The Impact of Demand Uncertainty on Consumer Subsidies...

MANAGEMENT SCIENCEVol. 62, No. 5, May 2016, pp. 1235–1258ISSN 0025-1909 (print) � ISSN 1526-5501 (online) http://dx.doi.org/10.1287/mnsc.2015.2173

© 2016 INFORMS

The Impact of Demand Uncertainty onConsumer Subsidies for Green Technology Adoption

Maxime C. CohenOperations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139,

[email protected]

Ruben LobelThe Wharton School, University of Pennsylvania, Philadelphia, Pennsylvania 19104,

[email protected]

Georgia PerakisSloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139,

[email protected]

This paper studies government subsidies for green technology adoption while considering the manufacturingindustry’s response. Government subsidies offered directly to consumers impact the supplier’s production

and pricing decisions. Our analysis expands the current understanding of the price-setting newsvendor model,incorporating the external influence from the government, who is now an additional player in the system. Wequantify how demand uncertainty impacts the various players (government, industry, and consumers) whendesigning policies. We further show that, for convex demand functions, an increase in demand uncertainty leadsto higher production quantities and lower prices, resulting in lower profits for the supplier. With this in mind,one could expect consumer surplus to increase with uncertainty. In fact, we show that this is not always thecase and that the uncertainty impact on consumer surplus depends on the trade-off between lower prices andthe possibility of underserving customers with high valuations. We also show that when policy makers such asgovernments ignore demand uncertainty when designing consumer subsidies, they can significantly miss thedesired adoption target level. From a coordination perspective, we demonstrate that the decentralized decisionsare also optimal for a central planner managing jointly the supplier and the government. As a result, subsidiesprovide a coordination mechanism.

Keywords : government subsidies; green technology adoption; newsvendor; cost of uncertainty; supply chaincoordination

History : Received January 3, 2014; accepted November 24, 2014, by Yossi Aviv, operations management.Published online in Articles in Advance September 14, 2015.

1. IntroductionRecent developments in green technologies have cap-tured the interest of the public and private sectors. Forexample, electric vehicles (EVs) historically predategasoline vehicles but have only received significantinterest in the last decade (see Eberle and Von Helmot2010 for an overview). In the height of the economicrecession, the U.S. government passed the Ameri-can Recovery and Reinvestment Act (ARRA) of 2009,which granted a tax credit to consumers who pur-chased electric vehicles. Besides boosting the U.S.economy, this particular tax incentive was aimed atfostering further research and scale economies in thenascent electric vehicle industry. In December 2010,the all-electric car, Nissan Leaf, and General Motors’plug-in hybrid, Chevy Volt, were both introduced inthe U.S. market. After a slow first year, sales startedto pick up, and most major car companies are now in

the process of launching their own versions of electricvehicles.

More recently, in 2012 Honda introduced the FitEV model and observed low customer demand. Afteroffering sizable leasing discounts, Honda quickly soldout in Southern California (Hirsch and Thevenot2013). It is not uncommon to read about waiting listsfor Tesla’s new Model S or the Fiat 500e while otherEVs are sitting unwanted in dealer parking lots. Bothstories of supply shortages or oversupply have beencommonly attributed to electric vehicle sales. At theroot of both these problems is demand uncertainty.Sallee (2011) studied the supply shortages and cus-tomer waiting lists shortly after Toyota launched thehybrid electric Prius in 2002. When launching a newproduct, it is hard to know how many units customerswill request. In addition, finding the correct pricepoint is also not a trivial task, especially with the pres-ence of a government subsidy. In fact, understanding

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Cohen, Lobel, and Perakis: Impact of Demand Uncertainty on Consumer Subsidies1236 Management Science 62(5), pp. 1235–1258, © 2016 INFORMS

demand uncertainty should be a first-order consider-ation for manufacturers and policy makers alike.

For the most part, the subsidy design literature ingreen technologies has not studied demand uncer-tainty (see, for example, Benthem et al. 2008, Atasuet al. 2009, Alizamir et al. 2013, and Lobel and Perakis2013). In practice, demand uncertainty has also oftenbeen not considered. As suggested in private com-munication with several sponsors of the MIT EnergyInitiative,1 policy makers often ignore demand uncer-tainty when designing consumer subsidies for greentechnology adoption (Stauffer 2013). The purpose ofthis paper is to study whether incorporating demanduncertainty in the design of subsidy programs forgreen technologies is important. In particular, weexamine how governments should set subsidies whenconsidering the manufacturing industry’s responseunder demand uncertainty. We show that demanduncertainty plays a significant role in the system’s wel-fare distribution and should not be overlooked.

Consider the following two examples of green tech-nologies: electric vehicles and solar panels. By theend of 2013, more than 10 GW of solar photovoltaic(PV) panels had been installed in the United States,producing an annual amount of electricity roughlyequivalent to two Hoover Dams. Although still anexpensive generation technology, this large level ofinstallation was only accomplished via the supportof local and federal subsidy programs, such as theSunShot Initiative. In 2011, U.S. Secretary of EnergySteven Chu announced that the goal of the Sun-Shot Initiative by 2020 is to reduce the total cost ofPV systems by 75%, or an equivalent of $1 a watt(U.S. Department of Energy 2012), at which pointsolar technology will be competitive with traditionalsources of electricity generation. Even before this fed-eral initiative, many states had been actively promot-ing solar technology with consumer subsidies in theform of tax rebates or renewable energy credits.

Similarly, federal subsidies were also introduced tostimulate the adoption of electric vehicles through theARRA. As we previously mentioned, General Motors(GM) and Nissan have recently introduced afford-able electric vehicles in the U.S. market. GM’s ChevyVolt was awarded the most fuel-efficient compact carwith a gasoline engine sold in the United States, asrated by the U.S. Environmental Protection Agency(2012). However, the price tag of the Chevy Volt isstill considered high for its category. The cumulativesales of the Chevy Volt in the United States sinceit was launched in December 2010 until September2013 amount to 48,218. It is likely that the $7,500government subsidy offered to each buyer through

1 See http://mitei.mit.edu/about/external-advisory-board (acces-sed January 3, 2014).

federal tax credit played a significant role in the salesvolume. The manufacturer’s suggested retail price(MSRP) of GM’s Chevy Volt in September 2013 was$39,145, but the consumer was eligible for $7,500 taxrebates so that the effective price reduced to $31,645.The amount of consumer subsidies has remained con-stant since the launch in December 2010 until the endof 2013. This seems to suggest that, to isolate theimpact of demand uncertainty without complicatingthe model, it is reasonable to consider a single periodsetting.

In this paper, we address the following ques-tions: How should governments design green sub-sidies when facing an uncertain consumer market?How does the uncertain demand and subsidy pol-icy decision affect the supplier’s price (MSRP) andproduction quantities? Finally, what is the resultingeffect on consumers? In practice, policy makers oftenignore demand uncertainty and consider average val-ues when designing consumer subsidies. This igno-rance may be caused by the absence (or high cost) ofreliable data, among other reasons. We are interestedin understanding how the optimal subsidy levels,prices, and production quantities, as well as consumersurplus, are affected when one explicitly considersdemand uncertainty relative to the case when demandis approximated by its deterministic average value.

Although the government designs subsidies tostimulate the adoption of new technologies, the man-ufacturing industry responds to these policies withthe goal of maximizing its own profit. In this paper,we model the supplier as a price-setting newsvendorthat responds optimally to the government subsidy.More specifically, the supplier adjusts its productionand price depending on the level of consumer sub-sidies offered by the government to the consumer.This study also helps us to expand the price-settingnewsvendor model while accounting for the externalinfluence of the government.

In our model, the government is assumed to havea given adoption target for the technology. This ismotivated by several examples of policy targets forelectric vehicles and solar panels. For instance, in the2011 State of the Union, U.S. President Barack Obamamentioned the following goal: “With more researchand incentives, we can break our dependence on oilwith biofuels, and become the first country to havea million electric vehicles on the road by 2015” (U.S.Department of Energy 2011, p. 2). Another exampleof such an adoption target has been set for solar pan-els in the California Solar Incentive (CSI) program,which states: “The CSI program has a total budget of$2.167 billion between 2007 and 2016 and a goal toinstall approximately 1,940 MW of new solar gener-ation capacity” (CSI 2007). Hence, in our model, we

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Cohen, Lobel, and Perakis: Impact of Demand Uncertainty on Consumer SubsidiesManagement Science 62(5), pp. 1235–1258, © 2016 INFORMS 1237

optimize the subsidy level to achieve a given adop-tion target level while minimizing government expen-diture. In §3, we discuss alternative models for thegovernment (such as maximizing the total welfare) aswell as consider the subsidy program budget, emis-sion reductions, and social welfare.

We then quantify the impact of demand uncer-tainty on government expenditures, firm profit, andconsumer surplus. We further characterize who bearsthe cost of uncertainty depending on the structureof the demand model. Finally, we study the sup-ply chain coordination (i.e., when the governmentowns the supplier) and show that subsidies coordi-nate the overall system. More precisely, we show thatthe price paid by consumers, as well as the produc-tion level, coincides in both the centralized (wherethe supplier is managed/owned by the government)and the decentralized (where the supplier and gov-ernment act separately) models.

1.1. ContributionsGiven the recent growth of green technologies, sup-ported by governmental subsidy programs, this paperexplores a timely problem in supply chain man-agement. Understanding how demand uncertaintyaffects subsidy costs, as well as the economic surplusof suppliers and consumers, is an important part ofdesigning sensible subsidy programs. The main con-tributions of this paper are as follows.

• Demand uncertainty does not always benefit consu-mers2 nonlinearity plays a key role. As uncertainty in-creases, quantities produced increase, whereas theprice and the supplier’s profit decrease. In general,demand uncertainty benefits consumers in terms ofeffective price and quantities. One might hence expectthe aggregate consumer surplus to increase withuncertainty. In fact, we show that this is not alwaystrue. We observe that the effect of uncertainty onconsumer surplus depends on the demand form. Forexample, for linear demand uncertainty increases theconsumer surplus, whereas for isoelastic demand theopposite result holds. Depending on the demand pat-tern, the possibility of not serving customers withhigh valuations can outweigh the benefit of reducedprices for the customers served.

• By ignoring demand uncertainty, the governmentwill undersubsidize and miss the desired adoption target.Through the case of the newly introduced Chevy Voltby General Motors in the U.S. market, we measureby how much the government misses the adoptiontarget by ignoring demand uncertainty. We show thatwhen the supplier takes into account demand uncer-tainty information while the government considersonly average information on demand, the resultingexpected sales can be significantly below the desiredtarget adoption level.

• The cost of demand uncertainty is shared betweenthe supplier and the government. We analyze who bearsthe cost of demand uncertainty between governmentand supplier, which we show depends on the profitmargin of the product. In general, the governmentexpenditure increases with the added inventory risk.For linear demand models, the cost of demand uncer-tainty shifts from the government to the supplier asthe adoption target increases or the production costdecreases.

• Consumer subsidies are a sufficient mechanism to co-ordinate the government and the supplier. We comparethe optimal policies to the case where a central plan-ner manages jointly the supplier and the government.We determine that the price paid by the consumersand the production levels coincide for both the decen-tralized and the centralized models. In other words,consumer subsidies coordinate the supply chain interms of price and quantities.

2. Literature ReviewOur setting is related to the newsvendor problem,which has been extensively studied in the literature(see, e.g., Porteus 1990, Winston 1994, Zipkin 2000,and the references therein). An interesting extensionthat is even more related to this research is the price-setting newsvendor (see Petruzzi and Dada 1999 andYao et al. 2006). More recently, Kocabıyıkoglu andPopescu (2011) identified a new measure of demandelasticity, the elasticity of the lost sales rate, to gener-alize and complement assumptions commonly madein the price-setting newsvendor. Kaya and Özer (2012)provide a good survey of the literature on inventoryrisk sharing in a supply chain with a newsvendor-like retailer, which is closer to our framework. Nev-ertheless, our problem involves an additional player(the government) that interacts with the supplier’sdecisions and complicates the analysis and insights.Most previous works on the stochastic newsvendorproblem treat the additive and multiplicative modelsseparately (e.g., in Petruzzi and Dada 1999) or focusexclusively on one case, with often different conclu-sions regarding the price of demand uncertainty. Inour problem, however, we show that our conclusionshold for both demand models.

In the traditional newsvendor setting, the produc-tion cost is generally seen as the variable cost ofproducing an extra unit from raw material to fin-ished good. In capital-intensive industries such aselectric vehicles, the per-unit cost of capacity invest-ment in the manufacturing facility is usually muchlarger than the per-unit variable cost. For this reason,we define the production quantities of the supplier tobe a capacity investment decision, similar to Cachonand Lariviere (1999).

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Another stream of research related to our paperconsiders social welfare and government subsidies inthe area of vaccines (see, e.g., Arifoglu et al. 2012,Mamani et al. 2012, Taylor and Xiao 2014). Arifogluet al. (2012) study the impact of yield uncertainty,in a model that represents both supply and demand,on the inefficiency in the influenza vaccine supplychain. They show that the equilibrium demand can begreater than the socially optimal demand. Taylor andXiao (2014) assume a single supplier with stochas-tic demand and consider how a donor can use salesand purchase subsidies to improve the availability ofvaccines.

Among those studying the design of subsidies forgreen technologies, Carlsson and Johansson-Stenman(2003) examine the social benefits of electric vehicleadoption in Sweden and report a pessimistic outlookfor this technology in the context of net social wel-fare. Avci et al. (2014) show that adoption of elec-tric vehicles has societal and environmental benefits,as long as the electricity grid is sufficiently clean.Benthem et al. (2008) develop a model for optimizingsocial welfare with solar subsidy policies in California.These two papers assume nonstrategic industry play-ers. While considering the manufacturer’s response,Atasu et al. (2009) study the use of a take-back sub-sidy and product recycling programs. In a similar wayto the papers mentioned above, they optimize socialwelfare of the system, assuming a known environ-mental impact of the product. Our work focuses ondesigning optimal policies to achieve a given adop-tion target level, which can be used to evaluate thewelfare distribution in the system. In this paper, wealso incorporate the strategic response of the indus-try into the policy-making decision. Also consider-ing an adoption-level objective, Lobel and Perakis(2013) study the problem of optimizing subsidy poli-cies for solar panels and present an empirical study ofthe German solar market. The paper shows evidencethat the current feed-in-tariff system used in Germanymight not be efficiently using the positive networkexternalities of early adopters. Alizamir et al. (2013)also tackle the feed-in-tariff design problem, compar-ing strategies for welfare maximization and adoptiontargets. Finally, Raz and Ovchinnikov (2015) present aprice-setting newsvendor model for the case of pub-lic interest goods. The authors compare, for the caseof linear demand, different government interventionmechanisms and study under what conditions the sys-tem is coordinated in terms of welfare, prices, and sup-ply quantities. On the other hand, in this paper weinvestigate the impact of demand uncertainty on thevarious players of the system for nonlinear demandsand model explicitly the strategic response of the man-ufacturer to the subsidy policy.

Numerous papers in supply chain managementfocus on linear demand functions. Examples includeAnand et al. (2008) and Erhun et al. (2011) and thereferences therein. These papers study supply chaincontracts where the treatment mainly focuses on lin-ear inverse demand curves. In this paper, we showthat the impact of demand uncertainty on the opti-mal policies differs for some classes of nonlineardemand functions relative to linear models. In par-ticular, we observe that the effect of demand uncer-tainty depends on whether demand is convex (ratherthan linear) with respect to the price. In addition,the demand nonlinearity plays a key role on the con-sumer surplus.

As mentioned before, our paper also contributesto the literature on supply chain coordination (seeCachon 2003 for a review). The typical supply chainsetting deals with a supplier and a retailer, whoact independently to maximize individual profits.Mechanisms such as rebates (Taylor 2002) or rev-enue sharing (Cachon and Lariviere 2005) can coordi-nate the players to optimize the aggregate surplus inthe supply chain. Liu and Özer (2010) examine howwholesale price, quantity flexibility, or buybacks canincentivize information sharing when introducing anew product with uncertain demand. Lutze and Özer(2008) study how a supplier should share demanduncertainty risk with the retailer when there is a lead-time contract. In Granot and Yin (2005, 2007, 2008),the authors study different types of contracts in aStackelberg framework using a price-setting newsven-dor model. In particular, Granot and Yin (2008) ana-lyze the effect of price and order postponements in adecentralized newsvendor model with multiplicativedemand, wherein the manufacturer possibly offers abuyback rate. In our setting, the government and thesupplier are acting independently and could perhapsadversely affect one another. Instead, we show thatthe subsidy mechanism is sufficient to achieve a coor-dinated outcome. Chick et al. (2008) and Mamani et al.(2012) have looked at supply chain coordination ingovernment subsidies for vaccines. Nevertheless, aswe discussed above, the two supply chains are fairlydifferent.

Without considering demand uncertainty, there isa significant amount of empirical work in the eco-nomics literature on the effectiveness of subsidy poli-cies for hybrid and electric vehicles. For example,Diamond (2009) shows that there is a strong relation-ship between gasoline prices and hybrid adoption.Chandra et al. (2010) show that hybrid car rebates inCanada created a crowding out of other fuel-efficientvehicles in the market. Gallagher and Muehlegger(2011) argue that sales tax waivers are more effec-tive than income tax credits for hybrid cars. The in-crease in hybrid car sales from 2000 to 2006 is mostly

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explained by social preferences and increasing gaso-line prices. Aghion et al. (2012) show that the autoindustry innovates more in clean technologies whenfuel prices are higher. Jenn et al. (2013) determine thatincentives are only effective when the amount is suffi-ciently large. For plug-in electric cars, Sierzchula et al.(2014) argue that financial incentives, charging infras-tructure, and local presence of production facilitiesare strongly correlated with electric vehicle adoptionrates across different countries.

Also in economics literature, one can find a vastnumber of papers that consider welfare implicationsand regulations for a monopolist (see, e.g., Train1991). There is also a relevant stream of literature onmarket equilibrium models for new product introduc-tion (see, e.g., Huang and Sošic 2010). However, mostof these papers do not consider demand uncertainty.

The issue of how demand uncertainty creates amismatch in supply and demand has been mostlyresearched in the operations management literature;therefore, we mainly focus our literature survey inthis area. There are some exceptions in economics,such as Sallee (2011), who essentially argues that con-sumers captured most of the incentives for the Toy-ota Prius, while the firm did not appropriate anyof that surplus despite a binding production con-straint. Sallee (2011) shows that there was a shortageof vehicles manufactured to meet demand when thePrius was launched. This reinforces our motivationfor studying a newsvendor model in this context.

Also considering demand uncertainty, Fujimotoand Park (1997) show that an export subsidy (asopposed to a tax) is the equilibrium government strat-egy for a duopoly where each firm is in a differ-ent country and is uncertain about demand in theother country. In a slightly different setting, Boadwayand Wildasin (1990) argue that subsidies can be usedto protect workers from uncertain industry shocks,when there is limited labor mobility.

Some works on electricity peak-load pricing andcapacity investments address the stochastic demandcase (see Crew et al. 1995 for a review on that topic).In this context, it is usually assumed that the sup-plier knows the willingness to pay of customers andcan therefore decline the ones with the lowest valu-ations in the case of a stockout. In our application,however, one cannot impose such an assumption, andthe demand model follows a general price-dependentcurve while the customers arrive randomly and areserved according to a first-come-first-served logic.

The remainder of the paper is structured as fol-lows. In §3, we describe the model. In §4, we considerboth additive and multiplicative demand with pricing(price-setter model), analyze special cases, and finallystudy the effect of demand uncertainty on consumer

surplus. In §5, we study the supply chain coordina-tion, and in §6, we consider a different mechanismwhere the government subsidizes the manufacturer’scost. Finally, in §7, we present some computationalresults, and our conclusions are reported in §8. Theproofs of the different propositions and theorems arerelegated to the appendix.

3. ModelWe model the problem as a two-stage Stackelberggame where the government is the leader and thesupplier is the follower (see Figure 1). We assumea single time period model with a unique supplierand consider a full-information setting. The govern-ment decides the subsidy level r per product, and thesupplier follows by setting the price p and produc-tion quantities q to maximize his or her profit. Thesubsidy r is offered from the government directly tothe end consumer. We consider a general stochasticdemand function that depends on the effective pricepaid by consumers, z = p− r , and on a random vari-able �, denoted by D4z1�5. Once demand is realized,the sales level is determined by the minimum of sup-ply and demand; that is, min4q1D4z1 �55. It shouldbe noted that this single period model is particularlysuitable for policies with a short time horizon, suchas one year. For policies with longer time horizons, adynamic model with prices, quantities, and subsidieschanging over time would perhaps be more realistic.For the purpose of studying the impact of demanduncertainty, it is sufficient to look at a single periodwithout the added complexity of time dynamics.

The selling price p can be viewed as the MSRP—that is, the price the manufacturer recommends forretail. Additionally, in industries where productionlead time is long and incurs large fixed costs, we con-sider the production quantities to be equivalent tothe capacity investment built in the manufacturingfacility.

The goal of our model is to study the overall impactof demand uncertainty. To isolate this effect, we con-sider a single period monopolist model. These mod-eling assumptions are reasonable approximations forthe Chevy Volt, which we use in our numerical anal-ysis. Note that, since the introduction of electric vehi-cles, the MSRP for the Chevy Volt and the subsidylevel have remained fairly stable. Consumer subsidieswere posted before the introduction of these products

Figure 1 Order of Events: (1) Subsidies, (2) Price and Quantity, and(3) Sales

Government

Customermin(q, D (z, �))

Supplier

1 3

2

r

p, q

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and have remained unchanged ($7,500) since it waslaunched in December 2010. We assume the supplieris aware of the amount of subsidy offered to con-sumers before starting production. The supplier mod-eling choice is motivated by the fact that consumersubsidies for EVs started at a time when very fewcompetitors were present in the market and the prod-uct offerings were significantly different. The ChevyVolt is an extended-range midpriced vehicle, the Nis-san Leaf is a cheaper, all-electric alternative, and theTesla Roadster is a luxury sports car. These productsare also significantly different from traditional gaso-line engine vehicles so that they can be viewed asprice-setting firms within their own niche markets.

Given a marginal unit cost, c, and consumer sub-sidy level, r , announced by the government, the sup-plier faces the following profit maximization problem:

ç= maxq1p

{

p · Ɛ6min4q1D4z1 �557− c · q}

0 (1)

Denote as ç the optimal expected profit of the sup-plier. Note that the marginal cost c may incorporatethe manufacturing cost (such as material and labor)as well as the cost of building an additional unit ofmanufacturing capacity. Depending on the applica-tion setting, the cost of building capacity can be moresignificant than the per-unit manufacturing cost. Ifthere is no demand information gained between thebuilding of capacity and the production stage, thencapacity is built according to the planned production.Therefore, we can assume both of these costs to becombined in c. Furthermore, we can extend the modelto incorporate a salvage value v for each unsold unit,such that v < c, or an underage cost u as a penalty forunmet demand. These extensions do not qualitativelyaffect our results. They simply shift the newsvendorproduction quantile. To keep the exposition simple,we assume that salvage value and underage cost areboth zero: v = u= 0.

We consider the general case for which the sup-plier decides on both the price (MSRP) and the pro-duction quantities (i.e., the supplier is a price setter).An alternative case of interest is the one for whichthe price is exogenously given (i.e., the supplier is aprice taker and decides only production quantity). Asmentioned before, we consider the early stages of theEV market as a good application of the monopolistprice-setting model. It should be noted that a simi-lar analysis can be done for the simpler price-takermodel, which might be more appropriate in a differ-ent context.

We assume that the government is introducingconsumer subsidies r in order to stimulate sales toreach a given adoption target. We denote by â thetarget adoption level, which is assumed to be com-mon knowledge. Conditional on achieving this tar-get in expectation, the government wants to minimize

the total cost of the subsidy program. Define Exp asthe minimal expected subsidy expenditure, which isdefined through the following optimization problem:

Exp = minr

{

r · Ɛ6min4q1D4z1 �557}

s.t. Ɛ6min4q1D4z1 �557≥ â1

r ≥ 00 (2)

In what follows, we discuss the modeling choices forthe government in more detail.

3.1. Government’s ConstraintsThe adoption-level constraint used in (2) is motivatedby real policy-making practice. For example, Presi-dent Obama stated the adoption target of one mil-lion electric vehicles by 2015 (see U.S. Departmentof Energy 2011). More precisely, the government isinterested in designing consumer subsidies so as toachieve the predetermined adoption target. An addi-tional possibility is to incorporate a budget constraintfor the government in addition to the adoption tar-get. In various practical settings, the government mayconsider both requirements (see, for example, CSI2007). Incorporating a budget constraint in our settingdoes not actually affect the optimal subsidy solutionof the government problem (assuming that the budgetdoes not make the problem infeasible). In addition,one can show that there exists a one-to-one correspon-dence between the target adoption level and the min-imum budget necessary to achieve this target. Hence,we will only solve the problem with a target adoptionconstraint, but the problem could be reformulated asa budget allocation problem with similar insights.

Given that actual sales are stochastic, the constraintused in our model meets the adoption target inexpectation:

Ɛ6min4q1D4z1 �557≥ â0 (3)

Our results can be extended to the case where thegovernment aims to achieve a target adoption levelwith some desired probability (chance constraint)instead of an expected value constraint. Such a model-ing choice will be more suitable when the governmentis risk averse and is given by

�(

6min4q1D4z1 �557≥ â)

≥ã1 (4)

where ã represents the level of conservatism of thegovernment. For example, when ã= 0099, the govern-ment is more conservative than when ã= 009. We notethat the insights we gain are similar for both classesof constraints (3) and (4); therefore, in the remainderof this paper, because of space limitations, we focuson the case of an expected value constraint.

Note that one can consider a constraint on green-house gas reduction instead of an adoption target.

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If the government has a desired target on emissionsreduction, it can be translated to an adoption target inEV sales, for example. In particular, one can computethe decrease in carbon emissions between a gasolinecar and an electric vehicle (see, e.g., Arar 2010). Inother words, the value of â is directly tied to a valueof a carbon emissions reduction target. The resultsin this paper remain valid with a constraint on theexpected sales amount. More generally, if we set a tar-get on any increasing function of sales, the results alsoremain the same.

3.2. Government’s ObjectiveTwo common objectives for the government are tominimize expenditures or to maximize the welfarein the system. In the former, the government aimsto minimize only its own expected expenditures,given by

Exp = r · Ɛ6min4q1D4z1 �5570 (5)

Welfare can be defined as the sum of the expectedsupplier’s profit (denoted by ç and defined in (1))and the consumer surplus (denoted by CS) net theexpected government expenditures:

W =ç+ CS − Exp0 (6)

The consumer surplus is formally defined in §4.3 andaims to capture the consumer satisfaction. Interest-ingly, one can show that, under some mild assump-tions, both objectives are equivalent and yield thesame optimal subsidy policy for the government. Theresult is summarized in the Proposition 1.

Proposition 1. Assume that the total welfare is a con-cave and unimodal function of the subsidy r . Then, thereexists a threshold value â ∗ such that for any given valueof the target level above this threshold, i.e., â ≥ â ∗, bothproblems are equivalent.

Proof. Since the welfare function is concave andunimodal, there exists a unique optimal uncon-strained maximizer solution. If this unconstrainedsolution satisfies the adoption-level target, the con-strained problem is solved to optimality. However,if the target adoption level â is large enough, thissolution is not feasible with respect to the adop-tion constraint. Since the expected sales increase withrespect to r , one can see that the optimal solutionof the constrained welfare maximization problem isobtained when the adoption-level constraint is exactlymet. Otherwise, by considering a larger subsidy level,one still satisfies the adoption constraint but does notincrease the welfare. Consequently, both problems areequivalent and yield the same optimal solution forwhich the adoption constraint is exactly met. �

In conclusion, if the value of the target level â issufficiently large, both problems (minimizing expen-ditures in (5) and maximizing welfare in (6)) are

equivalent. Note that the concavity and unimodalityassumptions are satisfied for various demand modelsincluding the linear demand function. In particular,for linear demand, the threshold â ∗ can be character-ized in closed form and is equal to twice the optimalproduction with zero subsidy and therefore satisfiedin most reasonable settings. Furthermore, for smalleradoption target levels, Cohen et al. (2015b) show that,even for multiple products in a competitive environ-ment, the gaps between both settings (minimizingexpenditures versus maximizing welfare) are small(if not zero) so that both problems yield solutionsthat are close to one another. For the remainder ofthe paper, we assume the government objective is tominimize expenditures while satisfying an expectedadoption target, as in (2). This modeling choice wasfurther motivated by private communications withsponsors of the MIT Energy Initiative.

Besides minimizing the subsidy cost, another objec-tive for the government often is maximizing thepositive environmental externalities of the greentechnology product. Assume there is a positive ben-efit, denoted by pCO2

, for each ton of CO2 emissionavoided by each unit sold of the green product. Byintroducing a monetary value to emissions, one canconsider a combined government objective of mini-mizing the subsidy program cost, minus the benefitof emission reduction; i.e.,

r · Ɛ6min4q1D4z1 �557− pCO2· Ɛ6min4q1D4z1 �5570 (7)

As in Proposition 1, we can show that if there is anadoption target larger than a certain threshold â , thenthe objectives in (5) and (7) are equivalent and yieldthe same outcomes. In particular, if â ≥ â , the optimalsubsidy policy will be defined by the tightness of theadoption target constraint. Alternatively, if the sub-sidy level r required to reach â is significantly largerthan the price of carbon, pCO2

, the optimal subsidyis defined by the target constraint. Given a certainadoption target level â , there is a threshold level pCO2

such that for any price of carbon below this level,pCO2

≤ pCO2, the optimal subsidy policy is defined by

the adoption target constraint. We next show that foran EV such as the Chevy Volt, a conservative estimatefor the price of carbon emission mitigated for each EVsold is much lower than this threshold level.

As we mentioned above, the positive externalitiesof EVs correspond to the reductions in CO2 emissionsthroughout their lifetime, converted to U.S. dollars.Following the analysis in Arar (2010), the emissionrate per unit of energy is 755 kg of CO2 per MWh.By using the calculations in Cohen et al. (2015b), anestimate for an EV gas emission reduction is approx-imately 5003 tons of CO2. To convert this number toU.S. dollars, we use the value assigned by the U.S.

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Environmental Protection Agency to a ton of CO2.The value for 2014 is $23.3 per ton of CO2)−1 so thatthe monetary positive externality of an EV is equal to$1,172. One can see that the positive externality foran EV is smaller than the consumer subsidies (equalto $7,500), and therefore, it is sufficient to minimizeexpenditures, as described in (5).

With the formal definitions of the optimizationproblems faced by the supplier (1) and the govern-ment (2), in the next section we analyze the optimaldecisions of each party and the impact of demanduncertainty.

4. The ModelFor products such as electric vehicles, for which thereare only a few suppliers in the market, it is reason-able to assume that the selling price (MSRP) of theproduct is an endogenous decision of the firm. Inother words, p is a decision variable chosen by thesupplier in addition to the production quantity q. Inthis case, the supplier’s optimization problem can beviewed as a price-setting newsvendor problem (see,e.g., Petruzzi and Dada 1999). Note, however, that inour problem the solution also depends on the gov-ernment subsidy. In particular, both q and p are deci-sion variables that should be optimally chosen by thesupplier for each value of the subsidy r set by thegovernment. To keep the analysis simple and to beconsistent with the literature, we consider separatelythe cases of a stochastic demand with additive or mul-tiplicative uncertainty. In each case, we first considergeneral demand functions and then specialize to lin-ear and isoelastic demand models that are common inthe literature. Finally, we compare our results to thecase where demand is approximated by a determinis-tic average value and draw conclusions about the costof ignoring demand uncertainty.

In practice, companies very often ignore demanduncertainty and consider average values when takingdecisions such as price and production quantities. Asa result, we are interested in understanding how theoptimal subsidy levels, prices, and production quanti-ties are affected when we explicitly consider demanduncertainty relative to the case when demand is justapproximated by its deterministic average value. Forexample, the comparison may be useful in quantify-ing the value of investing some large efforts in devel-oping better demand forecasts.

We next present the analysis for both additive andmultiplicative demand uncertainty.

4.1. Additive NoiseDefine additive demand uncertainty as follows:

D4z1�5= y4z5+ �0 (8)

Here, y4z5 = Ɛ6D4z1 �57 is a function of the effectiveprice z = p − r and represents the nominal determin-istic part of demand and � is a random variable withcumulative distribution function (CDF) F�.

Assumption 1. We impose the following conditions ondemand functions with additive noise:

• Demand depends only on the difference between p andr , denoted by z.

• The deterministic part of the demand function y4z5 ispositive, twice differentiable, and a decreasing function ofz and hence invertible.

• When p = c and r = 0, the target level cannot beachieved; i.e., y4c5 < â .

• The noise � is a random variable with zero mean:Ɛ6�7= 0.

We consider that the demand function representsthe aggregate demand for all the consumers in themarket during the entire horizon. As a result, theassumption that the target level cannot be achievedif the product is sold at cost and there are no sub-sidies translates to the fact that the total number ofconsumers will not reach the desired adoption targetlevel without government subsidies. Under Assump-tion 1, we characterize the solution of problems (1)and (2) sequentially. First, we solve the optimal quan-tity q∗4p1 r5 and price p∗4r5 offered by the supplier as afunction of the subsidy r . By substituting the optimalsolutions of the supplier problem, we can solve thegovernment problem defined in (2). Note that prob-lem (2) is not necessarily convex, even for very simpleinstances, because the government needs to accountfor the supplier’s best response p∗4r5 and q∗4p1 r5.Nevertheless, one can still solve this using the tight-ness of the target adoption constraint. Because of thenonconvexity of the problem, the tightness of theconstraint cannot be trivially assumed. We formallyprove the constraint is tight at optimality in Theo-rem 1. Using this result, we obtain the optimal sub-sidy of the stochastic problem (2), denoted by rsto. Theresulting optimal decisions of price and quantity aredenoted by psto = p∗4rsto5 and qsto = q∗4psto1 rsto5. Fromproblems (1) and (2), the optimal profit of the sup-plier is denoted by çsto and government expendituresby Expsto.

We consider problems (1) and (2), where demand isequal to its expected value; that is, Ɛ6D4z1 �57 = y4z5.We denote this deterministic case with the subscript“det,” with optimal values: rdet1 pdet1 qdet1 zdet1 çdet1Expdet. We next compare these metrics in the deter-ministic versus stochastic case.

Theorem 1. Assume that the following condition issatisfied:

2y′4z5+4p−c5·y′′4z5+c2

p3·

1f�4F

−1� 44p−c5/p55

<00 (9)

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The following holds:1. The optimal price of problem (1) as a function of r is

the solution of the following nonlinear equation:

y4p− r5 + Ɛ

[

min(

F −1�

(

p− c

p

)

1 �

)]

+ y′4p− r5 · 4p− c5= 00 (10)

In addition, using the solution from (10), one can computethe optimal production quantity:

q∗4p1 r5= y4p− r5+ F −1�

(

p− c

p

)

0 (11)

2. The optimal solution of the government problem isobtained when the target adoption level is exactly met.

3. The optimal expressions follow the followingrelations:

zsto = y−14â −K4psto55≤ zdet = y−14â51

qsto = â + F −1�

(

psto − c

psto

)

−K4psto5≥ qdet = â1

where K4psto5 is defined as

K4psto5= Ɛ

[

min(

F −1�

(

psto − c

psto

)

1 �

)]

0 (12)

If, in addition, the function y4z5 is convex,

psto = c+â

�y′4zsto5�≤ pdet = c+

â

�y′4zdet5�1

çsto =â 2

�y′4zsto5�− c · 4qsto − â5≤çdet =

â 2

�y′4zdet5�0

Remark 1. Note that, for a general function y4p−r5,one cannot derive a closed-form solution of (10)for p∗4r5. Consequently, one cannot find a closed-formexpression for the optimal price psto for a generaladditive demand. This is consistent with the fact thatthere does not exist a closed-form solution for theprice-setting newsvendor. However, one can use (10)to characterize the optimal solution and even numer-ically compute the optimal price by using a binarysearch method (see more details in the appendix).

Similarly, one cannot generally express K4psto5 inclosed form. Instead, K4psto5 represents a measure ofthe magnitude of the noise that depends on the pricepsto and the noise distribution. The measure K4psto5 ismainly used to draw insights on the impact of demanduncertainty on the optimal decision variables.

Assumption (9) guarantees the uniqueness of theoptimal price as a function of the subsidies, becauseit implies the strict concavity of the profit functionwith respect to p. In case this condition does not hold,problem (1) is still numerically tractable (see Petruzziand Dada 1999). For the remainder of this paper, we

will assume that condition (9) is satisfied. For the caseof linear demand, we discuss relation (15), which is asufficient condition that is satisfied in many reason-able settings.

Note that the optimal ordering quantity in (11)is expressed as the expected demand plus the opti-mal newsvendor quantile 4p − c5/p related to thedemand uncertainty. The government can ensure thatthe expected sales achieve the desired target adop-tion level â by controlling the effective price z. Whendemand is stochastic, to achieve expected sales of â ,the government must encourage the supplier to pro-duce a higher quantity than â to compensate for thedemand scenarios where stockouts occur. The addi-tional production level is captured by K4psto5.

The optimal price p is characterized by the optimal-ity condition written in (10) that depends on the costand on the price elasticity evaluated at that optimalprice p, denoted by Ed4p5. This can be rewritten so thatthe marginal cost equals the marginal revenue; i.e.,c = p41 − 1/Ed4p55. Even without knowing a closed-form expression for the optimal price, we can stillshow that the optimal price decreases in the presenceof demand uncertainty and so does the firm profit.

Remark 2. The results of Theorem 1 can be gener-alized to describe how the optimal variables (i.e., z, q,p, and ç) change as demand uncertainty increases.Instead of comparing the stochastic case to the deter-ministic case (i.e., where there is no demand uncer-tainty), one can instead consider how the optimalvariables vary in terms of the magnitude of the noise(for more details, see the proof of Theorem 1 in theappendix). In particular, the quantity that captures theeffect of demand uncertainty is K4psto5.

Since the noise � has zero mean, the quantity K4psto5in (12) is always nonpositive. In addition, when thereis no noise (i.e., � = 0 with probability 1), K4psto5 = 0and the deterministic scenario is obtained as a spe-cial case. For any intermediate case, K4psto5 is negativeand nonincreasing with respect to the magnitude ofthe noise. For example, if the noise � is uniformly dis-tributed, the inverse CDF can be written as a linearfunction of the standard deviation � as follows:

F −1�

(

p− c

p

)

= �√

3 ·

(

2 ·p− c

p− 1

)

0 (13)

Therefore, K4psto5 scales monotonically with the stan-dard deviation for uniform demand uncertainty. Inother words, all the comparisons of the optimal vari-ables (effective price, production quantities, etc.) aremonotonic functions of the standard deviation of thenoise. This result is true for a large class of commondistributions that can be parameterized by the stan-dard deviation, such as uniform, normal, and expo-nential. As a result, one can extend our insights in a

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continuous fashion with respect to the magnitude ofthe noise. For example, the inequality of the effectiveprice is given by zsto = y−14â − K4psto55. This equa-tion is nonincreasing with respect to the magnitudeof K4psto5 and is maximized when there is no noise(deterministic demand) so that zdet = zsto = y−14â5. Ingeneral, as the magnitude of the noise increases, thegaps between the optimal decision variables increase(see plots of optimal decisions as functions of the stan-dard deviation of demand uncertainty in Figure 4).For more general demand distributions, the relation-ship with the standard deviation is not as simple.The quantile F −1

� 44p − c5/p5 may not move monoton-ically with the standard deviation, for example withnonunimodal distributions.

Remark 3. The solutions of the optimal quantity qand the effective price z provide another interestinginsight. Theorem 1 states that, when demand is uncer-tain, the consumers are better off in terms of effectiveprice and production quantities (this is true for anydecreasing demand function). Furthermore, the sell-ing price and the profit of the supplier are lower inthe presence of uncertainty, assuming that demand isconvex. These results imply that the consumers are, ingeneral, better off when demand is uncertain. Never-theless, as we will show in §4.3, this is not always thecase when we use the aggregate consumer surplus asa metric.

By focusing on a few demand functions, we canprovide additional insights. We will first consider thelinear demand case, which is the most common inthe literature. The simplicity of this demand formenables us to derive closed-form solutions and adeeper analysis of the impact of demand uncertainty.Note that the insights can be quite different for non-linear demand functions. The results presented inTheorem 1 justify the need for considering nonlinearfunctions as well. For this reason, we later considerthe isoelastic demand case and compare it to the lin-ear case.

The impact of demand uncertainty on the subsidylevel r and the overall government expenditure isharder to observe for a general demand form. Toexplore this further, we focus on the cases of linearand isoelastic demands. For both cases, we show thatthe subsidy increases with the added inventory riskcaptured by K4psto5.

4.1.1. Linear Demand. In what follows, we quan-tify the effect of demand uncertainty on the subsidylevel and the expected government expenditures. Wecan obtain such results for specific demand models,among them the linear demand model. Define the lin-ear demand function as

D4z1�5= d−� · z+ �1 (14)

where d and � are given positive parameters that rep-resent the maximal market share and the price elastic-ity, respectively. Note that, for this model, a sufficientcondition for assumption (9) to hold is given by

�>1

2c · infx f�4x50

For example, if the additive noise is uniformly distri-buted, i.e., � ∼ U6−a21 a273 a2 > 0 (note that since thenoise is uniform with zero mean, it has to be symmet-ric), we obtain

�>a2

c0 (15)

One can see that by fixing the cost c, condition (15)is satisfied if the price elasticity � is large relative tothe standard deviation of the noise. Next, we deriveclosed-form expressions for the optimal price, produc-tion quantities, subsidies, profit, and expenditures forboth deterministic and stochastic demand models andcompare the two settings.

Theorem 2. The closed-form expressions and compar-isons for the linear demand model in (14) are given by

psto = c+â

�= pdet1

qsto = â + F −1�

(

psto − c

psto

)

−K4psto5≥ qdet = â1

rsto =2â�

+ c−d

�−

1�

·K4psto5≥ rdet =2â�

+ c−d

�1

çsto =â 2

�− c · 4qsto − â5≤çdet =

â 2

�1

Expsto = â · rsto ≥ Expdet = â · rdet0

We note that the results of Theorem 2 can be pre-sented in a more general continuous fashion asexplained in Remark 2. Surprisingly, the optimal priceis the same for both the deterministic and stochas-tic models. In other words, the optimal selling priceis not affected by demand uncertainty for lineardemand. On the other hand, with increased quanti-ties, the expected profit of the supplier is lower underdemand uncertainty. At the same time, the optimalsubsidy level and expenditures increase with uncer-tainty. Therefore, both the supplier and governmentare worse off when demand is uncertain. Corollary 1and the following discussion provide further intu-ition of how this cost of demand uncertainty is sharedbetween the supplier and the government.

Corollary 1. (1) The difference in quantity, qsto −qdet,decreases in c and increases in â .

(2) The difference in subsidy, rsto − rdet, increases in cand decreases in â .

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(3) Assume that � has support 6a11 a27. Then, the opti-mal subsidy for the stochastic and deterministic demandsrelates as follows:

rdet ≤ rsto ≤ rdet +�a1�

�0

Corollary 1 can be better understood in terms of theoptimal service level for stochastic demand, denotedby � = 4psto − c5/psto. Note that � is an endogenousdecision of the supplier, which is a function of theoptimal price psto. For linear demand, the optimal ser-vice level can be simplified as � = â/4c� + â5. Thisservice level is decreasing in the cost c but increasingwith respect to the target adoption â .

On one hand, when the optimal price is signifi-cantly higher than the production cost, i.e., psto � c,the high profit margin encourages the supplier to sat-isfy a larger share of demand by increasing its pro-duction. In this case, the supplier has incentives tooverproduce and bear more of the inventory risk. Thegovernment may then set low subsidies, in fact thesame as in the deterministic case, which guaranteethat the average demand meets the target. On theother hand, when psto is close to c (low profit mar-gin), the supplier has no incentives to bear any riskand produces quantities to match the lowest possibledemand realization. In this case, the government willbear all the inventory risk by increasing the value ofthe subsidies.

Note that as production cost c increases, the re-quired subsidy is larger for both stochastic and de-terministic demands, meaning the average subsidyexpenditure is higher. At the same time, the ser-vice level � decreases, and from Corollary 1, the gapbetween rsto and rdet increases. The supplier’s costincrease amplifies the cost of demand uncertainty forthe government.

A similar reasoning can be applied to the targetadoption level. As â increases, the overall cost ofthe subsidy program increases, as expected. Interest-ingly, the service level � also increases. From Corol-lary 1, the production gap between qsto and qdetwidens while the subsidy gap between rsto and rdetshrinks. Effectively, the burden of demand uncer-tainty is transferred from the government to the sup-plier as â increases. This means that a higher targetadoption will induce the product to be more prof-itable. This will make the supplier take on more ofthe inventory risk, consequently switching who bearsthe cost of demand uncertainty.

Part (3) of Corollary 1 shows that the govern-ment subsidy decision is bounded by the worst-casedemand realization normalized by the price sensitiv-ity. In other words, it provides a guarantee on the gapbetween the subsidies for stochastic and deterministicdemands.

In conclusion, by studying the special case of alinear demand model, we obtain the following addi-tional insights: (i) The optimal price does not dependon demand uncertainty. (ii) The optimal subsidy setby the government increases with demand uncer-tainty. Consequently, the introduction of demanduncertainty decreases the effective price paid by con-sumers. In addition, the government will spend morewhen demand is uncertain. (iii) The cost of demanduncertainty is shared by the government and the sup-plier and depends on the profit margin (equivalently,service level) of the product. As expected, lower/higher margins mean the supplier takes less/moreinventory risk. Therefore, increasing the adoption tar-get or decreasing the manufacturing cost will shift thecost of demand uncertainty from the government tothe supplier.

4.2. Multiplicative NoiseIn this section, we consider a demand with a mul-tiplicative noise (see, for example, Granot and Yin2008). The nominal deterministic part is assumed tobe a function of the effective price, denoted by y4z5:

D4z1�5= y4z5 · �0 (16)

Assumption 2. We impose the following conditions ondemand functions with multiplicative noise:

• Demand depends only on the difference betweenp and r , denoted by z.

• The deterministic part of the demand function y4z5 ispositive, twice differentiable, and a decreasing function of zand hence invertible.

• When p = c and r = 0, the target level cannot beachieved; i.e., y4c5 < â .

• The noise � is a positive and finite random variablewith mean equal to 1: Ɛ6�7= 1.

One can show that the results of Theorem 1 holdfor both additive and multiplicative demand models.The proof for multiplicative noise follows a similarmethodology and is not repeated because of spacelimitations. We next consider the isoelastic demandcase to derive additional insights on the optimalsubsidy.

Isoelastic demand models are very popular in var-ious application areas. In particular, a large numberof references in economics consider such models (see,e.g., Simon and Blume 1994, Pindyck and Rubinfeld2001) as well as revenue management (see Talluri andVan Ryzin 2006). Isoelastic demand is also sometimescalled the log–log model, and its main property is thatelasticities are constant for any given combination ofprice and quantities. In addition, it does not requireone to know a finite upper limit on price. For moredetails, see, for example, Huang et al. (2013). Vari-ous papers on oligopoly competition consider isoelas-tic demand (see, e.g., Puu 1991, Lau and Lau 2003,Beard 2015). Puu (1991) studies the dynamics of two

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competing firms in a market in terms of Cournot’sduopoly theory. Lau and Lau (2003), consider (amongothers) an iso-elastic demand in a multiechelon inven-tory/pricing setting and show that the results mightdiffer depending on the demand shape. Anotherapplication that uses isoelastic demands relates tocommodity pricing (see, e.g., Deaton and Laroque1992). Finally, practitioners and researchers have usedisoelastic demand models for products in retail such asgroceries, fashion (see, e.g., Capps 1989, Pindyck andRubinfeld 2001, Andreyeva et al. 2010), and gasoline(e.g., Bentzen 1994).

4.2.1. Isoelastic Demand. Define the isoelasticdemand as

y4z5= d · z−� 4� > 150 (17)

The isoelastic model considered in the literatureusually assumes that � > 1 in order to satisfy theincreasing price elasticity property (see, e.g., Yao et al.2006). Note that the function y4z5 is convex withrespect to z for any value �> 1. Therefore, the resultsfrom Theorem 1 hold. Using this particular demandstructure, we obtain the following additional resultson the optimal subsidy.

Proposition 2. For the isoelastic demand model in (17),we have

rsto ≥ rdet0

We note that the result of Proposition 2 can bepresented in a continuous fashion, as explained inRemark 2. Note also that this allows us to recoverthe same results as the linear additive demand modelregarding the impact of demand uncertainty on thesubsidies. These two cases show that the subsidyincreases with demand uncertainty.

4.3. Consumer SurplusIn this section, we study the effect of demand uncer-tainty on consumers using consumer surplus as ametric. For that purpose, we compare the aggregatelevel of consumer surplus under stochastic and deter-ministic demand models. The consumer surplus is aneconomic measure of consumer satisfaction calculatedby analyzing the difference between what consumersare willing to pay and the market price. For a generaldeterministic price demand curve, the consumer sur-plus is denoted by CSdet and can be computed as thearea under the demand curve above the market price(see, e.g., Vives 2001):

CSdet =

∫ qdet

04D−14q5− zdet5 dq =

∫ zmax

zdet

D4z5dz0 (18)

We note that, in our case, the market price is equal tothe effective price paid by consumers z= p−r . Denoteas D−14q5 the effective price that will generate demandexactly equal to q. Note that zdet and qdet represent the

optimal effective price and production, whereas zmaxcorresponds to the value of the effective price thatyields zero demand. The consumer surplus representsthe surplus induced by consumers that are willing topay more than the posted price.

When demand is uncertain, however, defining theconsumer surplus (denoted by CSsto) is somewhatsubtler because of the possibility of a stockout. Sev-eral papers on peak-load pricing and capacity invest-ments by a power utility under stochastic demandpartially address this modeling issue (see Brown andJohnson 1969, Carlton 1986, Crew et al. 1995). Nev-ertheless, the models developed in this literature arenot applicable to the price-setting newsvendor. Morespecifically, Brown and Johnson (1969) assume thatthe utility power facility has access to the willingnessto pay of the customers so that it can decline the oneswith the lowest valuations. This assumption is not jus-tifiable in our setting where a first-come-first-servedlogic with random arrivals is more suitable. Raz andOvchinnikov (2015) study a price-setting newsvendormodel for public goods and consider the consumersurplus for linear additive stochastic demand.

For general stochastic demand functions, the con-sumer surplus CSsto4�5 is defined for each realizationof demand uncertainty �. If there is no supply con-straint, considering the effective price and the realizeddemand, the total amount of potential consumer sur-plus is defined as

CSmax4�5=

∫ zmax4�5

zsto

D4z1�5dz0

Figure 2 displays the area under the demand curves(linear and isoelastic) that defines the maximum con-sumer surplus CSmax4�5 for a given demand realiza-tion �. Note that the actual consumer surplus willbe a fraction of this maximum surplus, based on thefraction of customers that are actually served. Sincecustomers are assumed to arrive in a first-come-first-served manner, irrespective of their willingness topay, under certain demand realizations some propor-tion of these customers will not be served becauseof stockouts. The proportion of served customersunder one of these demand realizations is givenby the ratio of actual sales over potential demand:min4D4zsto1 �51 qsto5/D4zsto1 �5. Therefore, the consumersurplus can be defined as the total available surplustimes the proportion of that surplus that is actuallyserved:

CSsto4�5= CSmax4�5 ·min4D4zsto1 �51 qsto5

D4zsto1 �50 (19)

We note that, in this case, the consumer surplusis a random variable that depends on the demandthrough the noise �. Note that we are interested in

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Figure 2 Consumer Surplus for Stochastic Demand, Before Demand Rationing

�

�

�

�

�

�

�

�

�

�

comparing CSdet to the expected consumer surplusƐ�6CSsto4�57. For stochastic demand, (19) has a similarinterpretation to its deterministic counterpart. Never-theless, we also incorporate the possibility that a con-sumer who wants to buy the product does not findit available. As we will show, the effect of demanduncertainty on consumer surplus depends on thestructure of the nominal demand function. In partic-ular, we provide the results for the two special caseswe have considered in the previous section and showthat the effect is opposite. For the linear demand func-tion in (14), we have

CSdet =

∫ qdet

04D−14q5− zdet5 dq =

q2det

2�=

â 2

2�0 (20)

For the isoelastic demand from (17), we obtain

CSdet =

∫ qdet

04D−14q5− zdet5 dq

=d

�− 1·

(

d

â

)41−�5/�

4� > 150 (21)

One can then show the following results regarding theeffect of demand uncertainty on the consumer surplusfor these two demand functions.

Proposition 3. For the linear demand model in (14),we have

Ɛ6CSsto7≥ CSdet0 (22)

For the isoelastic demand model in (17) with � > 1, wehave

Ɛ6CSsto7≤ CSdet0 (23)

The consumer surplus result in Proposition 3 isperhaps one of the most counterintuitive findingsof this paper. Proposition 3 shows that, under lin-ear demand, the expected consumer surplus is largerwhen considering demand uncertainty, whereas itis lower for the isoelastic model. We already haveshown in Theorem 1 that the effective price is lowerand that the production quantities are larger when

considering demand uncertainty relative to the deter-ministic model. In addition, this result was validfor both models (i.e., additive and multiplicativenoises for linear and nonlinear demand). As a result,demand uncertainty benefits overall the consumersin terms of effective price and available quantities.With this in mind, one could expect consumer surplusto increase with uncertainty. However, when com-paring the consumer surplus using Equation (19) forstochastic demand, we obtain that, for the isoelas-tic demand, consumers are in aggregate worse offwhen demand is uncertain. On one hand, demanduncertainty benefits the consumers since it lowers theeffective price and increases the quantities. On theother hand, demand uncertainty introduces a stock-out probability because some of the consumers maynot be able to find the product available. These twofactors (effective price and stockout probability) affectthe consumer surplus in opposite ways.

For isoelastic demand, the second factor is dom-inant, and therefore the consumer surplus is lowerwhen demand is uncertain. In particular, the isoelas-tic demand admits some consumers that are willingto pay a very large price. If these consumers experi-ence a stockout, it will reduce drastically the aggre-gate consumer surplus. In fact, the gap between thestochastic and deterministic consumer surplus widenswhen K4psto5 is smaller. This happens when the profitmargin is low, meaning that there is more inventoryrisk for the supplier. For linear demand, the domi-nant factor is not the stockout probability, and conse-quently, the consumer surplus is larger when demandis uncertain. We note that this result is related to thestructure of the nominal demand rather than the noiseeffect. For example, if we were to consider a lineardemand with a multiplicative noise, we would havethe same result as for the linear demand with additivenoise.

Next, we compare and contrast our findings onproduction quantity, price, and profit as well as con-sumer surplus against what is already known in the

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literature about the classical price-setting newsvendorproblem. This way, we can investigate the impact ofincorporating the government as an additional playerin the system. In the classical price-setting newsven-dor, there does not exist a closed-form expressionfor the optimal price and production even for simpledemand forms. However, one can still compare theoutcomes between stochastic and deterministic sce-narios. We compare the results of Theorems 1 and 2to the classical price-setting newsvendor (i.e., with-out the government). The optimal price, quantity, andprofit can be found in a similar way to that in thispaper. First, one can show that the optimal price fol-lows the same relation as in our paper; i.e., psto ≤ pdet.Note that, in the classical model, p is equivalent to theeffective price paid by consumers, and therefore, sim-ilar insights apply (see Theorem 1). However, the rela-tion for the optimal quantity differs. More precisely,the inequality on quantity depends on the criticalnewsvendor quantile being larger or smaller than 1.For symmetric additive noises, if the profit margin isbelow 005 (this is usually the case for the EV indus-try), the supplier will not take the overstocking risk,and the optimal quantity decreases with respect tothe magnitude of the noise. As a result, the optimalquantity relation will be opposite tothe one we obtainin this paper, where the government is an additionalplayer in the supply chain. In addition, the resultson optimal profits agree with the case of the paper(again, assuming that the profit margin is below 005)so that the expected profits for stochastic demand arelower relative to the case where demand is determin-istic, as expected. In conclusion, the effect of demanduncertainty for the classical price-setting newsven-dor (assuming that the profit margin is below 005)states that quantity, price, and profit are all lowerwhen demand is stochastic. When comparing to Theo-rem 1, we first observe that the results do not dependon the profit margin. In addition, the optimal quan-tity follows the opposite relation, whereas the priceand profit follow the same one. Therefore, in our set-ting, the government is bearing some uncertainty risktogether with the supplier and incentivizes the sup-plier to overproduce in order to make sure that theadoption target is achieved as expected. Finally, onecan do a similar analysis for the expected consumersurplus. However, the analysis is not straightforwardand depends on the demand function, the structure ofthe noise (additive or multiplicative), and the capacityrationing rule (see Cohen et al. 2015a). Indeed, whenthe government is present in the supply chain, it canhelp by increasing the production and, consequently,by inducing larger consumer surplus in expectation.Note that the profit is still lower, because the stochas-tic scenario remains more risky for the supplier.

5. Supply Chain CoordinationIn this section, we examine how the results changein the case where the system is centrally managed. Inthis case, one can imagine that the government andthe supplier take coordinated decisions together. Thecentral planner needs to decide the price, the subsidy,and the production quantities simultaneously. Thissituation may arise when the firm is owned by thegovernment. We study the centrally managed prob-lem as a benchmark to compare to the decentralizedcase developed in the previous sections. In particular,we are interested in understanding if the decentral-ization will have an adverse impact on either partyand, more importantly, if it will hurt the consumers.We show in this section that this is not the case. Infact, the decentralized problem achieves the same out-come as the centralized problem; hence, governmentsubsidies act as a coordinating mechanism, as far asconsumers are concerned. Supply chain coordinationhas been extensively studied in the literature. In par-ticular, some of the supply chain contracting literature(see, e.g., Cachon 2003) discusses mechanisms thatcan be used to coordinate operational decisions suchas price and production quantities.

Define the central planner’s combined optimizationproblem to maximize the firm’s profits minus govern-ment expenditures as follows:

maxq1 z

{

z · Ɛ6min4q1D4z1 �557− c · q}

s0t0 Ɛ6min4q1D4z1 �557≥ â0(24)

Note that, in this case, we impose the additionalconstraint p ≥ c so that the selling price has to be atleast larger than the cost. Indeed, for the centralizedversion, it is not clear that this constraint is automat-ically satisfied by the optimal solution as it was inthe decentralized setting. Our goal is to show howthe centralized solutions for q, p, and r compare totheir decentralized counterparts from §4. We considerboth deterministic and stochastic demand models andfocus on additive uncertainty under Assumption 1.

Theorem 3. The optimal effective price z = p − r andproduction level q are the same in both the decentralizedand centralized models. Therefore, consumer subsidies area sufficient mechanism to coordinate the government andthe supplier.

Note that for problem (24), one can only solve forthe effective price and not p and r separately. In par-ticular, there are multiple optimal solutions for thecentralized case and the decentralized solution hap-pens to be one them. If the government and the sup-plier collude into a single entity, this does not affectthe consumers in terms of effective price and pro-duction quantities. Therefore, the consumers are not

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affected by the coordination. This result might be sur-prising because one could think that the coordina-tion would add additional information and powerto the central planner, as well as mitigate some ofthe competition effects between the supplier and thegovernment. However, in the original decentralizedproblem, the government acts as a quantity coordi-nator in the sense that the optimal solutions in bothcases are obtained by the tightness of the target adop-tion constraint.

6. Subsidizing theManufacturer’s Cost

In this section, we consider a different incentive mech-anism where the government offers subsidies directlyto the manufacturer (as opposed to the end con-sumers). In particular, our goals are (i) to studywhether the impact of demand uncertainty and mostof our insights are preserved if the government wereto use a cost subsidy mechanism and (ii) to comparethe outcomes of both mechanisms. Offering subsidiesdirectly to the manufacturer can be implemented bypartially sharing the cost of production or in the formof loans or free capital to the supplier. An exampleof this type of subsidy was the $249 million federalgrant provided to battery maker A123 Systems underthe ARRA to increase manufacturing of batteries forelectric and hybrid vehicles (Buchholz 2010). Thisgrant was later criticized after the company declaredbankruptcy, along with another failed subsidy pro-gram to solar panel manufacturer Solyndra (Healy2012). Knowing that this type of subsidy mechanismis also used in practice, in addition to consumer sub-sidies, we hope to do a similar analysis to observe theimpact of demand uncertainty. Below, we formalizethe model for this setting and provide the results. Wethen summarize our findings as well as compare bothsettings.

The government still seeks to encourage green tech-nology adoption. Instead of offering rebates to theend consumers, the government provides a subsidy,denoted by s ≥ 0, directly to the manufacturer. Notethat this mechanism does not have a direct impacton the demand function, which depends only on theselling price p and not explicitly on s. As before, thegovernment leads the game by solving the followingoptimization problem:

mins

8s · q4p1 s59

s.t. Ɛ6min4q1D4p1 �557≥ â1

s ≥ 00 (25)

Note that in this case, the government subsidizes thetotal produced units instead of the total expected soldunits as before.

Given a subsidy level s announced by the govern-ment, the supplier faces the following profit maxi-mization problem. Note that c denotes the cost ofbuilding an additional unit of manufacturing capacity,as before:

ç= maxq1p

{

p · Ɛ6min4q1D4p1 �557− 4c− s5 · q}

0 (26)

For simplicity, we assume a linear and additivedemand, but one can extend the results for non-linear demand models as well as for multiplicativeuncertainty. However, to keep the analysis simple, wepresent the results for the linear case, given by

D4p1�5= d−� · p+ �0 (27)

First, we study the impact of demand uncer-tainty on the decision variables for the cost subsidymechanism (denoted by CSM). Second, we comparethe outcomes for both mechanisms and elaborateon the differences. The results on the impact of de-mand uncertainty are summarized in the followingtheorem.

Theorem 4. Assume a linear demand as in (27). Thecomparisons for the cost subsidy mechanism are given by

psto =d− â

�+

1�

·K ′

� ≤ pdet =d− â

�1

qsto = â + F −1�

(

psto − c+ ssto

psto

)

−K ′

� ≥ qdet = â1

ssto =2â�

+ c−d

�−

1�

·K ′

� ≥ sdet =2â�

+ c−d

�1

çsto =â 2

�− 4c− ssto5 · 4qsto − â5≤çdet =

â 2

�1

Expsto = qsto · ssto ≥ Expdet1= â · sdet1

where K ′� = Ɛ6min4F −1

� 44psto − c + ssto5/psto51 �57. Notethat ssto is not given in a closed-form expressionbecause both sides depend on ssto, and one needsto solve a nonlinear fixed point equation. The proofof Theorem 4 is not reported here because of spacelimitations (it is based on a methodology similar toTheorem 2). One can see that the impact of demanduncertainty on all the decision variables is the sameas in the consumer rebates mechanism (see Theo-rem 2). Next, we compare the outcomes of both mech-anisms under deterministic and stochastic demands.The results are summarized below.

• The amount of subsidy (per unit) paid by thegovernment follows the following relation:

ssto ≤ rsto3 sdet = rdet0

• The price paid by the consumers follows the fol-lowing relation:

pCSMsto ≥ psto − rsto3 pCSM

det = pdet − rdet0

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• The production quantities follow the followingrelation:

qCSMsto ≥ qsto3 qCSM

det = qdet0

• For the profit of the supplier and the governmentexpenditures, one cannot find the relation analytically.Instead, we study and compare the expressions com-putationally (see the discussion below).

Note that for the cost subsidy mechanism, the pricepaid by consumers is equal to p (since there is no sub-sidy to consumers), whereas for the consumer subsidymechanism, it is captured by p − r . We observe thatthe government can save money on the per-unit sub-sidy, but that does not mean that the overall expen-ditures are lower, because more units are potentiallysubsidized. In addition, the consumers are paying alarger price to compensate for this government sav-ings per unit. As a result, the consumers are worse offin terms of price but better off in terms of availablequantities. In addition, when the demand is determin-istic, all the outcomes are the same for both mech-anisms. However, in a stochastic setting, the typeof mechanism plays a key role in the risk sharingbetween the supplier and the government induced bythe demand uncertainty.

We next vary the different model parameters tocompare the outcomes for both mechanisms computa-tionally. We obtain the following results. As the vari-ance of the noise increases, the expected governmentexpenditures and the expected profit of the supplierunder the subsidy mechanism are higher when com-pared with the consumer subsidy mechanism. Con-sequently, although the optimal subsidy (per unit) islower, the overall subsidy program is actually morecostly to the government. Indeed, the government issubsidizing all the produced units instead of the soldones and therefore is bearing some of the overstockrisk from the demand uncertainty. Since the supplieris sharing this overstock risk with the government,the supplier can achieve higher profits on expectationdespite the fact that the subsidy is smaller by charg-ing a higher price to the consumers.

7. Computational ResultsIn this section, we present some numerical exam-ples that provide further insights into the resultsderived in §4. The data used in these experiments areinspired by the sales data of the first two years ofGeneral Motors’ Chevy Volt (between December 2010and December 2012). The total aggregate sales wasroughly equal to 35,000 (see Cobb 2013) electric vehi-cles, the listed price (MSRP) was $40,280, and the gov-ernment subsidy was set to $7,500. In addition, weassume a 10% profit margin so that the per-unit costof building manufacturing capacity is $36,000. Notethat we tested the robustness of all our results and

plots in this section with respect to c by varying theprofit margin from 005% to 20% and obtained verysimilar insights. For simplicity, we present here theresults using a linear demand with an additive Gaus-sian noise. We observe that our results, along with theanalysis. are robust with respect to the distribution ofdemand uncertainty. In fact, we obtain in our com-putational experiments the same insights for severaldemand distributions (including nonsymmetric ones).As discussed in §3, the government can either mini-mize expenditures or maximize the total welfare. Inparticular, the two objectives are equivalent and giverise to the same optimal subsidy policies for any tar-get level â above a certain threshold. In this case, thisthreshold is equal to 8601 and the condition is there-fore easily satisfied.

Throughout these experiments, we compute theoptimal decisions for both the deterministic andthe stochastic demand models by using the optimalexpressions derived in §4.1. We first consider a fixedrelatively large standard deviation � = 421000 (notethat when demand is close to the sales, this is equiv-alent to a coefficient of variation of 102) and plot theoptimal subsidy, production level, supplier’s profit,and government expenditures as a function of thetarget level â for both the deterministic and stochas-tic models. The plots are reported in Figure 3. Wehave derived in §4.1 a set of inequalities regardingthe relations of the optimal variables for determin-istic and stochastic demand models. The plots allowus to quantify the magnitude of these differences andstudy the impact of demand uncertainty on the opti-mal policies. One can see from Figure 3 that the opti-mal production levels are not strongly affected bydemand uncertainty (even for large values of �) whenthe target level â is set close to the expected salesvalue of 35,000. However, the optimal value of thesubsidy is almost multiplied by a factor of 2 whendemand uncertainty is taken into account. In otherwords, when the government and the supplier con-sider a richer environment that accounts for demanduncertainty, the optimal subsidy nearly doubles.

One can see that the optimal production quanti-ties for deterministic and stochastic cases differ verylittle, whereas the subsidy and profit show signifi-cantly higher discrepancy. By looking at the closed-form expressions for linear demand in Theorem 1,one can see that the difference in optimal quantity isequal to F −1

� 44psto − c5/psto5−K4psto5, whereas the dif-ference in optimal subsidy is proportional to K4psto5.Note that, in our case, the profit margin is relativelysmall (order of 0.1), and therefore the quantile value,F −1� 44psto − c5/psto5, is likely to be negative. In partic-

ular, in our example, c = 361000 and â ranges from20,000 to 55,000. As a result, since we assume a sym-metric noise distribution, the quantile is always neg-ative. Consequently, the difference in quantities is

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Figure 3 (Color online) Optimal Values as a Function of theTarget Level

r*

2.0×104

×104 ×104

×104 ×104

×104

1.5

1.0

0.5

0

Π∗

×108

4

3

2

1

0

Exp

∗

×108

10

8

6

4

2

0

2 3 4

�

5 6 2 3 4

DeterministicStochastic

�

5 6

2 3 4

�

5 6 2 3 4

�

5 6

q*

6

5

4

3

2

clearly smaller than the difference in subsidies. Oneinteresting interpretation relies on the fact that thecost of uncertainty in production quantity is sharedbetween the supplier and the government. Indeed,the government wants to incentivize the supplier toincrease production in order to reach the adoptiontarget and therefore is willing to share some of theuncertainty risk so that qsto is not far from qdet. Finally,the profit discrepancy is larger than the quantity dis-crepancy because it is equal to the same differencescaled by the cost c (see Theorem 1). In our example,c = 361000, and therefore, we can see a more signifi-cant difference.

This raises the following interesting question. Whathappens if the government ignores demand uncer-tainty and decides to undersubsidize by using theoptimal value from the deterministic model? It is clearthat in this case, since the real demand is uncer-tain, the expected sales will not attain the desiredexpected target adoption. We address this question in

Figure 4 (Color online) Relative Normalized Differences in Subsidies (Left) and Supplier’s Profit (Right)

ΠΠ

Π

�

�

�

�

the remainder of this section. We first plot the sub-sidies and the supplier’s profit as a function of thestandard deviation of the noise that represents a mea-sure of the demand uncertainty magnitude.

More precisely, we plot in Figure 4 the relative dif-ferences in subsidies (i.e., 4rsto − rdet5/rdet) as well asthe supplier’s profit as a function of the target level â(or, equivalently, the expected sales) for different stan-dard deviations of the additive noise varying from 35to 12,500. For â = 351000, this is equivalent to a coeffi-cient of variation varying between 00001 and 00357. Asexpected, one can see from Figure 4 that as the stan-dard deviation of demand increases, the optimal sub-sidy is larger, whereas the supplier’s profit is lower.As a result, demand uncertainty benefits consumersat the expense of hurting both the government andthe supplier.

Finally, we analyze by how much the governmentwill miss the actual target level (now â is fixed andequal to 35,000) by using the optimal policy assum-ing that demand is deterministic, rdet, instead of usingrsto. Recall that rsto ≥ rdet. In other words, the govern-ment assumes a simple average deterministic demandmodel whereas in reality demand is uncertain. Inparticular, this allows us to quantify the value ofusing a more sophisticated model that takes intoaccount demand uncertainty instead of simply ignor-ing it. Note that this analysis is different from theprevious comparisons in this paper, where we com-pared the optimal decisions as a function of demanduncertainty. Here, we assume that demand is uncer-tain with some given distribution but the governmentdecides to ignore the uncertainty. To this extent, weconsider two possible cases according to the modelingassumption of the supplier. First, we assume that thesupplier is nonsophisticated, in the sense that the sup-plier uses an average demand approximation modelas well (i.e., no information on demand distribution isused). In this case, both the supplier and the govern-ment assume an average deterministic demand but

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Figure 5 (Color online) Expected Sales Relative to Target

1.0

0.9

0.8

0.7

0.6

Sale

s re

lativ

e to

targ

et

0.5

0.40 0.1 0.2 0.3

Coefficient of variation

0.4 0.5 0.6 0.7

Nonsophisticated supplierSophisticated supplier

in reality demand is random. Second, the supplier ismore sophisticated. Namely, the supplier optimizes(over both p and q) by using a stochastic demandmodel together with the distribution information. Theresults are presented in Figure 5, where we vary thevalue of the coefficient of variation of the noise from0 to 007. When the government and the supplier areboth nonsophisticated, the government can poten-tially save money (by undersubsidizing) and still getclose to the target in expectation when demand uncer-tainty is not very large. As expected, when the sup-plier has more information on demand distribution(as is usually the case), the expected sales are far-ther from the target and the government could missthe target level significantly. If, in addition, demanduncertainty is large (i.e., coefficient of variation largerthan 1), the government misses the target in bothcases.

One can formalize the previous comparison ana-lytically by quantifying the gap by which thegovernment misses the target by ignoring demanduncertainty. In particular, let us consider the additivedemand model given in (8).

Proposition 4. Consider that the government ig-nores demand uncertainty when designing consumersubsidies.

(1) The government misses the adoption target (inexpectation) regardless of whether the supplier is sophisti-cated or not.

(2) The exact gaps are given by the following:• For a nonsophisticated supplier,

Ɛ6min4q1D4z1 �557= â + Ɛ6min401 �57≤ â0 (28)

• For a sophisticated supplier, assuming a linear de-mand model,

Ɛ6min4q1D4z1 �557

= â +12

· Ɛ

[

min(

F −1�

(

p∗ − c

p∗

)

1 �

)]

≤ â0 (29)

In addition, the optimal price of the sophisticated supplier,p∗, is lower than in the deterministic case; i.e., p∗ ≤ pdet.

(3) For low(high)profitmargins, thegapis larger (smaller)when the supplier is sophisticated (nonsophisticated).

Note that all the results of Proposition 4 (with theexception of Equation (29)) are valid for a generaldemand model. Nevertheless, when the supplier issophisticated, one needs to assume a specific model(in our case, linear) in order to compute the expectedsales. As expected, the previous analysis suggests thatthe target adoption will be missed by a higher marginas demand becomes more uncertain. When compar-ing the nonsophisticated and sophisticated cases, onecan see that in the former the government misses thetarget by Ɛ6min401 �57, whereas in the latter it missesby 1

2 · Ɛ6min4F −1� 44p∗ − c5/p∗51 �57. Consequently, this

difference depends not only on the distribution ofthe noise but also on the profit margin. Since the EVindustry has rather low profit margins, the gap maybe much larger when the supplier is sophisticated.Indeed, the sophisticated supplier decreases the price(relative to pdet). In addition, the supplier, who is notwilling to bear significant overstock risk due to thelow profit margin, reduces the production quantities.As a result, the expected sales are lower, and thereforethe government misses the adoption target level. Inconclusion, this analysis suggests that policy makersshould take into account demand uncertainty whendesigning consumer subsidies. Indeed, by ignoringdemand uncertainty, one can significantly miss thedesired adoption target.

8. ConclusionsWe propose a model to analyze the interaction be-tween the government and the supplier when design-ing consumer subsidy policies. Subsidies are oftenintroduced at the early adoption stages of green tech-nologies to help them become economically viablefaster. Given the high level of uncertainty in theseearly stages, we hope to have shed some light on howdemand uncertainty affects consumer subsidy poli-cies, as well as price and production quantity deci-sions from manufacturers and the end consumers.

In practice, policy makers often ignore demanduncertainty and consider only deterministic forecastsof adoption when designing subsidies. We demon-strate that uncertainty will significantly change howthese programs should be designed. In particular, weshow by how much the government misses the adop-tion target by ignoring demand volatility. Amongsome of our main insights, we show that the shape ofthe demand curve will determine who bears demanduncertainty risk. When demand is uncertain, quanti-ties produced will be higher and the effective pricefor consumers will be lower. For convex demand

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functions, prices will be lower, leading to lower indus-try profits.

Focusing on the linear demand model, we can de-rive further insights. For instance, to compensate foruncertain demand, quantities produced and subsidylevels are shifted by a function of the service level, i.e.,the profitability of the product. For highly profitableproducts, the supplier will absorb most of the demandrisk. When profit margins are smaller, the governmentwill need to increase the subsidy amount and pay alarger share for the risk.

When evaluating the uncertainty impact on con-sumers, we must consider the trade-off between lowereffective prices and the probability of a stockout(unserved demand). We again show that the shape ofthe demand curve plays an important role. For lin-ear demand, consumers will ultimately benefit fromdemand uncertainty. This is not the case, for instance,with an isoelastic demand model, where the possibil-ity of not serving customers with high valuations willoutweigh the benefits of decreased prices.

We also compare the optimal policies to the casewhere a central planner manages jointly the sup-plier and the government and tries to optimize theentire system simultaneously. We show that the opti-mal effective price and production level coincide inboth the decentralized and centralized models. Con-sequently, the subsidy mechanism is sufficient to coor-dinate the government and the supplier, and thecollusion does not hurt consumers in terms of priceand quantities.

There are interesting directions for future research.Our model focuses on a single period, which is moreapplicable for policies with short time horizons. Incases where the policy horizon is long, it would beinteresting to understand the effect of time dynam-ics on the actions of both government and supplier.It is not obvious how the frequency of policy adjust-ment will affect the outcome of the subsidy program,when considering a strategic response of the supplier.Additionally, introducing competition between sup-pliers could also be an interesting direction to extendthe current model.

AcknowledgmentsThe authors thank the department editor (Yossi Aviv), theassociate editor, and the two anonymous referees for theirinsightful comments, which have helped to improve thispaper. The authors also thank Gérard Cachon, Steve Graves,Sang-Hyun Kim, Jonathan Kluberg, and Matthieu Monschfor their valuable feedback, which has helped to improvethis paper. This research was partially supported by MITEnergy Initiative Seed funding as well as the National Sci-ence Foundation [Grant CMMI-1162034].

Appendix. Proofs

Proof of Theorem 1. We provide proofs for each part ofTheorem 1 in turn.

(1) Equations (10) and (11) are obtained by applyingthe first-order conditions on the objective function of prob-lem (1) with respect to q and then to p.

(2) We next prove the second claim about the fact thatthe optimal solution of the government problem is obtainedwhen the target adoption level is exactly met. Using con-dition (9), one can compute the optimal value of p∗4r5 byusing a binary search algorithm (note that Equation (10) ismonotonic in p for any given value of r). In particular, forany given r , there exists a single value p∗4r5 that satisfiesthe optimal Equation (10), and since all the involved func-tions are continuous, we may also conclude that p∗4r5 is acontinuous, well-defined function. As a result, the objectivefunction of the government when using the optimal pol-icy of the supplier is also a continuous function of r . Inaddition, the target level cannot be attained when r = 0 byAssumption 1. We then conclude that the optimal solutionof the government problem is obtained when the inequalitytarget constraint is tight. In addition, one can see that theexpected adoption target equation is monotonic in r so thatone can solve it by applying a binary section method.

(3) Finally, let us show the third part. For the deter-ministic demand model, we have qdet = y4zdet5 = â . Onthe other hand, when demand is stochastic, we haveƐ6min4qsto1D4zsto1 �557 = â so that we obtain qsto ≥ qdet.In addition, the above expression yields y4zsto5 = â −

K4psto5≥ â . Therefore, we obtain y4zsto5≥ y4zdet5. Since y4z5is nonincreasing with respect to z = p − r (from Assump-tion 1), we may infer the following relation for the effec-tive price: zdet ≥ zsto. We next compute the optimal price forthe deterministic model pdet by differentiating the supplier’sobjective function with respect to p and equate it with zero(first-order condition):

¡

¡pdet6qdet · 4pdet − c57= y′4zdet5 · 4pdet − c5+ y4zdet5= 00

One can see that, in both models, we have obtained thesame optimal equation for the price: y4z5 = −y′4z5 · 4p − c5.Namely, the optimal price satisfies p = c+â/�y′4z5�. We notethat the previous expression is not a closed-form expres-sion because both sides depend on the optimal price p.This is not an issue because our goal here is to comparethe optimal quantities in the two models rather than deriv-ing the closed-form expressions. Assuming that the deter-ministic part of demand y4z5 is a convex function, weknow that y′4z5 is a nondecreasing function and then 0 >y′4zdet5≥ y′4zsto5. We then have the following inequality forthe optimal prices: psto ≤ pdet. We note that the optimal sub-sidy in both models does not follow such a clear relation,and it will actually depend on the specific demand function.We next proceed to compare the optimal supplier’s profit inboth models. For the deterministic demand model, the opti-mal profit is given by çdet = qdet · 4pdet − c5= â 2/�y′4zdet5�. Inthe stochastic model, the expression of the optimal profit isgiven by

çsto = psto · Ɛ[

min4qsto1D4zsto1 �55]

− c · qsto

=â 2

�y′4zsto5�− c · 4qsto − â5≤çdet0 �

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Proof of Theorem 2. For the linear demand model in(14), the optimal solution of the supplier’s optimizationproblem has to satisfy the following first-order condition:

d+� · 4r + c− 2p5+c

p· F −1

�

(

p− c

p

)

+p− c

p· Ɛ

[

� � � ≤ F −1�

(

p− c

p

)]

= 00

Note that it does not seem easy to obtain a closed-formsolution for p∗4r5. In addition, since the previous equationis not monotone, one cannot use a binary search method.

Instead, one can express r as a function of p2 r=2p−

c− d/�+4a·c25/4�·p25. We next proceed to solve the gov-ernment optimization problem by using the tightness ofthe inequality target adoption constraint: Ɛ6min4q∗4p∗4r51 r51D4p∗4r5− r1�557 = � · 4p∗4r5 − c5 = â . One very interestingconclusion from this analysis is that we have a very sim-ple closed-form expression for the optimal price that is thesame for the deterministic case:

psto = pdet = c+â

�0 (30)

We can at this point derive the optimal supplier’s profit forboth models. In the deterministic case, the profit of the sup-plier is given by çdet = qdet · 4pdet − c5= â 2/�. In the stochas-tic model, the optimal profit is given by

çsto = psto · Ɛ6min4qsto1D4zsto1 �57− c · qsto

=â 2

�− c · 4qsto − â5≤çdet0

We next derive the optimal production level for both mod-els. In the deterministic case, we obtained that qdet = â . Forthe stochastic case, after substituting all the correspondingexpressions, we obtain

qsto = d+� · 4rsto − psto5+ F −1�

(

psto − c

psto

)

= â + F −1�

(

psto − c

psto

)

−K4psto5≥ qdet0

We finally compare the effect of demand uncertainty on theoptimal subsidy. One can show that after some appropriatemanipulations the optimal subsidy for the deterministic lin-ear demand model is given by rdet = 2â/�− d/�+ c. For thestochastic demand model, we have the following optimalequation: � · 42psto − rsto − c5 − d = K4psto5. Hence, one canfind the expression for the optimal subsidies as a functionof the optimal price psto: rsto = 2psto −c− d/�− 41/�5 ·K4psto5.By replacing psto = c+â/� from (30), we obtain rsto = 2â/�+

c− d/�− 41/�5 ·K4psto5= rdet − 41/�5 ·K4psto5≥ rdet. �

Proof of Corollary 1. We provide proofs for each partof the corollary in turn.

(1) We provide the proof of part (1) by using the factsd�/dc ≤ 0 and d�/4câ5 ≥ 0. We then show that the produc-tion gap widens as the service level � increases. Note thatqsto − qdet = F −1

� 4�5 − K4psto5 = Ɛ6max4F −1� 4�5 − �1057. Taking

the derivative with respect to the cost c, we obtain

d4qsto − qdet5

dc=

d4F −1� 4�55

d�·d�

dc·�=

1f 4F −1

� 4�55·d�

dc·�≤ 00

Similarly, for the target level â ,

d4qsto − qdet5

dâ=

d4F −1� 4�55

d�·d�

dâ·�=

1f 4F −1

� 4�55·d�

dâ·�≥ 00

(2) To prove part (2), we show that the subsidy gapdecreases with respect to�. Note that rsto − rdet = −K4psto5/�=

−Ɛ6min4F −1� 4�51 �57/�. Taking the derivative with respect to

the cost c, we obtain

d4rsto − rdet5

dc= −

1�

[

d4F −1� 4�55

d�·d�

dc· 41 −�5

]

= −1�

·1

f 4F −1� 4�55

·d�

dc· 41 −�5≥ 00

Similarly, for the target level â ,

d4rsto − rdet5

dâ= −

1�

[

d4F −1� 4�55

d�·d�

dâ· 41 −�5

]

= −1�

·1

f 4F −1� 4�55

·d�

dâ· 41 −�5≤ 00

(3) We next present the proof of part (3). We assume that� is an additive random variable with support 6a11 a27, notnecessarily with a symmetric probability density function.For the linear demand model from (14), we have

rsto = rdet −1�

·K4psto50 (31)

First, let us prove the first inequality by showing that theterm on the right in Equation (31) is nonpositive for ageneral parameter y. We have Ɛ6min4y1 �57 = y · �4y ≤ �5 +

Ɛ6� � � < y7 · �4� < y5. Now, let us divide the analysis intotwo different cases according to the sign of y. If y ≤ 0, weobtain Ɛ6min4y1 �57 = y ·�4y ≤ �5+�4� < y5 · Ɛ6� � � < y7 ≤ 0.In the previous equation, both terms are nonpositive. Forthe case where y > 0, we have Ɛ6min4y1 �57 < Ɛ6�7= 0. There-fore, Ɛ6min4F −1

� 44p−c5/p51 �57≤ 01 showing the first inequal-ity, rdet ≤ rsto. We now show the second inequality. We knowfrom the optimality that p ≥ c. Let us evaluate the expres-sion of rsto in (31) for different values of p. If p = c, weobtain F −1

� 44p− c5/p5= F −1� 405= a1 < 0. Then, we have rsto =

rdet − 41/�5 · Ɛ6min4a11 �57 = rdet − a1/� > rdet. If p � c, weobtain F −1

� 44p−c5/p5→ F −1� 415= a2 > 0. Therefore, we obtain

rsto → rdet −1/� ·Ɛ6min4a21 �57= rdet − 41/�5 ·Ɛ6�7= rdet. Sincersto is continuous and nonincreasing in p for any p ≥ c, thesecond inequality holds. �

Proof of Proposition 2. By applying a similar method-ology to that in the proof of Theorem 1, one can derivethe following expressions (the steps are not reported forconciseness):

qdet = â1 qsto >â1

pdet = c+1�

·

(

d

â

)1/�

1

psto =F −1� 44psto − c5/psto5

K4psto5c+

1�

·

(

d

â·K4psto5

)1/�

1

rdet = c+

(

d

â

)1/�

·

(

1�

− 1)

1

rsto =F −1� 44psto − c5/psto5

K4psto5c+

(

d

â·K4psto5

)1/�

·

(

1�

− 1)

0

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Here, K4psto5 = Ɛ[

min4F −1� 44psto − c5/psto51 �57, so that the

above expressions for psto and rsto are not in closed form.Indeed, in this case, one cannot analytically derive a closed-form expression. However, we are still able to compare theoptimal subsidy between the deterministic and stochasticsettings. Since Ɛ6�7 = 1 and � ≥ 0, we have 0 ≤ K4psto5 ≤ 1.Consequently, one can see that, since �> 1, rsto ≥ rdet. �

Proof of Proposition 3. For the linear additive demandmodel presented in (14), one can compute the consumersurplus for given values of p, r , and q:

CSsto4�5 =

D4z1�52

2�if D4z1�5≤ q1

D4z1 �5 · q

2�if D4z1�5 > q1

=D4z1�5

2�· 6min4D4z1 �51 q570

Therefore, we have CSsto4�5 ≥ 6min4D4z1 �51 q572/42�5. Byapplying the expectation operator, we obtain

Ɛ6CSsto4�57 ≥Ɛ86min4D4z1 �51 q5729

2�≥

8Ɛ6min4D4z1 �51 q5792

2�

=â 2

2�= CSdet1

where the second inequality follows by Jensen’s inequal-ity (or the fact that the variance of any random variableis always nonnegative). The last equality follows from theprevious result that the target inequality constraint is tightat optimality.

We next compute the consumer surplus defined in (19) forthe isoelastic demand from (17). In particular, we observethat zmax4�5 = � for any value of � (if we assume that � isstrictly positive and finite). In addition, we have zsto = psto −

rsto = 64d/â5 · K4psto571/�. Note that K4psto5 is a determinis-

tic constant and does not depend on the realization of thenoise �. Therefore, when computing CSsto4�5 for a given �,since demand is multiplicative with respect to the noise, onecan see that � cancels out and that simplifies the calculation.We obtain

CSsto4�5 =1

�− 1

[

d

â·K4psto5

]41−�5/�

·

[

d

âK4psto5

]

min4D4zsto1 �51 qsto50

Then, by taking the expectation operator, we obtain

Ɛ6CSsto4�57 =d

�− 1·

(

d

â

)41−�5/�

·K4psto51/�

= CSdet · 6K4psto571/�0

Here, we have used the fact that the inequality adoption con-straint is tight at optimality; that is, Ɛ6min4D4zsto1 �51 qsto57= â . In addition, this is the only term that depends on thenoise �. Since we have 0 ≤K4psto5≤ 1, one can conclude that,for any �> 1, Ɛ6CSsto4�57≤ CSdet. �

Proof of Theorem 3. We present first the proof for thedeterministic demand model and then the one for stochas-tic demand. Let us first consider the unconstrained opti-mization problem faced by the central planner. If demand

is deterministic, the objective function is given by J 4p1 r5=

q4p1 r5 · 4p− r − c5. We assume that demand is a function ofthe effective price (denoted by z); that is, q4p1 r5= y4p− r5=

y4z5. Next, we compute the unconstrained optimal solu-tion denoted by z∗ by imposing the first-order condition:dJ 4z5/dz = 0 ⇒ z∗ = c − y4z∗5/y′4z∗5. Although, we did notderive a closed-form expression for z∗, we know that itshould satisfy the above fixed point equation. We now showthat any unconstrained optimal solution is infeasible forthe constrained original problem since it violates the targetinequality constraint:

q4p∗1 r∗5= y4z∗5= y

(

c−y4z∗5

y′4z∗5

)

≤ y4c5 < â0

We used the facts that demand is positive, differenti-able, and a decreasing function of the effective price (seeAssumption 1). In addition, since we assumed that the tar-get level cannot be achieved without subsidies, we haveshown that the unconstrained optimal solution is not feasi-ble. Therefore, we conclude that the target inequality con-straint has to be tight at optimality: q4pdet1 rdet5= â . In otherwords, the optimal effective price and production level arethe same as in the decentralized model.

We now proceed to present the proof for the case wheredemand is stochastic. Let us consider the constrained opti-mization problem faced by the central planner. We denoteas J the objective function (multiplied by −1) and denote as�i; i = 11213 the corresponding Karush-Kuhn-Tucker (KKT)multipliers of the three constraints. The KKT optimalityconditions are then given by

¡J

¡q−�2 ·�4q ≤D5= 01

¡J

¡p−�1 −A ·�2 = 01

¡J

¡r−�3 −B ·�2 = 00

(32)

Here, A and B are given by

A=¡

¡pƐ6min4q1D4z1 �557= y′4z5 · FD4z1�54q51

B =¡

¡rƐ6min4q1D4z1 �557= −y′4z5 · FD4z1�54q5= −A0

If, in addition, the noise is additive, we have FD4z1�54q5 =

F�4q − y4z55. We also have

¡J

¡q= c− z · 61 − FD4z1�54q571

¡J

¡p= −Ɛ6min4q1D4z1 �557− z ·A= −

¡J

¡r0

We note that the last two equations are symmetric andhence equivalent. Equivalently, the central planner decidesonly on the effective price z = p − r and not p and rseparately. We next assume that �1 = �3 = 0. This corre-sponds (from the complementary slackness conditions) withthe assumption that both corresponding constraints are nottight. Indeed, clearly the optimal subsidies may be assumedto be strictly positive since we assumed that, when r = 0, theadoption constraint cannot be satisfied. We further assumethat the supplier wants to achieve positive profits, so thatthe optimal price is strictly larger than the cost. Therefore,

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the KKT conditions (both stationarity and complementaryslackness) can be written as follows:

c− 4z+�25 · 61 − F�4q − y4z557= 01 (33)

−4z−�25 · y′4z5 · F�4q − y4z55= Ɛ6min4q1D4z1 �5571 (34)

�2 · 4â − Ɛ6min4q1D4z1 �5575= 00 (35)

We now have two possible cases depending on the valueof �2. Let us first investigate the case where �2 = 0. FromEquation (33), we have F�4q − y4z55 = 4z − c5/z. By usingEquation (34), we obtain z = c − 4Ɛ6min4q1D4z1 �5575/y′4z5.Now, we have

Ɛ6min4q1D4z1�557

=y

(

c−Ɛ6min4q1D4z1�557

y′4z5

)

+Ɛ

[

min(

F −1�

(

p−c

p

)

1�

)]

0

Since we assume that the function y4z5 is a decreasingfunction of the effective price z and that both q andD4z1�5 are nonnegative, we obtain Ɛ6min4q1D4z1 �557 <y4c5 + Ɛ6min4F −1

� 44p − c5/p51 �57 ≤ y4c5. In the last step, weused the fact that Ɛ6min4F −1

� 44p− c5/p51 �57 ≤ 0. Therefore,we conclude that Ɛ6min4q1D4z1 �557 < â . In other words, thesolution is not feasible since it violates the target inequalityconstraint. Hence, we must have �2 > 0, and the inequalityconstraint is tight at optimality: Ɛ6min4q1D4z1 �557= â . Now,by using Equation (33), we obtain F�4q−y4z55= 4z+�2 − c5/4z+�25. We then substitute the above expression in Equa-tion (34): z = c − �2 − â/y′4z5. Now, we have â = y4z5 +

Ɛ6min4F −1� 44z+ �2 − c5/4z+ �2551 �57. By expressing the pre-

vious equation in terms of the effective price, we obtainâ = y4z5+ Ɛ6min4F −1

� 4−4â/y′4z55/4c − 4â/y′4z55551 �57. There-fore, one can solve the previous equation and find the opti-mal effective price z. We note that this is exactly the sameequation as in the decentralized case, so that the effectiveprices are the same. The optimal production levels are givenby q = y4z5+ F −1

� 44−4â/y′4z555/4c− 4â/y′4z5555. Similarly, theequations are the same in both the decentralized and cen-tralized models so that the optimal production levels areidentical. Finally, we just need to show that �2 > 0 in orderto complete the proof. We have

â = y

(

c−�2 −â

y′4z5

)

+ Ɛ

[

min(

F −1�

(

−â/y′4z5

c− â/y′4z5

)

1 �

)]

≤ y

(

c−�2 −â

y′4z5

)

<y4c−�250

If we assume by contradiction that �2 < 0, we obtain â <y4c − �25 < y4c5. This is a contradiction so that �2 > 0, andthe proof is complete. �

Proof of Proposition 4. We first consider the scenariowhere the supplier is nonsophisticated. In this case, the opti-mal decision variables are still rdet, qdet, and pdet. However,in reality, demand is uncertain, and therefore the expectedsales are given by

Ɛ6min4qdet1D4zdet1 �557 = Ɛ6min4qdet1y4zdet5+ �57

= â + Ɛ6min401 �57≤ â0 (36)

Here, we have used the fact that qdet = y4zdet5= â .

Next, we assume that the supplier is sophisticated. Notethat, in this case, the optimal subsidies set by the govern-ment are still equal to rdet. In other words, the governmentdoes not have any distributional information on demanduncertainty and believes that neither does the supplier.In particular, the subsidies are set such that y4pdet − rdet5= â . However, the supplier, who is sophisticated, uses dis-tributional information on demand uncertainty to decidethe optimal price and production. In a similar way to that inEquations (10) and (11) from Theorem 1, the optimal pricewhen r = rdet can be obtained as the solution of the follow-ing nonlinear equation:

y4p− rdet5+ Ɛ

[

min(

F −1�

(

p− c

p

)

1 �

)]

+ y′4p− rdet5 · 4p− c5= 00 (37)

In addition, one can compute the optimal production levelas follows:

q∗4p1 rdet5= y4p− rdet5+ F −1�

(

p− c

p

)

0

For the linear demand model, Equation (37) becomes d −

� · 42p− rdet −c5+Ɛ6min4F −1� 44p−c5/p51 �57= 0. Equivalently,

the optimal price denoted by p∗ follows the following rela-tion:

p∗= c+

â

�+

12�

· Ɛ

[

min(

F −1�

(

p∗ − c

p∗

)

1 �

)]

0

Note that the optimal price when demand is deterministicis equal to pdet = c+â/�, and therefore p∗ ≤ pdet. As a result,the expected demand is given by

y4p∗− rdet5 = d−� · 4p∗

− rdet5

= â −12

· Ɛ

[

min(

F −1�

(

p∗ − c

p∗

)

1 �

)]

≥ â0

We next proceed to compute the expected sales:

Ɛ6min4q∗1D4p∗− rdet1 �557

= Ɛ

[

min(

y4p∗− rdet5+ F −1

�

(

p− c

p

)

1y4p∗− rdet5+ �

)]

= â +12

· Ɛ

[

min(

F −1�

(

p∗ − c

p∗

)

1 �

)]

≤ â0 �

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The Impact of Demand Uncertainty on Consumer Subsidies...

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