case clearly the optimum values for the f, nE are the following: nG = nE , E
f G = k G nG /N d. E Case 2: If nE (1) denotes the maximum nE corresponding to f = 1 (i.e. nE (1) = then f G = 1, nG = nE (1). E Combined, the two cases complete the proof.
1N d ) kG
and still
GF2 nG E
<0
p. 34
Online Appendix Theorem 1. Proof: Proposition 2 shows that f G f 0 . We consider the two cases f G = f 0 and f G > f 0 separately, and prove that both cases lead to the relation nS > nG . E E Case 1: f G = f 0 . Using the inequality in Corollary 2 (i.e. k S < k G ) and using the denitions of k G , k S it immediately follows that nS > nG , as desired. E E Case 2: f G > f 0 . (proof by contradiction) Assume to the contrary that nS nG . First of all we obtain E E the sign of
GF2 nE nG . E
As in the proof of Proposition 3, there are two cases for nG . If the condition in case E
GF2 nE nG E
1 of Proposition 3 holds, then following relation is true:
= 0. If case 2 holds, then
GF2 nE nG E
0. In either case, the
GF2 nE On the other hand, GF2 nE b = ( + pa ) d
f 0N d nG E
nG E
0
(25)
kG
nG E
0
f 0N d
ufU (u)du + pr
0 kG 0 kG
ufU (u)du + pr k G ufU (u)du + pr k G
kG kG
fU (u)du fU (u)du
b nS E ( + pa ) ufU (u)du + pr d 0 b c = ( + pa )( b ) + c + pr k G d pa = pr k G
kG d
fU (u)du
fU (u)du > 0
The inequality in the second line comes from the assumption nS nG , and with Assumption 2. The third E E line is valid by (5) and Proposition 2. But the last inequality contradicts (25), so nG nS is false. E E
Proposition 4.
Proof: The proof of Theorem 1 shows that there does not exist a wholesale contract which
coordinate this supply chain. That proof proceeded in two cases. The rst case requires nS > nG . For full E E coordination, we require nS = nG for some pr . In case 2, nS = nG for some pr implies that E E E E which would not be true for the optimizer of GF .
GF nE nG E
> 0,
p. 35
Online Appendix Theorem 2. Proof: First we show that f e f 0 by showing that optimum value for GF1 for f [0, f 0 ] is always obtained at f 0 . By replacing f =
ke k e nE Nd
we get GF1 to be only a function of nE :
ke ke
GF1 (nE ) = b
(M
0
nE k e nE u )fU (u)du + b(M N ) d Nd
ke 0
fU (u)du fU (u)du + pe nE
+(pa + pr )nE
ufU (u)du + (pa + pr )(k e nE )
Now by taking the derivative of GF1 with respect to nE we obtain that: GF1 nE b = d
ke 0
ufU (u)du
ke
b e k d
ke
fU (u)du
ke
+(pa + pr )
0
ufU (u)du + (pa + pr )k e
S e
fU (u)du + pe (26)
k k b + pa ) = ( ufU (u)du + pr ufU (u)du d 0 0 b + ( + pa + pr )k e fU (u)du + pe d ke b = c + (c pe ) + ( + pa + pr )k e fU (u)du + pe d ke b = ( + pa + pr )k e fU (u)du, d ke
(27) (28)
in which (26) is obtained because k e = k S , and (27) is obtained using Proposition 2 and (14). On the other hand (28) is negative by Assumption 3, so that GF1 is decreasing for all eligible nE . Hence f 0 and the corresponding nE (i.e. nE =
f 0N d ke
=
f 0N d ) kS
are optimal in this case. So f e f 0 . Because k e = k S , it
immediately follows that ne nS . E E Now we show that the optimum of GF2 , for f [f 0 , 1], also occurs at f 0 , completing the proof. Note that f f 0 and k e = k S imply that nE nS . Consider GF2 . E GF2 (nE ) = b
f 0Nd nE
0
nE u (M )fU (u)du + pa nE d
ke
f 0Nd nE
0
ufU (u)du + pa f 0 N d
ke
f 0N d nE
fU (u)du
+pr nE
0
ufU (u)du + pr (k e nE )
fU (u)du + pe nE
p. 36
Online Appendix The derivative is nonnegative, GF2 nE b = ( + pa ) d b = ( + pa ) d b ( + pa ) d = pr k e
ke
f 0Nd nE
ke
0
f 0Nd nE
ufU (u)du + pr
0
ufU (u)du + pr k e
ke ke
ke
fU (u)du + pe (29) (30) (31)
0
f 0Nd nS E
ufU (u)du + c + pr k e ufU (u)du + c + pr k e
fU (u)du fU (u)du
0
fU (u)du 0
(29) comes from (14). As before, (30) comes from Assumption 2 and the fact that nE nS . Finally, (31) is E true by Proposition 2. The last inequality shows that the optimum value for GF2 occurs at f 0 hence f e = f 0 and because of the fact that k e = k S , we obtain ne = nS . E E Proposition 6. Proof: The proof resembles the proof of Proposition 2, except for the change in role of f 0 to f , and the denitions of SF 1, SF1 and SF 2, SF2 . We rst show that the optimum value of SF1 always occurs at the border, i.e. f = f , by examining the KKT condition for SF1 :
bT (f ) b Nd
fNd nE
fNd nE
fU (u)du + pa N d
fNd nE
fU (u)du + = 0
fNd nE
0
nE u T( )ufU (u)du + pa Nd (f f ) = 0 ;
0
ufU (u)du + c = 0
0
If f < f , then by the convexity of T (f ) and the denition in (1), we conclude that bT (f ) + pa N d < 0. So the rst equation forces > 0, then by the third equation we obtain f = f . So the optimum value for SF1 occurs at the border which is f . Since SF does not change as f varies in [f , 1], we have shown the rst part of the claim. The optimum value for n in this case can be obtained using the second equation E of the KKT conditions and the fact that f = f . Namely, the optimum nE solves the following equation:
fNd n E
0
n u b T ( E ) + pa ufU (u)du + c = 0, as claimed. Nd Nd
It is now enough to show that in the second case where f f , the same relation holds for the optimum production level. To show this, note that rst of all, SF2 is a functio...