On the Effectiveness of Uniform Subsidies in Increasing ... Subsidies MS.pdfLevi, Perakis, and...

MANAGEMENT SCIENCEArticles in Advance, pp. 1–18ISSN 0025-1909 (print) � ISSN 1526-5501 (online) http://dx.doi.org/10.1287/mnsc.2015.2329

© 2016 INFORMS

On the Effectiveness of Uniform Subsidies inIncreasing Market Consumption

Retsef Levi, Georgia PerakisSloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02142

{[email protected], [email protected]}

Gonzalo RomeroRotman School of Management, University of Toronto, Toronto, Ontario M5S 3E6, Canada,

[email protected]

We study the problem faced by a central planner trying to increase the consumption of a good, such asnew malaria drugs in Africa. The central planner allocates subsidies to its producers, subject to a budget

constraint and endogenous market response. The policy most commonly implemented in practical applications ofthis problem is uniform, in the sense that it allocates the same per-unit subsidy to every firm, primarily because ofits simplicity and perceived fairness. Surprisingly, we identify sufficient conditions of the firms’ marginal costssuch that uniform subsidies are optimal, even if the firms’ efficiency levels are arbitrarily different. Moreover, thisinsight is usually preserved even if the central planner is uncertain about the specific market conditions. Further inmany cases, uniform subsidies simultaneously attain the best social welfare solution. Additionally, simulationresults in relevant settings where uniform subsidies are not optimal suggest that they induce an early optimalmarket consumption. On the other hand, if the firms face a fixed cost of entry to the market, then the performanceof uniform subsidies can be significantly worse, suggesting the need for an alternative policy in this setup.

Keywords : subsidies; budget constraint; Cournot competitionHistory : Received September 27, 2013; accepted July 29, 2015, by Yossi Aviv, operations management Published

online in Articles in Advance.

1. IntroductionIn this work, we study the important setting in whicha central planner aims to impact a given market. Specif-ically, the central planner’s goal is to increase theaggregate market consumption of a good by providingcopayment subsidies, which are paid for each unitthat is produced to competing (profit-maximizing)heterogeneous firms. The motivation to provide suchsubsidies stems from the positive societal externalitiesthat can be obtained by increasing the market consump-tion and from the fact that, left alone, the resultingmarket equilibrium induced by the selfish competingproducers might not be socially optimal. A currentexample is the recent efforts around the production ofinfectious disease treatments to the developing world,such as antimalarial drugs (e.g., Arrow et al. 2004), andvaccines (e.g., Snyder et al. 2011).

Furthermore, typically the central planner makesits subsidy allocation in the presence of a budget con-straint, which is often determined before the actualsubsidy allocation decision. For example, in some casesthe central planner could be a foundation that raised acertain amount of money to address a related issue,and it is then facing the challenge of how to allo-cate the budget toward copayment subsidies. Another

challenge typically faced by the central planner is thatthe intervention in the market through the allocationof subsidies will likely change the market equilibriuminduced by the competing producers. Hence, to opti-mally allocate the subsidies, the central planner has totake into account these complex dynamics.

In this paper, we propose a novel modeling frame-work to study strategic and operational issues relatedto copayment subsidies allocation. The models that wedevelop explicitly capture the setting of a central plan-ner aiming to maximize the market consumption of agood, in the presence of a budget constraint, and marketcompetition between heterogeneous profit-maximizingfirms. The firms are heterogeneous in terms of theirrespective efficiency and cost structure. This is mod-eled through firm-specific marginal cost functions. Themodels that we develop fall into the class of mathemat-ical programs with equilibrium constraints (MPECs).They are relatively general and capture different coststructures and inverse demand functions, as well asa range of market dynamics of quantity competitiontypical of the settings being studied. For example, themodels capture as special cases Cournot competitionwith linear demand, as well as Cournot competitionunder yield uncertainty with linear demand and linear

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Levi, Perakis, and Romero: On the Effectiveness of Uniform Subsidies2 Management Science, Articles in Advance, pp. 1–18, © 2016 INFORMS

marginal cost functions. MPEC models are typicallycomputationally challenging, both to solve optimallyand to analyze; see, e.g., Luo et al. (1996). However, byreformulating these problems, we are able to developtractable mathematical programs that provide upperbounds on the optimal objective value and allowthe development of efficient algorithms. Even moreimportantly, they allow analysis of the effectivenessof practical policies. In particular, this paper focusesattention on the effectiveness of commonly used uni-form copayments, in which the per-unit copayment isthe same for all competing firms in the market. Thecommon use of uniform copayments, in spite of theexistence of heterogeneous firms, each with poten-tially different efficiency levels, is primarily drivenby the simplicity of implementation, as well as somenotion of fairness. This paper addresses the importantquestion of to what extent uniform copayments areeffective in increasing market consumption, comparedto potentially more sophisticated policies that couldallow the copayment to be firm specific. Through themathematical programming upper bound relaxationthat we develop, this paper provides some surprisinginsights. First, we can show that, for a large class offirm-specific cost structures, uniform copayments arein fact optimal. That is, there is no loss of efficiency inusing uniform copayments in these settings comparedto any other possible copayment allocation. Second, thisinsight is maintained even if one considers the case inwhich there exists uncertainty about the future marketstate, and the central planner has to set up the subsidiesbefore the realization of the market condition. Third, inmany cases uniform subsidies do not only obtain theoptimal (maximal) market consumption but at the sametime obtain the best social welfare solution. Fourth,in other relevant settings where uniform subsidiesare not optimal, extensive computational experimentssuggest that they still perform, on average, very closeto optimal. Finally, we also identify conditions wherethe performance of uniform subsidies is not as good.

To demonstrate the applicability of the model andthe relevance of the issues studied in the paper, wenext discuss in detail the case of antimalarial drugs.

1.1. Application: Global Subsidy forAntimalarial Drugs

A motivating example, where the setting modeled inthis paper is observed in practice, is the global fightagainst malaria. This has been a long-standing challengefor the healthcare industry. It is estimated that in 2012approximately 200 million cases occurred worldwideand more than 600,000 people died of malaria; seeWorld Health Organization (2013). To make mattersworse, recently chloroquine, the traditional drug fortreating malaria, has become less effective becauseof growing resistance to this medication. Artemisinin

combination therapies (ACTs) have been identified asthe successor drugs to chloroquine for the treatmentof malaria; however, they are at least 10 times morecostly; see White (2008).

In 2004 the Institute of Medicine (IoM) reviewedthe economics of antimalarial drugs. It identified thatseveral manufacturers compete in an unregulatedmarket and concluded that the most effective way ofensuring access to ACTs for the greatest number ofpatients would be to provide a centralized subsidyto the producers. The goal would be achieving ahigh overall coverage of ACTs. Moreover, the IoMrecognized that firms had not invested in producingACTs on the scale needed to supply Africa becausethere had been no assured market; therefore, the globalcapacity to produce ACTs was quite limited; see Arrowet al. (2004).

In this context, in 2007 the Roll Back Malaria Part-nership and the World Bank developed the AffordableMedicines Facility for malaria (AMFm), a concreteinitiative to improve access to safe, effective, and afford-able antimalarial medicines. In 2008, the Global Fundstarted hosting the AMFm, which began operations inJuly 2010. By July 2012, the AMFm had managed abudget of US$336 million—pledged by UNITAID, thegovernments of the United Kingdom and Canada, andthe Bill & Melinda Gates Foundation—to pursue itsmain objective: increasing the consumption of ACTs,as detailed in the evaluation report by the AMFmIndependent Evaluation Team (2012).

The policy proposed by AMFm is consistent withthe common practice of giving a uniform copayment.Specifically, each firm receives the same copaymentfor each unit sold, regardless of their individual char-acteristics. Moreover, there are 11 firms participatingin the AMFm program, ranging from large pharma-ceuticals like Novartis (having manufacturing plantsin the United States and China) and Sanofi (havingmanufacturing plants in Germany and Morocco) tosmaller firms with manufacturing plants in Uganda,India, and Korea (for more details see A2S2 2014, themarket intelligence aggregator funded by UNITAID).Note that the firms receiving the uniform copaymentsare highly heterogeneous, both in their market size andlocation-wise. One additional relevant characteristic ofthe AMFm program is that all the ACT manufacturersthat receive copayments commit to supply antimalarialdrugs on a no-profit/no-loss basis; see the report byBoulton (2011). Giving the right incentives to anti-malarials producers can increase access to them, hence,it has the potential to have a significant impact on thisglobal problem, see Arrow et al. (2004).

1.1.1. Results and Contributions. The main contri-butions of this paper are the following.

New Modeling Framework for a Subsidy Allocation Prob-lem. We introduce a general optimization framework to

Levi, Perakis, and Romero: On the Effectiveness of Uniform SubsidiesManagement Science, Articles in Advance, pp. 1–18, © 2016 INFORMS 3

analyze subsidy allocation problems with endogenousmarket response, under a budget constraint on the totalamount of subsidies the central planner can pay. Thecentral planner’s objective is to maximize the marketconsumption of a good. Our models allow generalinverse demand and marginal cost functions, assumingonly that the inverse demand function is decreasing inthe market consumption and that the firms’ marginalcosts are increasing. These are standard assumptions inthe literature, and even more general than assumptionsusually considered.

Sufficient Conditions for the Optimality of UniformCopayments. We compare uniform copayments to theoptimal, and potentially differentiated, copayment allo-cation, which provides more flexibility but is potentiallyharder to implement. The main result of this papershows that uniform copayments are in fact optimal fora large family of marginal cost functions. This result issurprising, considering that firms are heterogeneous,and particularly since the assumptions on the inversedemand function are very general (essentially onlymonotonicity and continuity). More importantly, itestablishes sufficient conditions such that the policythat is frequently being used in practice is actuallyoptimal. Additionally, we provide sufficient conditionsfor uniform copayments to simultaneously maximizethe social welfare.

Incorporation of Market State Uncertainty. We extendthe models by assuming that the central planner doesnot know the exact market state with certainty (i.e.,the specific inverse demand function is uncertain),but it has a set of possible scenarios and beliefs onthe likelihood that each scenario will materialize. Wemodel this setting as a stochastic MPEC, where thecentral planner decides its copayment allocation policywith the objective of maximizing the expected marketconsumption. This model is considerably harder toanalyze; see Patriksson and Wynter (1999). However,we show that uniform copayments are still optimal inthis setting, for a fairly large family of firms’ marginalcost functions. Moreover, the analysis suggests that thecentral planner only needs to consider the scenariowith the highest market consumption at equilibrium,regardless of the exact distribution over the differentmarket states.

Tractable Upper Bound Problems (UBPs). Based onan innovative mathematical programming reformula-tion of our model, we develop tractable upper boundproblems. These are used extensively in the analysismentioned above. In addition, we use them to con-duct a numerical study of the performance of uniformcopayments in relevant settings where they are notoptimal. Specifically, we consider Cournot competitionwith linear demand and constant marginal costs, aswell as more general settings with nonlinear demand,and nonlinear marginal cost. The results obtained on

data generated at random suggest that the market con-sumption induced by uniform subsidies is on averagevery close to optimal. We believe that the innovativereformulation of the model, and the resulting upperbounds, would be useful in the study of additionalimportant research questions.

Limitations of Uniform Copayments. We identify setupswhere the performance of uniform subsidies is notas good. The most important one of those is the casewhere the firms face a fixed cost of entry to the market.In this case, we find that uniform subsidies are notoptimal even in the simple setup where firms havelinear marginal costs and face a linear demand. Specifi-cally, their relative performance in maximizing marketconsumption can be as low as 63%. This suggests theneed for an alternative policy in this setup.

2. Literature ReviewThe subject of taxes and subsidy allocation and inci-dence has a vast literature in the economics community.Fullerton and Metcalf (2002) present a thorough reviewof classical and recent results in this area. This paper isclosely related to the study of subsidies in imperfectcompetition models. However, the traditional approachin this literature assumes homogeneous firms andfocuses on studying the impact of taxes, or subsidies,on the number of firms participating in the market ina symmetric equilibrium, because it is directly relatedto the ability to pass taxes forward to the consumer;see Fullerton and Metcalf (2002). More generally, whendoing comparative statics analysis in oligopoly models,it is fairly common to focus on symmetric equilib-ria with homogeneous firms in order to obtain moreprecise insights; see, e.g., Vives (2001). In contrast, inour model we take an operational view: we assumeheterogeneous firms that produce a commodity, andwe focus on the specific subsidy allocation among them.Additionally, we consider a budget constraint in thetotal amount of funding that can be allocated to thesesubsidies.

The strategic trade policy literature, particularlythe “third market model,” studies a related problem;see Brander (1995). In this model, n home firms and n∗

foreign firms export a commodity to a third market,where the market price is set through Cournot competi-tion with constant marginal costs. The government canallocate subsidies to the home firms, increasing theirprofit at the expense of the foreign competitors. Thegovernment’s utility is equal to the profit earned by thehome firms, minus the cost of the subsidy payments.We focus here on the case with heterogeneous firms.In this setting, Collie (1993) and Long and Soubeyran(1997) assume a uniform subsidy and study its effectin the market shares of the firms. Later, Leahy andMontagna (2001) assume a linear demand function


and derive closed-form expressions for the optimalsubsidies. They conclude that the optimal subsidy pol-icy is generally not uniform, and that the governmentshould allocate higher subsidies to more efficient firms.This result is consistent with our numerical study in §7,where uniform subsidies are not optimal for Cournotcompetition with linear demand and constant marginalcosts. Nonetheless, we find computational evidencethat the relative performance of uniform subsidiesis very good. In contrast, our model assumes moregeneral increasing marginal costs, and the analysisobtains conditions under which uniform subsidies areoptimal.

More recently, Cohen et al. (2014) show in a random-ized controlled trial in Kenya that a high subsidy forACT antimalarial drugs dramatically increases access tothem. This is an important empirical insight indicatingthat subsidies for ACTs, as the one considered as amotivation in §1.1, are actually effective in practice.Similarly, Dupas (2014) also show that short-run subsi-dies for an antimalarial bed net had a positive impacton the willingness to pay for it a year later. This resultsuggests that short-run subsidy programs, such as theone also considered toincrease the consumption ofACTs, are expected to be beneficial in the long runas well.

MPECs are very hard to solve and analyze. In par-ticular, even the simplest case with linear objectiveand linear constraints is NP-hard; see Luo et al. (1996).Moreover, stochastic mathematical programs with equi-librium constraints (SMPECs) can be even harder tosolve in practice; see Patriksson and Wynter (1999). Inthis context, the best that we can hope for is to identifyinteresting structure in particular cases that might leadto structural or algorithmic results. The copaymentallocation problem (CAP) in §§3 and 4 and its variantwith market uncertainty (stochastic copayment alloca-tion problem (SCAP)) in §5 are particular cases of anMPEC and an SMPEC, respectively. In both cases, ourmain methodological contribution is the identificationof a special structure that allows us to prove surprisingstructural results, such as the optimality of uniformsubsidies. Examples in the literature that study similarmodels include DeMiguel and Xu (2009) and Adidaand DeMiguel (2011). Recently, Correa et al. (2014)find sufficient conditions for the existence of markupequilibria for marginal cost functions very similar tothe ones where uniform copayments are optimal in ourmodel. On the other hand, the problem of controllingand reducing the contagion of infectious diseases hasbeen studied in the operations management litera-ture as mainly focusing on the analysis of a vaccine’smarkets, particularly the influenza vaccine, its supplychain coordination (e.g., Chick et al. 2008, Mamaniet al. 2012), and the market competition under yielduncertainty (e.g., Deo and Corbett 2009, Arifoglu et al.2012), as opposed to our interest in subsidy allocation.

In particular, we consider the case of allocating subsi-dies to Cournot competitors under yield uncertainty,showing that if the demand and the marginal costs arelinear, then uniform copayments are optimal.

The problem of allocating subsidies to increase themarket consumption of new antimalarial drugs is alsostudied by Taylor and Xiao (2014). However, they con-sider the case of one manufacturer selling to multipleheterogeneous retailers facing stochastic demand. Theiranalysis focuses on the placement of the subsidy bythe central planner in the supply chain, comparingthe possibility of subsidizing either sales or purchases(from the retailer’s point of view). They conclude thatthe central planner should only subsidize purchases,which is equivalent to subsidizing the manufacturer inour setting. Furthermore, they characterize the orderup to the level of the retailers, which is decreasing inthe wholesale price. Therefore, this model can be char-acterized by an arbitrary decreasing inverse demandfunction from the manufacturer’s point of view. Ourpaper complements this work by focusing on the effec-tiveness of uniform subsidies. The combined messageof these two papers to the policy makers is that notonly allocating copayments to manufacturers makessense as a strategy to maximize the market consump-tion but, even if the manufacturers are heterogeneous,the very simple and practical policy of allocating thesame copayments to each firm will most likely obtainmost of the potential benefits. Another growing streamof work in the operations management literature is onethat studies the problem of a central planner decidingrebates that are directed to the consumers, with thegoal of incentivizing the adoption of green technology(see Aydin and Porteus 2009, Lobel and Perakis 2012,Cohen et al. 2015, Krass et al. 2013). In contrast to thatstream of work, this paper is motivated by a differentset of practical applications and focuses on copaymentsthat are allocated to the producers.

3. ModelIn this section, we introduce a mathematical program-ming formulation of the subsidy allocation problem.We then use this formulation to obtain a relaxation ofthe problem, which provides an upper bound on thelargest market consumption that can be induced withthe available budget.

We consider a market for a commodity composedby n≥ 2 heterogeneous competing firms. Each firmi ∈ 811 0 0 0 1 n9 decides its output qi independently, withthe goal of maximizing its own profit. We assumethat the introduction of subsidies in the market willinduce an increase in the market consumption, andthat the firms do not have the installed capacity toprovide all of it. This implies that capacity is scarce inthe market. We model this effect by assuming that the


marginal cost of each firm is increasing. Specifically, weassume that the firms have a firm-specific, nonnegative,increasing, and differentiable marginal cost function ontheir production quantity, denoted by hi4qi5. Consumersare described by an inverse demand function P4Q5,where Q ≡

∑ni=1 qi is the market consumption. We

assume that P4Q5 is nonnegative, decreasing, anddifferentiable in 601 Q7, where Q is the smallest valuesuch that P4Q5= 0.

The assumption on the market equilibrium dynamicsis that each firm participating in the market equilibriumproduces up to the point where its marginal costequals the market price; firms that do not participatein the market equilibrium must have a marginal cost ofproducing any positive amount that is larger than themarket price. This can be expressed in the followingcondition:

For each i1j1 if qi>01 then hi4qi5=P4Q5≤hj4qj50(1)

Assuming a decreasing inverse demand function, andincreasing firms’ marginal cost functions, ensure thatthere exists a unique market equilibrium; see Marcotteand Patriksson (2006). Some special cases of the marketequilibrium condition (1) include imperfect marketcompetition models, such as Cournot competition withlinear demand and Cournot competition under yielduncertainty, with linear demand and linear marginalcosts. Another special case of condition (1) is the modelwhere firms act as price takers and compete in quantity,for any decreasing inverse demand function. In partic-ular, the latter special case captures the behavior ofACT manufacturers discussed in the motivation in §1.1,where all the firms receiving copayments operate on ano-profit/no-loss basis. More generally, the price-takingassumption is a reasonable approximation wheneverthe firms in the market have little market power, e.g.,when there are many firms competing in the market orwhen firms face a threat of entry to the market; seeTirole (1988).

The generality of an arbitrary decreasing inversedemand function allows the modeling of complexdemand mechanisms that have been considered in theoperations management literature. One such example isthe case of multiple competing retailers under demanduncertainty. Specifically, Bernstein and Federgruen(2005) have shown that, in a model where each retailerchooses its retail price and its order quantity and facesmultiplicative random demand (the distribution ofwhich may depend on its own retail price as well asthose of the other retailers), there exists a unique Nashequilibrium in which all the retailer prices decreasewhen the wholesale price is reduced. Moreover, underadditional mild assumptions, this leads to each equilib-rium order quantity decreasing in the wholesale price,resulting in a decreasing inverse demand function.

In other words, the general assumptions in our modelabout the inverse demand function could potentiallycapture relatively complex models of how the price tothe final consumer is reduced when the central plannerallocates copayments to the manufacturers. Specifically,as long as the model induces a demand to the suppliersthat is decreasing in the wholesale price, then it can beconsidered as a special case of a general decreasingdemand function; therefore our results apply.

3.1. Copayment Allocation ProblemWe will refer to the problem faced by the centralplanner as the copayment allocation problem, and wewill denote its mathematical programming formulationby (CAP). Problem (CAP) is a particular case of aStackelberg game, or a bilevel optimization problem.In the first stage, the central planner allocates a givenbudget B > 0, in the form of copayments yi ≥ 0, to eachfirm i ∈ 811 0 0 0 1 n9, per each unit provided in the market.Moreover, the central planner anticipates that, in thesecond stage, the equilibrium output of each firm willsatisfy a modified version of the equilibrium condition.The difference in the market equilibrium condition isgiven by the fact that, from firm i’s perspective, theeffective price for each unit sold is now P4Q5+ yi or,equivalently, that its marginal cost is reduced by yi.

The central planner’s objective is to maximize themarket consumption. Note that, in the setting we study,this is equivalent to maximizing the consumer surplus,which is equal to

∫ Q

0 P4x5dx−P4Q5Q. Specifically, thederivative on the consumer surplus with respect tothe equilibrium market consumption is −P ′4Q5Q> 0,which is positive for any decreasing inverse demandfunction. This is the appropriate objective in manyapplications, where the central planner is a suprana-tional authority, like the World Bank, whose maininterest is effectively maximizing the market consump-tion, say, of an infectious disease treatment, withoutexplicitly taking into account the additional surplusobtained by local producers (see Arrow et al. 2004 forfurther discussion on this topic). Additionally, in §6,we will analyze the case where the central planner’sobjective is to maximize social welfare, including boththe consumer and the producer surplus.

Finally, let us emphasize that the central plannercan only allocate copayments and can never chargetaxes for the units produced in the market. In otherwords, the copayments being allocated have to benonnegative. Next is a formulation of the copaymentallocation problem:

maxy1q1Q

Q

s.t.n∑

i=1

qiyi ≤ B (2)

yi ≥ 01 for each i ∈ 811 0 0 0 1n93 (3)


n∑

i=1

qi =Q3 (4)

qi ≥ 01 for each i ∈ 811 0 0 0 1n93 (5)

P4Q5+ yi = hi4qi51 for each i ∈ 811 0 0 0 1n90 (6)

This is a valid formulation even if there are firms thathave a positive marginal cost of producing any positiveamount, which prevents them from participating inthe market equilibrium. Namely, if for some firm iwe have hi405≥ P4Q5, then we can just set qi = 0 andyi = hi405−P4Q5≥ 0. This is without loss of generality,because setting qi = 0 ensures that firm i does nothave any impact on the budget constraint (2), and thenonnegativity constraint on the copayment yi ensuresthat the market equilibrium condition is satisfied. Inother words, constraint (6) does not imply that everyfirm has to participate in the market equilibrium.

From the equilibrium condition given in con-straint (6), it follows that we can replace all the copay-ment variables yi by hi4qi5− P4Q5. Namely, we canreformulate the copayment allocation problem as if thecentral planner were deciding the output of each firm,as long as there exist feasible copayments that cansustain the outputs chosen as the market equilibrium.The feasibility of the copayments will be given by boththe budget constraint (2) and the nonnegativity of thecopayments (3). We summarize this observation in thefollowing proposition.

Proposition 1. The copayment allocation problem facedby the central planner can be formulated as follows:

(CAP) maxq1Q

Q

s.t.n∑

j=1

qjhj4qj5−P4Q5Q≤B (7)

hi4qi5≥P4Q51 for each i∈8110001n93 (8)n∑

j=1

qj =Q3 (9)

qi ≥01 for each i∈8110001n90 (10)

The copayments that the central planner must allocate toinduce outputs q are yi4q5= hi4qi5− P4Q5, for each i.

Constraint (7) is equivalent to budget constraint (2).Note that it has a budget balance interpretation; i.e.,the total cost in the market minus the total revenuein the market has to be less or equal than the budgetintroduced by the central planner. Constraint (8) isequivalent to the nonnegativity of the copayments (3).

3.2. Special CasesOur model is fairly general. In particular, in this sectionwe discuss some well-known imperfect competitionmodels that are captured as special cases.

3.2.1. Cournot Competition with Linear Demand.The classical oligopoly model proposed by Cournot isdefined in a very similar setting. The only difference isthat, given all the other firms’ production levels, eachfirm sets its output qi at a level such that it maximizesits profit çi, where çi = P4Q5qi −

∫ qi0 hi4xi5 dxi. If we

assume that P4Q5 is decreasing and hi4qi5 are increasing,for each i, as well as P ′4Q5+ qiP

′′4Q5≤ 0, then thereexists a unique market equilibrium defined by thesolution to the first-order conditions of the firms’ profitmaximization problem; see Vives (2001). Namely, atequilibrium, each firm sets its output at a level suchthat

For each i1 if qi > 01 then¡çi

¡qi= 01

or equivalently1 P4Q5= hi4qi5− P ′4Q5qi0 (11)

In the equilibrium condition (11), the marginal costmust be equal to the marginal revenue, whereas, in theequilibrium condition (1), the marginal cost must beequal to the market price. Moreover, the term P ′4Q5qiis not independent for each firm.

Now, for the commonly assumed special case wherethe inverse demand function is linear, i.e., P4Q5= a−bQ,it follows that P ′4Q5= −b. Define hi4qi5≡ hi4qi5+bqi, foreach i; then we can rewrite the equilibrium conditionas follows. For each i, if qi > 0 then P4Q5= hi4qi5. Thisequilibrium condition is a special case of condition (1)but for a modified cost function hi4qi5.

3.2.2. Cournot Competition Under Yield Uncer-tainty with Linear Demand and Linear MarginalCosts. We consider the Cournot competition underyield uncertainty model proposed in Deo and Corbett(2009). We assume that each firm i ∈ 811 0 0 0 1 n9 decidesits production target qi, although the actual output isuncertain and given by qi = �iqi, where �i is a randomvariable reflecting the random yield for firm i. Weassume that the random variables �i are identicallyand independently distributed (i.i.d.) for all firms, withƐ6�i7=�, and Var6�i7= �2. Additionally, we assume alinear inverse demand function P4Q5= a− bQ, whereQ =

∑ni=1 qi is again the market consumption.

We consider two marginal costs: (i) h4qi5 per unitof production target and (ii) h4qi5 per unit actuallyproduced. The first cost is driven by the amount ofraw materials needed for production, whereas thesecond cost corresponds to the cost of packaging theactual output. Finally, we assume Cournot competitionamong the firms. Namely, given the production targetof all the other firms, each firm sets its productiontarget qi to the level that maximizes its expected profit.We generalize the model used in Deo and Corbett(2009) in two ways. First, we consider heterogeneousfirms whereas Deo and Corbett consider homogeneousfirms. Second, Deo and Corbett assume a constant


marginal cost function and a fixed cost to enter themarket, whereas we assume more general marginal costfunctions. Moreover, we extend the model to include acentral planner allocating subsidies to the competingfirms, anticipating the market reaction to the subsidyallocation, and facing a budget constraint. To do so,we assume that both marginal cost functions are linear.Namely, we assume that h4qi5= giqi and h4qi5= giqi.Note that we consider heterogeneous firms, wheresome of them may be more efficient than the others,depending on the values of the firm-specific parametersgi and gi.

Let us start by considering the second-stage problem.Assume that the central planner allocates a copaymentyi ≥ 0 to each firm i ∈ 811 0 0 0 1n9. Each firm sets itsproduction target qi to the level that maximizes itsexpected profit, given by

Ɛ

[

P4Q5qi + yiqi −∫ qi

0h4xi5 dxi −

∫ qi

0h4xi5 dxi

]

= Ɛ

[(

a− bn∑

i=1

�iqi

)

�iqi + yi�iqi − gi

q2i

2− gi

�2i q

2i

2

]

0

The expectation is taken with respect to the randomvariables �i. This is a concave maximization problemin qi; therefore, the first-order condition is sufficientfor optimality. To write the first-order condition in acompact form, define gi ≡ gi/�+ 44�2 +�25/�5gi + b�+

2b4�2/�5, hi4qi5 ≡ giqi, Q =∑n

j=1 qj , P 4Q5 = a − �bQ.Additionally, note that the expected market price hasthe following closed-form expression, Ɛ6P4Q57= a−

�b∑n

i=1 qi = a−�bQ= P 4Q5. Hence, we can write thefirst-order condition of the firms’ profit-maximizationproblem as P 4Q5= hi4qi5− yi.

To define the copayment allocation problem in thissetting, it remains for us to address how the yielduncertainty will be considered in the budget constraint.We consider two possible approaches that will leadto optimization problems with similar structure. First,assume that the central planner would like to find acopayment allocation, such that it satisfies the budgetconstraint in expectation; then we can write the budgetconstraint as Ɛ6

∑ni=1 qiyi7=�

∑ni=1 qiyi ≤ B. Alternatively,

assume that the central planner takes a robust approach.Namely, the central planner would like to satisfy thebudget constraint in each possible yield uncertaintyrealization. We will assume, for simplicity, that thei.i.d. random yields for each firm have a boundedsupport; i.e., �i ∈ 6�1 �7, for each i. Then, we can writethe budget constraint as �

∑ni=1 qiyi ≤ B.

Finally, noting that the objective that the central plan-ner is trying to maximize is Ɛ6Q7=�Q, we concludethat this is a special case of problem (CAP) with vari-ables 4q1 Q5 and functions hi4qi5= giqi, P 4Q5= a−�bQ.

3.3. An Upper Bound ProblemNote that under our assumptions, problem (CAP) isnot necessarily a convex optimization problem. Infact, we have only assumed that the marginal costfunctions hi4qi5 are increasing, for each i, and that theinverse demand function P4Q5 is decreasing. To gainsome insights into the structure of the optimal solution,we ignore the nonnegativity of the copayments andanalyze the following relaxation, which provides anupper bound on the market consumption that can beinduced with the available budget B.

(UBP) maxq1Q

Q

s.t.n∑

j=1

qjhj4qj5− P4Q5Q ≤ B (12)

n∑

j=1

qj =Q3 (13)

qi ≥ 01 for each i ∈ 811 0 0 0 1n90 (14)

This upper bound problem may still be noncon-vex, because of the budget constraint (12). However,Lemma 1 below asserts that at optimality the budgetconstraint is tight, and each of the active firms i musthave a value of 4hi4qi5qi5

′ equal to each other, and nolarger than any inactive firm. This property will havea central role in proving the main result of the nextsection.

Lemma 1. Assume that the marginal cost functionshi4qi5 are nonnegative and continuously differentiable in601 Q5 and that the inverse demand function P4Q5 is non-negative, decreasing, and differentiable in 601 Q7. Then, anyoptimal solution to the upper bound problem (UBP) mustsatisfy the budget constraint (12) with equality and alsosatisfy the following condition:

If qi > 01 then 4hi4qi5qi5′≤ 4hj4qj5qj5

′1

for each i1 j ∈ 811 0 0 0 1n90

The proof of Lemma 1 is in the online appendix (avail-able as supplemental material at http://dx.doi.org/10.1287/mnsc.2015.2329). Note that the assumptionthat the marginal cost functions are increasing is notnecessary for Lemma 1 to hold.

4. Optimality of Uniform CopaymentsThe result obtained in this section asserts that uniformcopayments are optimal for problem (CAP), for alarge class of marginal cost functions hi4qi5. Specifically,we show that if the marginal cost functions satisfyProperty 1 below, then uniform subsidies are optimal.

Property 1. For each i1 j and each qi1 qj ≥ 0, if hi4qi5 >hj4qj5, then 4hi4qi5qi5

′ 6= 4hj4qj5qj5′.


Next, we show that there exists a large class ofmarginal cost functions that satisfy Property 1. Con-sider the case in which hi4qi5= h4giqi5, where h4x5 isnonnegative, increasing, and differentiable over x ≥ 0and gi > 0 is a firm-specific parameter. This capturesthe setting where all firms use a similar technology butcan differ in their efficiency. Specifically, h4x5 modelsthe industry-specific marginal cost function, whereasgi > 0 models the efficiency of firm i.

In this setting there is no loss of generality in assum-ing that h405= 0. Specifically, any positive value forh405 will affect each firm in the same way; therefore, itwill only shift the market price by a constant that canbe rescaled to zero. This assumption implies that allfirms have a positive output in the market equilibrium,for any positive market price. Therefore, the underlyingassumption in order to show the optimality of uniformsubsidies is that all firms have already entered themarket before the subsidy is decided, and there is nosubsequent entry into or exit of firms from the market.This assumption is reasonable in our setting, wherethe subsidy is not permanent (it only applies untilthe budget is exhausted), and it is paid ex post to thefirms, for each unit already sold.

In this setting, any function h4x5 such that h4x5+h′4x5x is monotonic will satisfy Property 1. Specif-ically, for each such function we would have thathi4qi5 > hj4qj5 is equivalent, by definition, to h4giqi5 >h4giqj5. However, h4x5 increasing implies giqi >gjqj .Moreover, h4x5+h′4x5x monotonic implies h4giqi5+h′4giqi5giqi 6= h4gjqj5+h′4gjqj5gjqj , which is, again bydefinition, equivalent to 4hi4qi5qi5

′ 6= 4hj4qj5qj5′. Some

functions that satisfy this condition, and the respectivemarginal cost functions associated with them, are asfollow:

• h4x5= ex − 1, hi4qi5= egiqi − 1.• h4x5= xu, for u> 0, hi4qi5= giq

ui .

• h4x5= ln4x+ 15, hi4qi5= ln4giqi + 15.• Any polynomial with positive coefficients.

Specifically, all these functions have the property thath4x5x is convex over x ≥ 0; therefore, h4x5+h′4x5x isincreasing. Note that the marginal cost functions h4x5are allowed to be concave; e.g., h4x5= xu for 0 <u< 1,and h4x5= ln4x+ 15. Moreover, note that hi4qi5= giq

ui

corresponds to the unique homogeneous function ofdegree u> 0 in one variable.

4.1. Sufficient Optimality ConditionThe following theorem is the main result in this section.

Theorem 1. Assume that the marginal cost functionshi4qi5 are nonnegative, increasing, and continuously dif-ferentiable in 601 Q5; the inverse demand function P4Q5 isnonnegative, decreasing, and differentiable in 601 Q7. If themarginal cost functions satisfy Property 1, then uniformcopayments are optimal for problem (CAP).

Proof. The existence of an optimal solution to prob-lem (UBP) was shown in Lemma 1. Let 4q1Q5 be anoptimal solution to problem (UBP). We will show thatif the marginal cost functions satisfy Property 1, then4q1Q5 induces uniform copayments for every firmwith a positive output qi > 0. Moreover, 4q1Q5 is also afeasible solution of problem (CAP), therefore optimal.

From Lemma 1 it follows that 4q1Q5 is such thatthe budget constraint is binding, and for each i, j withqi > 0 and qj > 0, we must have 4hi4qi5qi5

′ = 4hj4qj5qj5′.

The assumption that the marginal cost functions satisfyProperty 1 implies that hi4qi5= hj4qj5, which implies thatuniform subsidies are optimal. Specifically, because thebudget constraint is tight, it follows that hi4qi5−P4Q5=

B/Q> 0 for every i such that qi > 0.It remains to show that the firms that do not par-

ticipate in the market equilibrium effectively havea marginal cost of producing zero units, which islarger than the induced market price. To do so, notethat 4q1Q5 is such that, for each i, j with qi > 0 andqj = 0, we have hj405−P4Q5≥ hi4qi5+h′

i4qi5qi −P4Q5≥

hi4qi5− P4Q5= B/Q> 0. The first inequality followsfrom Lemma 1, and the second inequality follows fromhi4qi5 increasing. The equality follows from the factthat the budget constraint is tight, and qi > 0.

Hence, 4q1Q5 is also feasible for problem (CAP),therefore optimal. Moreover, 4q1Q5 induces uniformcopayments. Therefore, uniform copayments are opti-mal for problem (CAP). �

This result is surprising, considering that the assump-tions on the inverse demand function are very general,particularly since firms can be heterogeneous and thecentral planner has the freedom to allocate differenti-ated copayments to each firm. The intuition behind itcomes from the market equilibrium condition and thebudget constraint. Essentially, if the central plannerallocates a larger copayment to a firm, then its resultingmarket share will increase, which is exactly the rateat which it will consume budget. This will in turnmake less budget available to the rest of the firms;therefore, their copayments would have to decrease.Theorem 1 shows that if the marginal cost functionssatisfy Property 1, then the net effect of this changewill never be positive.

In particular, Theorem 1 applies for the special caseswe considered in §3.2. For Cournot competition withlinear demand, uniform copayments are optimal for anymarginal cost functions hi4qi5, such that the functionshi4qi5= hi4qi5+ bqi satisfy Property 1. Specifically, if themarginal cost functions are linear, i.e., hi4qi5= giqi, foreach i, then uniform copayments are optimal. Similarly,for Cournot competition under yield uncertainty withlinear demand, if both marginal costs are linear, thenuniform subsidies are optimal. Note that in both caseswe allow for heterogeneous firms, where some can besignificantly more efficient than others.


5. Incorporating Market StateUncertainty

A natural extension of the model discussed in §3 is toconsider the setting where the central planner does notknow the market state (defined by the inverse demandfunction) with certainty but generally will have a setof possible market state scenarios and beliefs on thelikelihood that each scenario will materialize.

Specifically, we assume that the central planner hasa discrete description of the market state uncertainty,where each scenario s ∈ 811 0 0 0 1m9 is realized withprobability ps. Each scenario s is characterized by ascenario-dependent inverse demand function P s4Qs5.For each scenario s ∈ 811 0 0 0 1m9, we make assumptionsas in §3. Namely, we assume that each inverse demandfunction P s4Q5 is nonnegative, decreasing, and differen-tiable in 601 Qs7, where Qs is the smallest value suchthat P s4Qs5= 0. Similarly, for the market equilibriumcondition we assume that if scenario s is realized, thenfirms set their output qsi at a level such that, for eachi1 j , if qsi > 0, then hi4q

si 5= P s4Qs5≤ hj4q

sj 5.

Similar to §3, a formulation of the copayment alloca-tion problem under market state uncertainty can bewritten as follows:

max4qs1Qs 5s∈8110001m91y

m∑

s=1

Qsps

s.t.n∑

j=1

qsj yj ≤B1 for each s∈8110001m93 (15)

yi ≥01 for each i∈8110001n93 (16)n∑

j=1

qsj =Qs1 for each s∈8110001m93 (17)

qsi ≥01 for each i∈8110001n91

s∈8110001m93 (18)

hi4qsi 5−P s4Qs5−yi ≥01

for each i∈8110001n91 s∈8110001m93 (19)

qsi 4hi4qsi 5−P s4Qs5−yi5=01

for each i∈8110001n91 s∈8110001m90 (20)

The objective is to maximize the expected market con-sumption. Constraint (15) is the budget constraint foreach market state scenario. Constraint (16) correspondsto the nonnegativity of the copayments. Constraint (17)defines the market consumption for each scenario.Finally, constraints (18)–(20) are the complementarityconstraints, which tie together the different scenarios.They state that, in each scenario, each firm must satisfyone of the following two cases: either it participates inthe market equilibrium (in which case it produces thequantity that equates its marginal cost with the marketprice plus the copayment), or it is inactive (in which

case its marginal cost of producing any positive amountis strictly larger than the market equilibrium price plusthe copayment). Naturally, each firm must get the samecopayment in each possible scenario.

In other words, constraints (18)–(20) correspond tothe “nonanticipativity’’ constraints, and they state thatcopayments are a first-stage decision made by thecentral planner before the uncertainty is realized. This isprecisely what prevents us from using the copaymentsto eliminate the complementarity constraints from themodel formulation, similarly to Proposition 1. Thismakes the problem significantly harder to analyze.To somewhat simplify this formulation, we make theadditional assumption that producing zero units has amarginal cost of zero.

Proposition 2. If we additionally assume hi405= 0, foreach i ∈ 811 0 0 0 1n9, then the copayment allocation problemunder market state uncertainty faced by the central plannercan be rewritten as follows:

(SCAP)

max4qs1Qs 5s=110001m

m∑

s=1

Qsps

s.t.n∑

j=1

qsj hj4qsj 5−P s4Qs5Qs

≤B1

for each s∈8110001m93 (21)

hi4qsi 5≥P s4Qs51 for each i∈8110001n91

s∈8110001m93 (22)n∑

j=1


qsi ≥01 for each i∈8110001n91

s∈8110001m93 (24)

hi4qsi 5−P s4Qs5=hi4q

s′

i 5−P s′4Qs′51

for each i∈8110001n91 s1s′∈8110001m90 (25)

The copayments that the central planner must allocate toinduce outputs 8qs9s∈8110001m9 are

yi = hi4qsi 5− P s4Qs51

for each i ∈ 811 0 0 0 1n91 s ∈ 811 0 0 0 1m90 (26)

Proposition 2 states that, if the marginal cost ofproducing zero units is zero, then every firm willparticipate in the market equilibrium for any nonnega-tive market price. Therefore, Equation (26) holds, andwe can eliminate the variables yi from the problemformulation.

Like in Proposition 1, constraint (21) corresponds tothe budget constraint. Namely, for each scenario s, thetotal cost minus the total revenue in the market has tobe less than or equal to the budget introduced by the


central planner. Constraint (22) is the nonnegativity ofthe copayments: it ensures that the solution proposedby the central planner can be sustained as a marketequilibrium by allocating only subsidies, not taxes. Likebefore, the only constraint that ties all the scenariostogether is the nonanticipativity constraint (25), whichstates that each firm must get the same copayment ineach possible scenario.

This problem is still hard to analyze directly, whichmotivates us to develop a relaxation that provides anupper bound on the expected market consumption thatcan be induced with the available budget, as shownbelow. All the proofs in this section are presented inthe online appendix.

5.1. An Upper Bound ProblemWe start with a simple observation that is derived fromthe structure of problem (SCAP).

Lemma 2. For any feasible solution to problem (SCAP),without loss of generality, the scenarios can be renumbered,such that the following inequalities hold true:

P 14Q15≥ P 24Q25≥ · · · ≥ Pm4Qm53 (27)

hi4q1i 5≥ hi4q

2i 5≥ · · · ≥ hi4q

mi 51 for each i3 (28)

q1i ≥ q2

i ≥ · · · ≥ qmi 1 for each i3 (29)

Q1≥Q2

≥ · · · ≥Qm3 (30)n∑

j=1

q1j yj ≥

n∑

j=1

q2j yj ≥ · · · ≥

n∑

j=1

qmj yj1 (31)

where∑n

j=1 qsj yj is the total amount spent in copayments in

scenario s.

Let 4qs1Qs5∗s=110001m be an optimal solution to problem(SCAP), and assume that the scenarios are numberedsuch that Equations (27)–(31) hold. Then, we claimthat the solution to problem (SUBP) below provides anupper bound on the expected market consumption thatcan be induced with the available budget. Specifically,problem (SUBP) is derived from problem (SCAP) byadding constraints (36) and (37) below and replacingthe non-anticipativity constraint (25) with the relaxedversion (38).

(SUBP)

maxqs1Qs

m∑

s=1

Qsps

s.t.n∑

j=1

qsj hj4qsj 5−P s4Qs5Qs

≤B1

for each s∈8110001m93 (32)

hi4qsi 5≥P s4Qs51 for each i∈8110001n91

s∈8110001m93 (33)

n∑

j=1


qsi ≥01 for each i∈8110001n91 s∈8110001m93 (35)

P 14Q15≥P s4Qs51 for each s∈8110001m93 (36)

Q1≥Qs1 for each s∈8110001m93 (37)

hi4qsi 5−P s4Qs5≤hi4q

1i 5−P 14Q151

for each i∈8110001n91 s∈8110001m90 (38)

Problem (SUBP) is a valid relaxation of problem (SCAP).Specifically, the optimal solution of problem (SCAP),4qs1Qs5∗s=110001m, is feasible for problem (SUBP) andattains the same objective value. To argue the feasi-bility of solution 4qs1Qs5∗s=110001m for problem (SUBP),recall from Lemma 2 that s = 1 is the scenario thatattains the largest value for both the induced marketprice (see (27)) and the induced market consump-tion (see (30)) in solution 4qs1Qs5∗s=110001m. It followsthat adding constraints (36) and (37) does not cut offsolution 4qs1Qs5∗s=110001m. Finally, solution 4qs1Qs5∗s=110001msatisfies constraint (38) with equality.

5.2. Optimality of Uniform CopaymentsIn this section, we consider again the setting whereall firms use a similar technology but can differ intheir respective efficiency, similar to the assumptions in§4. Specifically, we consider the case in which hi4qi5=

h4giqi5, where h(x) is nonnegative, increasing, anddifferentiable over x > 0, and gi > 0 is a firm-specificparameter. The function h4x5 models the industry-specific marginal cost function, whereas gi models theefficiency of firm i. Recall from §4 that we can assume,without loss of generality, that h405= 0; therefore, wewill refer to problem (SCAP), and its upper boundproblem (SUBP).

We now present sufficient conditions to ensure thatuniform subsidies maximize the expected market con-sumption in this setting. Specifically, we show that if thefirms’ marginal cost functions satisfy Property 2 below,then uniform subsidies are optimal for problem (SCAP).

Property 2. The function h4x5 is convex, such that forany x1 > x2 ≥ 0 and x1 > x3 > x4 ≥ 0, if h4x25/h4x15 >h4x45/h4x35, then h′4x25/h

′4x15 > h′4x45/h′4x35.

Note that Property 2 implies Property 1, discussedin §4. Specifically, h4x5 increasing and convex impliesthat h4x5 + h′4x5x is increasing. This is a sufficientcondition for Property 1 to hold.

Remark 1. The functions hi4qi5 = giqmi , for m ≥ 1,

and hi4qi5= egiqi − 1, satisfy Property 2.

Note, from Remark 1, that from the examples ofmarginal cost functions that satisfy Property 1 givenin §4, all the examples that are also convex satisfy


Property 2 as well. In this sense, the extra requirementsin Property 2, with respect to Property 1, are mainlydriven by the convexity assumption. Finally, note thatfunctions hi4qi5= giq

mi , for m≥ 1, are the unique convex

homogeneous functions in one variable.Theorem 2 below shows that there exists an optimal

solution to the upper bound problem (SUBP), such thatthe copayments induced in scenario s = 1, the one thatattains the largest market consumption (see (30)), andthe largest amount spent in copayments (see (31)) areuniform. This result will play a central role in provingthe main result in this section.

Theorem 2. Assume that the inverse demand functionP4Q5 is nonnegative, decreasing, and differentiable in 601 Q7.Assume that the marginal cost functions are given byhi4qi5= h4giqi5 for each i, for any increasing and continu-ously differentiable function h4x5, such that h405= 0. If h4x5satisfies Property 2, then there exists an optimal solutionto the upper bound problem (SUBP), 4qs1 Qs5s=110001m, suchthat, hi4q

1i 5− P 14Q15= y1 for each i ∈ 811 0 0 0 1 n9, for some

value y1 > 0.

To prove Theorem 2, we show several lemmasin the online appendix that are useful in the anal-ysis. Specifically, we consider the optimal solutionto problem (SUBP) with the smallest differencebetween 4maxi∈8110001n98hi4q

1i 595 and 4mini∈8110001n98hi4q

1i 595

(see Lemma 3 in the online appendix). Note that prov-ing Theorem 2 is equivalent to showing that thisdifference is zero. We assume by contradiction that thisdifference is strictly positive and show that then wecan construct another optimal solution with an evensmaller difference, a contradiction.

When constructing the modified optimal solution,Lemma 4 in the online appendix allows us to focusonly on constraint (38). On the other hand, using theconvexity assumption on h4x5, Lemma 5 in the onlineappendix provides bounds on the impact that themodifications to the optimal solution have on con-straint (38). These bounds allow us to complete theproof by arguing that the modified solution is feasi-ble and optimal, while attaining a smaller differencebetween the maximum marginal cost in scenario s = 1and the minimum marginal cost in scenario s = 1.

Theorem 3 below concludes this section characteriz-ing a family of firms’ marginal cost functions such thatuniform copayments are optimal, even if the centralplanner is uncertain about the market state. This familyincludes convex homogeneous functions of the samedegree.

Theorem 3. Assume that the inverse demand functionP4Q5 is nonnegative, decreasing, and differentiable in 601 Q7.Assume that the marginal cost functions are given byhi4qi5= h4giqi5 for each i, for any increasing and continu-ously differentiable function h4x5, such that h405= 0. If

h4x5 satisfies Property 2, then allocating the largest feasi-ble uniform copayment is an optimal solution for problem(SCAP).

This result is surprising because it shows that, withsome additional conditions, the optimality of uniformsubsidies is preserved, even if the central planner isuncertain about the market state. Different marketstates induce different inverse demand functions, whichcan be arbitrarily different. Moreover, the assumptionson the inverse demand functions of each scenario arevery mild. Specifically, we only assume that they aredecreasing. This is a very relevant setup because itcorresponds to a more realistic representation of theproblem faced in practice, where there are large uncer-tainties about different characteristics of the marketstate, which ultimately define the effective response ofthe demand side to different market prices.

Moreover, the analysis suggests that the central plan-ner only needs to consider the scenario with the highestmarket consumption at equilibrium (see (30)), i.e., sce-nario s = 1, regardless of the exact distribution over thedifferent market states. Specifically, Theorem 2 showsthat uniform subsidies are optimal for scenario s = 1 inthe relaxed upper bound problem (SUBP), whereasTheorem 3 shows that the uniform subsidies inducedby scenario s = 1 are, in fact, optimal for problem(SCAP). This insight suggests that the central planneronly needs to identify the scenario with the highestmarket consumption at equilibrium, and implement theuniform subsidies induced by it, as opposed to takinginto consideration beliefs on the likelihood that eachmarket state will be realized and the effect that thesubsidy allocation will have on each possible scenario.

6. Maximizing Social WelfareIn this section, we assume that the central planner’sobjective is in fact to maximize social welfare. Givensome � ∈ 40117, which represents the social cost of funds,the central planner’s problem of allocating subsidies tomaximize social welfare can be written as follows:

maxy1q1Q

∫ Q

0P4x5dx−

n∑

i=1

∫ qi

04h4xi5−yi5dxi−�

( n∑

i=1

qiyi

)

s.t. yi ≥01 for each i∈8110001n9 (39)n∑

i=1

qi =Q3 (40)

qi ≥01 for each i∈8110001n93 (41)

P4Q5+yi =hi4qi51 for each i∈8110001n90 (42)

The first two terms in the objective function correspondto the sum of the consumer and producer surplus,including the copayments yi. The third term in theobjective function corresponds to the social cost of


financing the subsidies. Note that in this problem thereis no budget constraint. Specifically, the social cost offunds � ∈ 40117 will induce a total amount invested insubsidies at optimality, which can be interpreted as theimplicit budget available. Constraint (39) states that thecentral planner is only allowed to allocate subsidies,not taxes, to the firms. Like in §3, constraint (42) doesnot imply that every firm has to participate in themarket equilibrium.

From the equilibrium condition given in con-straint (42), it follows again that we can replace all thecopayment variables yi by hi4qi5−P4Q5, as stated inthe proposition below.

Proposition 3. The social welfare maximization prob-lem can be written as follows:

(CAP-SW)

maxq1Q

≡

∫ Q

0P4x5dx−

n∑

i=1

∫ qi

0h4xi5 dxi

+ 41 − �5

( n∑

i=1

h4qi5qi − P4Q5Q

)

s0t0 h4qi5≥ P4Q51 for each i ∈ 811 0 0 0 1n93 (43)n∑

i=1

qi =Q3 (44)

qi ≥ 01 for each i ∈ 811 0 0 0 1n90 (45)

The copayments that the central planner must allocate toinduce outputs q are yi4q5= hi4qi5− P4Q5, for each i.

The first two terms in the objective function cor-respond to the sum of the consumer and producersurplus, with no subsidies. The third term correspondsto the increase in social welfare induced by subsidies,minus the social cost of financing them. Constraint (43)states that the central planner is only allowed to allocatesubsidies, not taxes, to the firms.

We will make the natural assumption that the socialcost of funds, � ∈ 40117, is such that objective function ofproblem (CAP-SW), denoted by SW4q1Q5, is coercive;1

therefore, there exists an optimal solution (see, e.g.,Bertsekas (1999)). Then, the budget B that the centralplanner spends in subsidies, in order to maximizesocial welfare, can be written as follows: let 4q∗1Q∗5be an optimal solution of problem (CAP-SW), thenB ≡

∑ni=1 h4q

∗i 5q

∗i − P4Q∗5Q∗.

6.1. Optimality of Uniform CopaymentsWe conclude this section by characterizing settingswhere, in addition to maximizing the market consump-tion, uniform copayments also maximize social welfare.Specifically, we show that if the marginal cost functionssatisfy Property 3 below, then uniform subsidies areoptimal for problem (CAP-SW).

1 SW4q1Q5 is coercive if SW4qk1Qk5−→ −� for any feasible sequencesuch that ��4qk1Qk5�� −→ �.

Property 3. For each i1 j and each qi1 qj ≥ 0, if hi4qi5 >hj4qj5 then hi4qi5/4h

′i4qi5qi5≥ hj4qj5/4h

′j4qj5qj5.

Two examples of marginal cost functions that satisfyProperty 3 are

• hi4qi5= giqui for u> 0.

• hi4qi5= ln4giqi + 15.Note that these marginal cost functions also satisfyProperty 1. Therefore, for these two marginal cost func-tions, uniform subsidies maximize both the consumersurplus and social welfare.

This leads to the main result of this section.

Theorem 4. Assume that the marginal cost functionshi4qi5 are nonnegative, increasing, and differentiable in601 Q5; the inverse demand function P4Q5 is nonnegative,decreasing, and differentiable in 601 Q7; and the social costof funds �∈ 40117 is such that it induces a finite centralplanner’s budget B. If the marginal cost functions satisfyProperty 3, then uniform copayments are optimal for thesocial welfare maximization problem (CAP-SW).

Proof. Let 4q∗1Q∗5 be the optimal solution of prob-lem (CAP-SW). First, we show that 4q∗1Q∗5 mustsatisfy

41 − �5 <hi4q

∗i 5

hi4q∗i 5+h′

i4q∗i 5q

∗i

1 for each i0 (46)

Specifically, the expression in the objective func-tion of problem (CAP-SW) related to the mar-ket consumption Q is strictly increasing in Q.Namely, 4¡/¡qi54

∫ Q

0 P4x5dx− 41 − �5P4Q5Q5= �P4Q5−41 − �5P ′4Q5Q> 0, where the inequality follows fromthe inverse demand function P4Q5 being nonnega-tive and decreasing. On the other hand, the remain-ing expression in the objective function of problem(CAP-SW), related to firm i’s output qi, is such that4¡/¡qi5441 − �5

∑ni=1 h4qi5qi −

∑ni=1

∫ qi0 h4xi5 dxi5= 41 − �5 ·

4hi4qi5+h′i4qi5qi5−hi4qi5. Assume for a contradiction

that Equation (46) does not hold. Namely, there existsan index i such that 41−�54hi4q

∗i 5+h′

i4q∗i 5q

∗i 5−hi4q

∗i 5≥ 0.

It follows that we can increase q∗i by � > 0 sufficiently

small and obtain a feasible solution that attains astrictly larger objective value. This is a contradiction tothe optimality of 4q∗1Q∗5.

Second, assume by contradiction that there existindexes i, j , with q∗

i > 0 and q∗j > 0, such that hi4q

∗i 5 >

hj4q∗j 5. The fact that the marginal cost functions

satisfy Property 3 implies that hi4q∗i 5/4h

′i4q

∗i 5q

∗i 5 ≥

hj4q∗j 5/4h

′j4q

∗j 5q

∗j 5. From direct algebraic manipulations

it follows that

hi4q∗i 5

hi4q∗i 5+h′

i4q∗i 5q

∗i

≤hi4q

∗i 5−hj4q

∗j 5

hi4q∗i 5+h′

i4q∗i 5q

∗i −hj4q

∗j 5−h′

j4q∗j 5q

∗j

0

4475

Now, Equations (46) and (47) imply that 41 − �5 <4hi4q

∗i 5− hj4q

∗j 55/4hi4q

∗i 5+ h′

i4q∗i 5q

∗i − hj4q

∗j 5− h′

j4q∗j 5q

∗j 5.


Therefore, we can transfer � > 0 sufficiently small fromq∗i to q∗

j and obtain the following positive marginalchange in the objective function, hi4q

∗i 5−hj4q

∗j 5− 41−�5 ·

4hi4qi5+h′i4qi5qi −hj4q

∗j 5−h′

j4q∗j 5q

∗j 5 > 0. Namely, there

exists a feasible solution with a strictly larger objectivevalue. This contradicts the optimality of 4q∗1Q∗5. Hence,we conclude that for each i, j , with q∗

i > 0 and q∗j > 0, it

must be the case that hi4q∗i 5= hj4q

∗j 5. Therefore, uniform

subsidies maximize social welfare. �

7. Numerical ResultsIn §4, we have identified conditions on the firms’marginal cost functions that guarantee the optimality ofuniform copayments to maximize the market consump-tion of a good. In this section we study the performanceof uniform copayments, in relevant settings where theyare suboptimal. Our goal here is to study numericallythe performance of uniform copayments on problemswith data generated at random.

To evaluate the relative performance of uniformsubsidies, we need to be able to compute the respectivemarket consumption induced by them. Proposition 4below addresses this issue for the special cases of ourmodel studied next in §§7.1 and 7.2.

Proposition 4. Assume that the marginal cost func-tions hi4qi5 are nonnegative, increasing, and differentiable in601 Q5 and that the inverse demand function P4Q5 is non-negative, decreasing, and differentiable in 601 Q7. Then, themarket equilibrium induced by the largest feasible uniformcopayment can be computed as the solution to the followingconvex optimization problem,

(UCAP) minq

{

n∑

j=1

∫ qj

0hj4xj5dxj −

∫ qn+1

0P4Q−xn+15dxn+1

−B ln4Q−qn+15}

s.t.n∑

j=1

qj +qn+1 =Q

qi ≥01 for each i0

Assuming that the inverse demand function P4Q5 isdecreasing and that the firms’marginal cost functionshi4qi5 are increasing implies that problem (UCAP) is aconvex optimization problem. On the other hand, inthe experimental settings we consider in this section, itwill always be the case that at least the upper boundproblem (UBP) is a convex optimization problem. Tosolve these problems we used CVX, a package forspecifying and solving convex programs; see Grantand Boyd (2012). We will denote by QU, QOPT, and QUB

the market consumption component of the optimalsolutions to problems (UCAP), (CAP), and (UBP),respectively.

7.1. Cournot Competition with Linear Demand andConstant Marginal Costs

The model presented in §3 captures Cournot competi-tion with linear demand and nondecreasing marginalcost functions hi4qi5. Specifically, this implies that themodified marginal cost function defined in §3.2, hi4qi5≡

hi4qi5+ bqi, is increasing. In particular, in this sectionwe consider constant marginal costs. Although the con-stant marginal costs case moves away from our scarceinstalled capacity assumption, it is a well-understoodmodel where uniform copayments are not optimal.Therefore, it is interesting to study the performance ofuniform copayments in this setting.

Specifically, in this section we assume that P4Q5=

a−bQ, and hi4qi5= ci, for each i. Therefore, the modifiedmarginal cost is hi4qi5= ci + bqi, for each i. Under theseassumptions, problem (CAP) is a convex optimizationproblem. Therefore, we solve both the uniform copay-ments allocation problem (UCAP) and problem (CAP),and we compare their objective functions directly. Weconsider four cases in the number of firms participatingin the market, n ∈ 82131101209. For each one of thesefour cases, we solve 1,000 instances of the problem.These instances are randomly generated, with parame-ters sampled from the following distributions: a, b areuniformly distributed in 601507 and ci are independentand uniformly distributed in 601 a7, for each i.

Figure 1 presents a box plot of the results for theratio QU/QOPT, and Table 1 presents some summarystatistics. It is interesting that the minimum value of theratio QU/QOPT never went below 91% in the simulationresults. Moreover, the mean and median values areabove 98% for each value of the number of firmsparticipating in the market n. This suggests that, in mostcases, the market consumption induced by uniformcopayments is fairly close to the market consumptioninduced by the optimal copayment allocation.

7.2. Price-Taking Firms with Nonlinear Demandand Nonlinear Marginal Costs

Now we consider a more general experimental setup,with nonlinear demand and nonlinear marginal costs,where the firms act as price takers. In this setting weassume that P4Q5 = a− bQm0 and hi4qi5 = ci + giq

mii

for each i. Under these assumptions, problem (CAP)is a nonconvex optimization problem. However, theupper bound problem (UBP) is convex. Therefore, wesolve both the uniform copayment allocation problem(UCAP) and the upper bound problem (UBP), and wecompare their objective functions.

We consider again four cases in the number of firmsparticipating in the market, n ∈ 82131101209. For eachone of these four cases, we solve 1,000 instances of theproblem. These instances are randomly generated, withparameters sampled from the following distributions:a, b are uniformly distributed in 601507. For each i, ci


Figure 1 Relative Performance of Uniform Copayments: CournotConstant Marginal Costs

1.00

U/ OPT

0.98

0.96

0.94

0.92

n = 2 n = 3 n = 10 n = 20

are independent and uniformly distributed in 601 a7, gi

are independent and uniformly distributed in 601507,and mi are independent and uniformly distributed in401207. Finally, m0 is uniformly distributed in 40137.

Note that P4Q5= a− bQm0 , m0 ∈ 40137 captures bothconvex and concave decreasing inverse demand func-tions. Similarly, hi4qi5= ci + giq

mii , mi ∈ 401207 for each

i captures both convex and concave marginal costfirms. The results for the ratio QU/QUB are displayedin Figure 2 and in Table 2. The minimum value of theratio QU/QUB never went below 70% in the simulationresults. Moreover, the mean and median values areabove 96% for each value of the number of firmsparticipating in the market n, where in this case weare not comparing directly to the optimal solution, butto an upper bound. This again suggests that, in mostcases, the market consumption induced by uniformcopayments is fairly close to the market consumptioninduced by the optimal copayment allocation.

7.3. Cournot Competition with NonlinearDemand and Nonlinear Marginal Costs

In this section, we extend the numerical study of theperformance of uniform subsidies for Cournot competi-tion with nonlinear demand. Recall from Equation (11)

Table 1 Summary of Relative Performance of Uniform Copayments:Cournot Constant Marginal Costs

Q U/Q OPT n = 2 n = 3 n = 10 n = 20

Min. 0.9360 0.9175 0.9182 0.94421st Qu. 0.9776 0.9734 0.9785 0.9828Median 0.9919 0.9845 0.9849 0.9876Mean 0.9860 0.9806 0.9834 0.98663rd Qu. 0.9983 0.9930 0.9902 0.9912Max. 1.0000 1.0000 1.0000 0.9991

Figure 2 Relative Performance of Uniform Copayments:Price-Taking Firms

1.00

0.95

0.90

0.85

0.80

0.75

0.70

n = 2 n = 3 n = 10 n = 20

U/ UB

that this model is not one of the special cases of formu-lation (CAP) given in §3.2, because it would correspondto each firm having a nonseparable marginal costfunction h4qi1Q5= hi4qi5− P ′4Q5qi, which depends onthe market output of all the other firms. Nonetheless,additional modeling techniques will allow us to numer-ically compute a bound on the performance of uniformsubsidies in the experiments for this case.

We consider the same nonlinear setting as in §7.2:the difference is that now firms are assumed to engagein Cournot competition. Under these assumptions,the modified nonseparable marginal cost function ofeach firm becomes h4qi1Q5= ci +giq

mii +m0bQ

m0−1qi.Moreover, we consider the natural generalizations forproblem (CAP) and the upper bound problem (UBP),where we directly replace the marginal cost functionhi4qi5 with the nonseparable function h4qi1Q5; hencewe skip the problem statements here.

Note that both problems (CAP) and (UBP) are non-convex for our experimental setup; however, we willbe able to solve problem (UBP) efficiently as fol-lows. First, note that any optimal solution to problem(UBP) must satisfy the budget constraint as equality.

Table 2 Summary of Relative Performance of Uniform Copayments:Price-Taking Firms

Q U/Q UB n = 2 n = 3 n = 10 n = 20



Second, note that the function TC4Q5 defined below isincreasing in Q.

TC4Q5 ≡ minq

n∑

i=1

h4qi1Q5qi

=

n∑

i=1

4ciqi + giqmi+1i +m0bQ

m0−1q2i 5

s.t.n∑

i=1

qi =Q1qi ≥ 0 for each i0

Finally, note that the total revenue in the marketP4Q5Q = aQ− bQm0+1 is concave for any m0 > 0. Let usdenote the total revenue-maximizing market outputby QM. From the first observation it follows that theoptimal objective value of problem (UBP), denoted byQUB, must satisfy P4QUB5QUB = TC4QUB5−B. Hence,if B ≥ TC4QM5−P4QM5QM then the functions P4Q5Qand TC4Q5−B have a unique intersection. Therefore,QUB can be computed efficiently using binary search,where a convex optimization problem must be solvedto evaluate TC4Q5 at each iteration.

On the other hand, we need to be able to computethe market consumption induced by uniform subsi-dies QU. However, from h4qi1Q5 nonseparable andasymmetric, it follows that QU cannot be computedby solving a convex optimization problem; see, e.g.,Correa and Stier-Moses (2011). Nonetheless, QU canbe computed by solving an asymmetric variationalinequality. Specifically, from Corollary 1 in Aghassiet al. (2006), it follows that 4qU1QU5 is the marketequilibrium induced by uniform subsidies if and onlyif there exists �U such that the solution 4qU1QU1�U5 isfeasible for problem (GUCAP) below, and it attains anobjective value of zero.

(GUCAP) minq1Q1�

n∑

i=1

hi4qi1Q5qi −n∑

i=1

P ′4Q5q2i

+

(

P4Q5+B

Q

)

4Q−Q5−�Q

s.t.n∑

i=1

qi =Q

qi ≥ 01 for each i3

�≤ hi4qi1Q51 for each i3

�≤ P4Q5+B

Q0

In fact, this implies that this solution is optimal forproblem (GUCAP) because the objective is nonnegativefor any feasible solution; see Aghassi et al. (2006). In oursetting, problem (GUCAP) is nonconvex. However, if anonlinear solver finds a feasible solution 4qU1QU1�U5with objective value equal to zero, then it follows thatQU is the market consumption induced by uniform

Figure 3 Relative Performance of Uniform Copayments: CournotNonlinear Demand

0.85

0.90

0.95

1.00

n = 2 n = 3 n = 10 n = 20

U/ UB

subsidies. We use LOQO to solve the smooth nonconvexproblem (GUCAP) (see Vanderbei (2006)), and in ourexperiments the solver finds the optimal solution in94% of the instances considered.

We consider the same experimental setup as in §§7.1and 7.2. The results for the ratio QU/QUB are displayedin Figure 3 and in Table 3. The minimum value of theratio QU/QUB never went below 83% in the simulationresults. Moreover, the mean and median values areabove 97% for each value of the number of firmsparticipating in the market n. These are relatively betterresults than the ones we obtained for price-taking firmsin the same setting, in §7.2. This suggests that, in mostcases, the market consumption induced by uniformcopayments is fairly close to the market consumptioninduced by the optimal copayment allocation, evenfor the setting of Cournot competition with nonlineardemand and nonlinear marginal costs considered inthis section.

8. Extensions and LimitationsIn this section, we show that there are instances ofproblem (CAP) where the market consumption inducedby uniform subsidies can be significantly lower thanthe one induced by optimal subsidies. Additionally, we

Table 3 Summary of Relative Performance of Uniform Copayments:Cournot Nonlinear Demand

Q U/Q OPT n = 2 n = 3 n = 10 n = 20



explore further extensions to our basic model, describedin §3. We conclude that uniform subsidies can be agood policy even if the firms have economies of scale.However, this does not extend to the case where eachfirm has a fixed cost of entry to the market, suggestingthe need of an alternative policy in this case.

8.1. Relatively Low Performance ofUniform Subsidies

Proposition 6 in the online appendix shows that thereare instances of problem (CAP) such that the perfor-mance of uniform subsidies can be as bad as inducingonly half of the optimal market consumption. Thedetails of these carefully constructed instances aredescribed in the proof in the online appendix. Theyconsist of only one efficient firm, and 4n− 15 homoge-neous inefficient firms, facing a linear inverse demandfunction. The budget is exactly balanced such thatthe uniform subsidies policy subsidizes the efficientfirm only (the inefficient firms are on the boundary ofjoining the market equilibrium), although it is optimalto subsidize the inefficient firms only. The structure ofthis type of instances is very particular, which partiallyexplains the good performance of uniform subsidiesin the numerical experiments in §7. Furthermore, theinstances where uniform subsidies have a low perfor-mance in §7 share a similar structure. Surprisingly, thissuggests that when all the firms are very different fromeach other, as opposed to having two clusters of firms,then the relative performance of uniform subsidies canbe expected to be better.

The relative performance of one-half can be approxi-mated arbitrarily closely as the number of firms in themarket n increases and as the inverse demand functionapproaches a constant. Moreover, we conjecture thatthis corresponds to the worst performance of uniformsubsidies for a large family of instances, includingarbitrary decreasing inverse demand functions and anyaffine increasing marginal cost functions.

8.2. Economies of ScaleWe consider the case where the firms have economiesof scale in their production. We model it by assumingdecreasing marginal costs. Note that then the unique-ness of the market equilibrium induced by any subsidyallocation is no longer guaranteed. Nonetheless, we canshow the following result in the spirit of Theorem 1.

Proposition 5. Assume that the marginal cost func-tions hi4qi5 are nonnegative, decreasing, and continuouslydifferentiable in 601 Q5, and that 4hi4qi5qi5

′ are monotonicin 601 Q5 as well. Assume that the inverse demand func-tion P4Q5 is nonnegative, decreasing, and differentiable in601 Q7. If the marginal cost functions satisfy Property 1,then uniform copayments are optimal for problem (CAP).

Proof. From the assumption that hi4qi5 are decreas-ing and continuously differentiable, 4hi4qi5qi5

′ are mono-tonic, and Property 1, it follows that hi405= hj405 foreach i1 j ∈ 811 0 0 0 1 n9. Specifically, assume that 4hj4qj5qj5

′

is decreasing and hi405 < hj405 for some i1 j ∈ 811 0 0 0 1 n9.Then there exists some qj > 0 such that hi405= hj4qj5+h′j4qj5qj > hj4qj5. Hence, for qi = 0 we have hi4qi5 > hj4qj5

and 4hi4qi5qi5′ = 4hj4qj5qj5

′, a contradiction of Property 1.The case where 4hj4qj5qj5

′ is increasing is analogous.The rest of the proof is exactly as in Theorem 1, except

that from the previous discussion there are no firmsthat do not participate in the market equilibrium. �

From Proposition 5 we conclude that the largestpossible market consumption can only be induced byuniform subsidies in this model, for any decreasinginverse demand function, and for any set of marginalcost functions that satisfy Property 1 and such that4hi4qi5qi5

′ are monotonic. The analogous examples tothe ones discussed in §4 are hi4qi5= c−giq

ui , hi4qi5=

c−egiqi −1, hi4qi5= c− ln4giqi +15, for any c > 0, and anypolynomial with negative coefficients. More generally,any positive and decreasing function hi4qi5, that canbe written as hi4qi5= h4giqi5 for some function h4x5such that h4x5x is concave, will satisfy the conditionsin Proposition 5.

However, as opposed to the model in §3, notethat these uniform subsidies may also induce alter-native market equilibria, potentially having marketconsumption levels that are smaller than the ones thatcan be induced by nonuniform subsidy allocations.Consider the following example with n = 2 homo-geneous firms such that hi4qi5 = 5 − qi for i ∈ 81129,P4Q5 = 9 − 2Q, and B = 1. Then, the optimal uni-form subsidy is yU = 00236, and the largest possiblemarket consumption is attained when only one firmparticipates in the market equilibrium and the budgetis exhausted; e.g., QU = qU

1 = 40236, qU2 = 0. However,

the same uniform subsidy also supports an alterna-tive market equilibrium where both firms participate,i.e., qU′

1 = qU′

2 = 10412 and QU′

= 20824, where the totalamount spend on subsidies is yUQU′

= 2/3 < 1. The lat-ter market consumption can be increased by allocatinga larger subsidy to any of the two firms, showing thatuniform subsidies may lead to a market consumptionthat is dominated by nonuniform subsidy allocations.This characteristic only gets worse when consideringmarkets with a larger number of heterogeneous firms.In summary, if each subsidy allocation is evaluated bythe largest market consumption it can induce, thenProposition 5 implies that uniform subsidies are thebest possible policy.

8.3. Fixed Cost of EntryWe additionally consider the case where each firmi ∈ 811 0 0 0 1n9 must pay a fixed cost Ki ≥ 0 to enter themarket; thus we include an entry game stage after the


central planner (the leader) has allocated the subsidies,and before the market competition between the firms.This type of entry game is usually analyzed under asymmetric equilibrium for homogeneous firms (see,e.g., Deo and Corbett (2009)), and, not surprisingly,uniform subsidies are optimal in that case. However,for heterogeneous firms, uniform subsidies may besuboptimal.

Consider the following example: let n = 2 andh14q15 = q1, K1 = 0, h24q25 = 0035q2, K2 = 603 (i.e., onerelatively inefficient firm with no fixed cost of entryand one efficient firm with a positive fixed cost ofentry). Let P4Q5= 10 −Q, and B = 1. From Theorem 1,if we ignore the fixed cost of entry, then the optimalsubsidy allocation is uniform. Both firms should par-ticipate in the induced market equilibrium; however,this solution is unattainable because it leads to theprofits of firm 2 being ç2 = 60207 < 603 = K2; hence,firm 2 does not enter the market. The best attainablemarket equilibrium induced by uniform subsidies isyU = 001962, where only firm 1 participates in themarket equilibrium and the budget is exhausted, lead-ing to QU = qU

1 = 50098, qU2 = 0. (If firm 2 enters the

market with this uniform subsidy, its profits at themarket equilibrium are ç2 = 60295 < 603 =K2.) How-ever, the subsidies y1 = 00079 and y2 = 0014 supporta market equilibrium where both firms participateand the budget is exhausted, i.e., q1 = 2004, q2 = 6, andQ= 8004 >QU, where the firms’ profits are ç1 = 2008and ç2 = 603. Moreover, the relative performance ofuniform subsidies in this instance is only 63.4%, incontrast to the good results in the extensive numericalexamples from §7. Hence, if a fixed cost of entry isadded to the model, then uniform subsidies are notoptimal even in the simple setting with linear demandand linear marginal costs.

9. ConclusionsWe provide a new modeling framework to analyzethe problem of a central planner injecting a budget ofsubsidies into a competitive market, with the objectiveof maximizing the market consumption of a good.The copayment allocation policy that is usually imple-mented in practice is uniform, in the sense that everyfirm gets the same copayment. A central question inthis paper is how efficient uniform copayments arecompared to the optimal subsidy allocation, assumingthat some firms could be significantly more efficientthan others.

Using our framework, we show that uniform copay-ments are in fact optimal for a large family of marginalcost functions. Moreover, we show that the optimalityof uniform copayments is preserved, under less generalconditions, in the case where the central planner isuncertain about the market state. Furthermore, weshow that uniform copayments also maximize the social

welfare for a large family of marginal cost functions.Additionally, we study the performance of uniformcopayments in relevant settings where they are notoptimal. Our simulation results suggest that the mar-ket consumption induced by uniform copayments isrelatively close to the market consumption induced bythe optimal copayment allocation. It is an interestingresearch question to explore whether there exist theo-retical bounds on the effectiveness of uniform subsidiesin these settings.

On the other hand, we also identify settings wherethe performance of uniform subsidies is not as good.In particular, if the firms face a fixed cost of entry tothe market, then uniform subsidies are not optimaleven if the firms have linear marginal costs and theyface a linear demand. This suggests the need for analternative policy in this setup, an interesting directionfor future research.

In summary, we present evidence that gives theoreti-cal support to the use of uniform copayments undersome conditions. Specifically, if the central planner’sintervention is concentrated in a short period of time,when the entry of new firms into the market that needto build their whole infrastructure is less relevant, thenit is very likely that the very simple uniform subsidypolicy will attain most of the potential benefits of moresophisticated allocation policies.

Supplemental MaterialSupplemental material to this paper is available at http://dx.doi.org/10.1287/mnsc.2015.2329.

AcknowledgmentsThe authors thank Yossi Aviv (the department editor handlingthis paper) as well as the anonymous associate editor andreviewers for their comments, which helped to substantiallyimprove both the content and exposition of this work. Theresearch of the first author is partially supported by theNational Science Foundation [Grant CMMI-0846554 (CAREERAward)] and by the Air Force Office of Scientific Research[Award FA9550-11-1-0150]. The research of the second authorwas partially supported by the National Science Foundation[Grants CMMI-1162034, CMMI-0758061, and CMMI-0824674]and the MIT Energy Initiative Seed Fund.

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