The Impact of Demand Uncertainty on Consumer Subsidies for Green Technology Adoption

Transcription

1 The Imact of Demand Uncertainty on Consumer Subsidies for Green Technology Adotion Maxime C. Cohen Oerations Research Center, MIT, Cambridge, MA 02139, Ruben Lobel The Wharton School, University of Pennsylvania, Philadelhia, PA 19104, Georgia Perakis Sloan School of Management, MIT, Cambridge, MA 02139, This aer studies government subsidies for green technology adotion while considering the manufacturing industry s resonse. Government subsidies offered directly to consumers imact the sulier s roduction and ricing decisions. Our analysis exands the current understanding of the rice-setting newsvendor model, incororating the external influence from the government who is now an additional layer in the system. We quantify how demand uncertainty imacts the various layers (government, industry and consumers when designing olicies. We further show that for convex demand functions, an increase in demand uncertainty leads to higher roduction quantities and lower rices, resulting in lower rofits for the sulier. With this in mind, one could exect consumer surlus to increase with uncertainty. In fact, we show this is not always the case and the uncertainty imact on consumer surlus deends on the trade-off between lower rices and the ossibility of under-serving customers with high valuations. We also show that when olicy makers such as governments ignore demand uncertainty when designing consumer subsidies, they can significantly miss the desired adotion target level. From a coordination ersective, we demonstrate that the decentralized decisions are also otimal for a central lanner managing jointly the sulier and the government. As a result, subsidies rovide a coordination mechanism. Key words : Government Subsidies, Green Technology Adotion, Newsvendor, Cost of Uncertainty, Suly Chain Coordination 1. Introduction Recent develoments in green technologies have catured the interest of the ublic and rivate sectors. For examle, electric vehicles (EV historically redate gasoline vehicles, but have only received significant interest in the last decade (see Eberle and Helmot (2010 for an overview. In the height of the economic recession, the US government assed the American Recovery and Reinvestment Act of 2009 which granted a tax credit for consumers who urchased electric vehicles. Besides boosting the US economy, this articular tax incentive was aimed at fostering further research and scale economies in the nascent electric vehicle industry. In December 2010, the allelectric car, Nissan Leaf, and the lug-in hybrid General Motors Chevy Volt were both introduced 1

2 2 Cohen, Lobel and Perakis: Consumer Subsidies with Industry Resonse for Green Technology Adotion in the US market. After a slow first year, sales started to ick u and most major car comanies are now in the rocess of launching their own versions of electric vehicles. More recently in 2012, Honda introduced the Fit EV model and observed low customer demand. After offering sizable leasing discounts, Honda quickly sold out in Southern California 1. It is not uncommon to read about waitlists for Tesla s new Model S or the Fiat 500e, while other EVs are sitting unwanted in dealer arking lots. Both stories of suly shortages or oversuly have been commonly attributed to electric vehicle sales. At the root cause of both these roblems is demand uncertainty. Sallee (2011 studied the suly shortages and customer wait lists shortly after Toyota launched the hybrid electric Prius in When launching a new roduct, it is hard to know how many units customers will request. In addition, finding the correct rice oint is also not a trivial task, esecially with the resence of a government subsidy. In fact, understanding demand uncertainty should be a first order consideration for both manufacturers and olicy-makers alike. For the most art, the subsidy design literature in green technologies has not studied demand uncertainty (see for examle, Benthem et al. (2008, Atasu et al. (2009, Lobel and Perakis (2013 and Alizamir et al. (2013. In ractice, demand uncertainty has also often been not considered. As suggested in rivate communication with several sonsors of the MIT Energy Initiative 2, olicy makers often ignore demand uncertainty when designing consumer subsidies for green technology adotion 3. The urose of this aer is to study whether incororating demand uncertainty in the design of subsidy rograms for green technologies is imortant. In articular, we examine how governments should set subsidies when considering the manufacturing industry s resonse under demand uncertainty. We show that demand uncertainty lays a significant role in the system s welfare distribution and should not be overlooked. Consider the following two examles of green technologies: electric vehicles and solar anels. By the end of 2013, more than 10GW of solar hotovoltaic (PV anels had been installed in the United States, roducing an annual amount of electricity roughly equivalent to two Hoover Dams. While still an exensive generation technology, this large level of installation was only accomlished due to the suort of local and federal subsidy rograms, such as the SunShot Initiative. In 2011, the US energy secretary Steven Chu announced that the goal of the SunShot Initiative by 2020 is to reduce the total cost of PV systems by 75%, or an equivalent of $1 a Watt (DOE 2012, at which oint solar technology will be cometitive with traditional sources of electricity generation. 1 Electric vehicles in short suly - Los Angeles Times - 06/05/ htt://articles.latimes.com/2013/jun/05/autos/la-fi-hy-autos-electric-cars-sold-out htt://mitei.mit.edu/about/external-advisory-board 3 Incentives for green technology adotion - Energy Futures, MIT Energy Initiative - htt://mitei.mit.edu/news/incentives-green-technology-adotion-getting-government-subsidies-right

3 Cohen, Lobel and Perakis: Consumer Subsidies with Industry Resonse for Green Technology Adotion 3 Even before this federal initiative, many states have been actively romoting solar technology with consumer subsidies in the form of tax rebates or renewable energy credits. Similarly, federal subsidies were also introduced to stimulate the adotion of electric vehicles through the Recovery Act. As we reviously mentioned, General Motors and Nissan have recently introduced affordable electric vehicles in the US market. GM s Chevy Volt was awarded the most fuel-efficient comact car with a gasoline engine sold in the US, as rated by the United States Environmental Protection Agency (EPA (2012. However, the rice tag of the Chevy Volt is still considered high for its category. The cumulative sales of the Chevy Volt in the US since it was launched in December 2010 until Setember 2013 amount to 48, 218. It is likely that the $7, 500 government subsidy offered to each buyer through federal tax credit layed a significant role in the sales volume. The manufacturer s suggested retail rice (MSRP of GM s Chevy Volt in Setember 2013 was $39, 145 but the consumer was eligible for $7, 500 tax rebates so that the effective rice reduced to $31, 645. The amount of consumer subsidies has remained constant since launch in December 2010 until the end of This seems to suggest that in order to isolate the imact of demand uncertainty without comlicating the model, it is reasonable to consider a single eriod setting. In this aer, we address the following questions. How should governments design green subsidies when facing an uncertain consumer market? How does the uncertain demand and subsidy olicy decision affect the sulier s rice (MSRP and roduction quantities? Finally, what is the resulting effect on consumers? In ractice, olicy makers often ignore demand uncertainty and consider average values when designing consumer subsidies. This ignorance may be caused by the absence (or high cost of reliable data, among other reasons. We are interested in understanding how the otimal subsidy levels, rices and roduction quantities as well as consumer surlus are affected when one exlicitly considers demand uncertainty relative to the case when demand is aroximated by its deterministic average value. While the government designs subsidies to stimulate the adotion of new technologies, the manufacturing industry resonds to these olicies with the goal of maximizing its own rofit. In this aer, we model the sulier as a rice-setting newsvendor that resonds otimally to the government subsidy. More secifically, the sulier adjusts its roduction and rice deending on the level of consumer subsidies offered by the government to the consumer. This study also hels us to exand the rice-setting newsvendor model while accounting for the external influence of the government. In our model, the government is assumed to have a given adotion target for the technology. This is motivated by several examles of olicy targets for electric vehicles and solar anels. For instance, in the 2011 State of the Union, US President Barack Obama mentioned the following goal:

4 4 Cohen, Lobel and Perakis: Consumer Subsidies with Industry Resonse for Green Technology Adotion With more research and incentives, we can break our deendence on oil with bio-fuels and become the first country to have a million electric vehicles on the road by 2015 (DOE Another examle of such adotion target has been set for solar anels in the California Solar Incentive (CSI rogram, which states that: The CSI rogram has a total budget of $2.167 billion between 2007 and 2016 and a goal to install aroximately 1,940 MW of new solar generation caacity (CSI Hence, in our model, we otimize the subsidy level to achieve a given adotion target level while minimizing government exenditure. In Section 3, we discuss alternative models for the government (such as maximizing the total welfare as well as consider the subsidy rogram budget, emission reductions and social welfare. We then quantify the imact of demand uncertainty on government exenditures, firm rofit and consumer surlus. We further characterize who bears the cost of uncertainty deending on the structure of the demand model. Finally, we study the suly-chain coordination i.e., when the government owns the sulier, and show that subsidies coordinate the overall system. More recisely, we show that the rice aid by consumers as well as the roduction level coincide in both the centralized (where the sulier is managed/owned by the government and the decentralized models (where sulier and government act searately. Contributions Given the recent growth of green technologies, suorted by governmental subsidy rograms, this aer exlores a timely roblem in suly chain management. Understanding how demand uncertainty affects subsidy costs, as well as the economic surlus of suliers and consumers, is an imortant art of designing sensible subsidy rograms. The main contributions of this aer are: Demand uncertainty does not always benefit consumers: Nonlinearity lays a key role. As uncertainty increases, quantities roduced increase whereas the rice and the sulier s rofit decrease. In general, demand uncertainty benefits consumers in terms of effective rice and quantities. One might hence exect the aggregate consumer surlus to increase with uncertainty. In fact, we show this is not always true. We observe that the effect of uncertainty on consumer surlus deends on the demand form. For examle, for linear demand, uncertainty increases the consumer surlus, whereas for iso-elastic demand the oosite result holds. Deending on the demand attern, the ossibility of not serving customers with high valuations can outweigh the benefit of reduced rices for the customers served. By ignoring demand uncertainty, the government will under-subsidize and miss the desired adotion target. Through the case of the newly introduced Chevy Volt by General Motors in the US market, we measure by how much the government misses the adotion target by ignoring demand uncertainty.

5 Cohen, Lobel and Perakis: Consumer Subsidies with Industry Resonse for Green Technology Adotion 5 We show that when the sulier takes into account demand uncertainty information while the government considers only average information on demand, the resulting exected sales can be significantly below the desired target adotion level. The cost of demand uncertainty is shared between the sulier and the government. We analyze who bears the cost of demand uncertainty between government and sulier, which we show deends on the rofit margin of the roduct. In general, the government exenditure increases with the added inventory risk. For linear demand models, the cost of demand uncertainty shifts from the government to the sulier as the adotion target increases or the roduction cost decreases. Consumer subsidies are a sufficient mechanism to coordinate the government and the sulier. We comare the otimal olicies to the case where a central lanner manages jointly the sulier and the government. We determine that the rice aid by the consumers and the roduction levels coincide for both the decentralized and the centralized models. In other words, consumer subsidies coordinate the suly-chain in terms of rice and quantities. 2. Literature Review Our setting is related to the newsvendor roblem which has been extensively studied in the literature (see, e.g., Zikin (2000, Porteus (1990, Winston (1994 and the references therein. An interesting extension that is even more related to this research is the rice-setting newsvendor (see Petruzzi and Dada (1999 and Yao et al. (2006. More recently, Kocabıyıkoğlu and Poescu (2011 identified a new measure of demand elasticity, the elasticity of the lost sales rate, to generalize and comlement assumtions commonly made in the rice-setting newsvendor. Kaya and Özer (2012 rovide a good survey of the literature on inventory risk sharing in a suly chain with a newsvendor-like retailer, which is closer to our framework. Nevertheless, our roblem involves an additional layer (the government that interacts with the sulier s decisions and comlicates the analysis and insights. Most revious works on the stochastic newsvendor roblem treat the additive and multilicative models searately (e.g., in Petruzzi and Dada (1999 or focus exclusively on one case, with often different conclusions regarding the rice of demand uncertainty. In our roblem however, we show that our conclusions hold for both demand models. In the traditional newsvendor setting, the roduction cost is generally seen as the variable cost of roducing an extra unit from raw material to finished good. In caital-intensive industries like solar anels and electric vehicles, the er-unit cost of caacity investment in the manufacturing facility is usually much larger than the er-unit variable cost. For this reason, we define the roduction quantities decision of the sulier in a way that includes both the caacity investment decision and the unit roduction, similar to Cachon and Lariviere (1999.

6 6 Cohen, Lobel and Perakis: Consumer Subsidies with Industry Resonse for Green Technology Adotion Another stream of research related to our aer considers social welfare and government subsidies in the area of vaccines (see, e.g., Arifoglu et al. (2012, Mamani et al. (2011 and Taylor and Xiao (2013. In Arifoglu et al. (2012, the authors study the imact of yield uncertainty, in a model that reresents both suly and demand, on the inefficiency in the influenza vaccine suly chain. They show that the equilibrium demand can be greater than the socially otimal demand. In Taylor and Xiao (2013, the authors assume a single sulier with stochastic demand and consider how a donor can use sales and urchase subsidies to imrove the availability of vaccines. Among aers that study the design of subsidies for green technologies, Carlsson and Johansson- Stenman (2003 examine the social benefits of electric vehicle adotion in Sweden and reort a essimistic outlook for this technology in the context of net social welfare. Avci et al. (2013 show that adotion of electric vehicles has societal and environmental benefits, as long as the electricity grid is sufficiently clean. In Benthem et al. (2008, the authors develo a model for otimizing social welfare with solar subsidy olicies in California. These two aers assume non-strategic industry layers. While considering the manufacturer s resonse, Atasu et al. (2009 study the use of a take-back subsidy and roduct recycling rograms. In a similar way as the revious aers mentioned above, they otimize social welfare of the system assuming a known environmental imact of the roduct. Our work focuses on designing otimal olicies to achieve a given adotion target level, which can be used to evaluate the welfare distribution in the system. In this aer, we also incororate the strategic resonse of the industry into the olicy making decision. Also considering an adotion level objective, Lobel and Perakis (2013 study the roblem of otimizing subsidy olicies for solar anels and resent an emirical study of the German solar market. The aer shows evidence that the current feed-in-tariff system used in Germany might not be efficiently using the ositive network externalities of early adoters. Alizamir et al. (2013 also tackle the feed-in-tariff design roblem, comaring strategies for welfare maximization and adotions targets. Finally, Ovchinnikov and Raz (2013 resent a rice setting newsvendor model for the case of ublic interest goods. The authors comare, for the case of linear demand, different government intervention mechanisms and study under what conditions the system is coordinated in terms of welfare, rices and suly quantities. On the other hand, in this aer we investigate the imact of demand uncertainty on the various layers of the system for non-linear demands and model exlicitly the strategic resonse of the manufacturer to the subsidy olicy. Numerous aers in suly-chain management focus on linear demand functions. Examles include Anand et al. (2008 and Erhun et al. (2011 and the references therein. These aers study suly chain contracts where the treatment mainly focuses on linear inverse demand curves. In this aer, we show that the imact of demand uncertainty on the otimal olicies differs for some classes of non-linear demand functions relative to linear models. In articular, we observe

7 Cohen, Lobel and Perakis: Consumer Subsidies with Industry Resonse for Green Technology Adotion 7 that the effect of demand uncertainty deends on whether demand is convex (rather than linear with resect to the rice. In addition, the demand non-linearity lays a key role on the consumer surlus. As mentioned before, our aer also contributes to the literature on suly chain coordination (see Cachon 2003 for a review. The tyical suly chain setting deals with a sulier and a retailer, who act indeendently to maximize individual rofits. Mechanisms such as rebates (Taylor 2002 or revenue-sharing (Cachon and Lariviere 2005 can coordinate the layers to otimize the aggregate surlus in the suly chain. Liu and Özer (2010 examine how wholesale rice, quantity flexibility or buybacks can incentivize information sharing when introducing a new roduct with uncertain demand. Lutze and Özer (2008 study how a sulier should share demand uncertainty risk with the retailer when there is a lead-time contract. In Granot and Yin (2005, 2007, 2008, the authors study different tyes of contracts in a Stackelberg framework using a rice-setting newsvendor model. In articular, Granot and Yin (2008 analyze the effect of rice and order ostonements in a decentralized newsvendor model with multilicative demand, wherein the manufacturer ossibly offers a buyback rate. In our setting, the government and the sulier are acting indeendently and could erhas adversely affect one another. Instead, we show that the subsidy mechanism is sufficient to achieve a coordinated outcome. Chick et al. (2008 and Mamani et al. (2011 have looked at suly chain coordination in government subsidies for vaccines. Nevertheless, as we discussed above the two suly chains are fairly different. Without considering demand uncertainty, there is a significant amount of emirical work in the economics literature on the effectiveness of subsidy olicies for hybrid and electric vehicles. For examle, Diamond (2009 shows that there is a strong relationshi between gasoline rices and hybrid adotion. Chandra et al. (2010 show that hybrid car rebates in Canada created a crowding out of other fuel efficient vehicles in the market. Gallagher and Muehlegger (2011 argue that sales tax waivers are more effective than income tax credits for hybrid cars. The increase in hybrid car sales from is mostly exlained by social references and increasing gasoline rices. Aghion et al. (2012 show that the auto industry innovates more in clean technologies when fuel rices are higher. Jenn et al. (2013 determine that incentives are only effective when the amount is sufficiently large. For lug-in electric cars, Sierzchula et al. (2014 argue that financial incentives, charging infrastructure, and local resence of roduction facilities are strongly correlated with electric vehicle adotion rates across different countries. Also in economics literature, one can find a vast amount of aers that consider welfare imlications and regulations for a monoolist (see, e.g., Train (1991. There is also a relevant stream of literature on market equilibrium models for new roduct introduction (see, e.g., Huang and Sošić (2010. However, most of these aers do not consider demand uncertainty.

8 8 Cohen, Lobel and Perakis: Consumer Subsidies with Industry Resonse for Green Technology Adotion The issue of how demand uncertainty creates a mismatch in suly and demand has been mostly researched in the oerations management literature, therefore we mainly focus our literature survey in this area. There are some excetions in economics, such as Sallee (2011, who essentially argues that consumers catured most of the incentives for the Toyota Prius, while the firm did not aroriate any of that surlus desite a binding roduction constraint. Sallee (2011 shows that there was a shortage of vehicles manufactured to meet demand when the Prius was launched. This reinforces our motivation for studying a newsvendor model in this context. Also considering demand uncertainty, Fujimoto and Park (1997 show that an exort subsidy (as oosed to a tax is the equilibrium government strategy for a duooly where each firm is in a different country and is uncertain about demand in the other country. In a slightly different setting, Boadway and Wildasin (1990 argue that subsidies can be used to rotect workers from uncertain industry shocks, when there is limited labor mobility. Some works on electricity eak-load ricing and caacity investments address the stochastic demand case (see Crew et al. (1995 for a review on that toic. In this context, it is usually assumed that the sulier knows the willingness to ay of customers and can therefore decline the ones with the lowest valuations in the case of a stock-out. In our alication however, one cannot imose such assumtion and the demand model follows a general rice deendent curve while the customers arrive randomly and are served according to a first-come-first-serve logic. The remainder of the aer is structured as follows. In Section 3, we describe the model. In Section 4, we consider both additive and multilicative demand with ricing (rice setter model, analyze secial cases and finally study the effect of demand uncertainty on consumer surlus. In Section 5, we study the suly-chain coordination and Section 6 considers a different mechanism where the government subsidizes the manufacturer s cost. Finally, Section 7 resents some comutational results and our conclusions are reorted in Section 8. The roofs of the different roositions and theorems together with the rice taker case are relegated to the Aendix. 3. Model We model the roblem as a two-stage Stackelberg game where the government is the leader and the sulier is the follower (see Figure 1. We assume a single time eriod model with a unique sulier and consider a full information setting. The government decides the subsidy level r er roduct and the sulier follows by setting the rice and roduction quantities q to maximize his/her rofit. The subsidy r is offered from the government directly to the end consumer. We consider a general stochastic demand function that deends on the effective rice aid by consumers, z = r, and on a random variable, denoted by D(z,. Once demand is realized, the sales level is determined by the minimum of suly and demand, that is, min(q, D(z,.

9 Cohen, Lobel and Perakis: Consumer Subsidies with Industry Resonse for Green Technology Adotion 9 Figure 1 Order of events: 1. Subsidies; 2. Price and Quantity; 3. Sales The selling rice can be viewed as the manufacturer s suggested retail rice (MSRP that is, the rice the manufacturer recommends for retail. Additionally, in industries where roduction lead time is long and incurs large fixed costs, we consider the roduction quantities to be equivalent to the caacity investment built in the manufacturing facility. The goal of our model is to study the overall imact of demand uncertainty. In order to isolate this effect, we consider a single eriod monoolist model. These modeling assumtions are reasonable aroximations for the Chevy Volt, which we use in our numerical analysis. Note that since the introduction of electric vehicles, the MSRP for the Chevy Volt and the subsidy level have remained fairly stable. Consumer subsidies were osted before the introduction of these roducts and have remained unchanged ($7, 500 since it was launched in December We assume the sulier is aware of the amount of subsidy offered to consumers before starting roduction. The sulier modeling choice is motivated by the fact that consumer subsidies for EVs started at a time where very few cometitors were resent in the market and the roduct offerings were significantly different. The Chevy Volt is an extended-range mid-riced vehicle, while the Nissan Leaf is a cheaer all-electric alternative and the Tesla Roadster is a luxury sorts car. These roducts are also significantly different from traditional gasoline engine vehicles so that they can be viewed as rice setting firms within their own niche markets. Given a marginal unit cost, c, and consumer subsidy level, r, announced by the government, the sulier faces the following rofit maximization roblem. Π = max q, E [ min(q, D(z, ] c q (1 Denote Π as the otimal exected rofit of the sulier. Note that the marginal cost c may incororate both the manufacturing cost (such as material and labor as well as the cost of building an additional unit of manufacturing caacity. Deending on the alication setting, the cost of building caacity can be more significant than the er-unit manufacturing cost. If there is no demand information gained between the building of caacity and the roduction stage, then caacity is built according to the lanned roduction. Therefore, we can assume both these costs to be combined in c. Furthermore, note that one can incororate a salvage value, s a c to our model without affecting our results. To kee the exosition simle, we assume that s a = 0.

10 10 Cohen, Lobel and Perakis: Consumer Subsidies with Industry Resonse for Green Technology Adotion We consider the general case for which the sulier decides uon both the rice (MSRP and the roduction quantities (i.e., the sulier is a rice setter. An alternative case of interest is the one for which the rice is exogenously given (i.e., the sulier is a rice taker and decides only roduction quantity. As mentioned before, we consider the early stages of the EV market as a good alication of the monoolist rice setting model. In contrast, the solar anel manufacturing industry is highly cometitive, as the to ten comanies share less than half of the US market. As a result, the solar manufacturing market is ossibly better reresented with a rice taker model. In this aer, we treat both settings, but focus on the more comlex rice setter model. Due to sace limitations, we relegate the rice taker case to the Aendix. We assume the government is introducing consumer subsidies, r, in order to stimulate sales to reach a given adotion target. We denote by Γ the target adotion level, which is assumed to be common knowledge. Conditional on achieving this target in exectation, the government wants to minimize the total cost of the subsidy rogram. Define Ex as the minimal exected subsidy exenditures, which is defined through the following otimization roblem: Ex = min r r E [ min(q, D(z, ] s.t. E [ min(q, D(z, ] Γ (2 r 0 In what follows, we discuss the modeling choices for the government in more detail. Government s constraints The adotion level constraint used in (2 is motivated by real olicy-making ractice. For examle, President Obama stated the adotion target of 1 million of electric vehicles by 2015 (see DOE More recisely, the government is interested in designing consumer subsidies so as to achieve the redetermined adotion target. An additional ossibility is to incororate a budget constraint for the government in addition to the adotion target. In various ractical settings, the government may consider both requirements (see for examle CSI Incororating a budget constraint in our setting does not actually affect the otimal subsidy solution of the government roblem (assuming the budget does not make the roblem infeasible. In addition, one can show that there exists a one-to-one corresondence between the target adotion level and the minimum budget necessary to achieve this target. Hence, we will only solve the roblem with a target adotion constraint, but the roblem could be reformulated as a budget allocation roblem with similar insights. Given that actual sales are stochastic, the constraint used in our model meets the adotion target in exectation: E [ min(q, D(z, ] Γ. (3

11 Cohen, Lobel and Perakis: Consumer Subsidies with Industry Resonse for Green Technology Adotion 11 Our results can be extended to the case where the government aims to achieve a target adotion level with some desired robability (chance constraint instead of an exected value constraint. Such a modeling choice will be more suitable when the government is risk-averse and is given by: ( [min(q, ] P D(z, Γ. (4 reresents the level of conservatism of the government. For examle, when = 0.99, the government is more conservative than when = 0.9. We note that the insights we gain are similar for both classes of constraints (3 and (4 and therefore in the remainder of this aer, due to sace limitations, we focus on the case of an exected value constraint. Note that one can consider a constraint on greenhouse gas reduction instead of an adotion target. If the government has a desired target on emissions reduction, it can be translated to an adotion target in EV sales, for examle. In articular, one can comute the decrease in carbon emissions between a gasoline car and an electric vehicle (see e.g., Arar (2010. In other words, the value of Γ is directly tied to a value of a carbon emissions reduction target. Therefore, considering a constraint on carbon emissions, the results in this aer remain valid. More generally, if we set a target on any increasing function of sales, the results also remain the same. Government s objective Two common objectives for the government are to minimize exenditures or to maximize the welfare in the system. In the former, the government aims to minimize only its own exected exenditures, given by: Ex = r E [ min(q, D(z, ]. (5 Welfare can be defined as the sum of the exected sulier s rofit (denoted by Π and defined in (1 and the consumer surlus (denoted by CS net the exected government exenditures: W = Π + CS Ex. (6 The consumer surlus is formally defined in Section 4.3 and aims to cature the consumer satisfaction. Interestingly, one can show that under some mild assumtions, both objectives are equivalent and yield the same otimal subsidy olicy for the government. The result is summarized in the following Proosition 1. Proosition 1. Assume that the total welfare is a concave and unimodal function of the subsidy r. Then, there exists a threshold value Γ such that for any given value of the target level above this threshold, i.e., Γ Γ, both roblems are equivalent.

12 12 Cohen, Lobel and Perakis: Consumer Subsidies with Industry Resonse for Green Technology Adotion Proof. Since the welfare function is concave and unimodal, there exists a unique otimal unconstrained maximizer solution. If this unconstrained solution satisfies the adotion level target, the constrained roblem is solved to otimality. However, if the target adotion level Γ is large enough, this solution is not feasible with resect to the adotion constraint. By using the non-decreasing roerty of the exected sales with resect to r (see Lemma 2 in the Aendix, one can see that the otimal solution of the constrained welfare maximization roblem is obtained when the adotion level constraint is exactly met. Otherwise, by considering a larger subsidy level, one still satisfies the adotion constraint but does not increase the welfare. Consequently, both roblems are equivalent and yield the same otimal solution for which the adotion constraint is exactly met. In conclusion, if the value of the target level Γ is sufficiently large, both roblems (minimizing exenditures in (5 and maximizing welfare in (6 are equivalent. Note that the concavity and unimodality assumtions are satisfied for various demand models including the linear demand function. In articular, for linear demand, the threshold Γ can be characterized in closed form and is equal to twice the otimal roduction with zero subsidy and therefore satisfied in most reasonable settings. Furthermore, for smaller adotion target levels, Cohen et al. (2013b show that even for multile roducts in a cometitive environment, the gas between both settings (minimizing exenditures versus maximizing welfare are small (if not zero so that both roblems yield solutions that are close to one another. For the remainder of the aer, we assume the government objective is to minimize exenditures, while satisfying an exected adotion target, as in (2. This modeling choice was further motivated by rivate communications with sonsors of the MIT Energy Initiative. Besides minimizing the subsidy cost, another objective for the government is often to maximize the ositive environmental externalities of the green technology roduct. Assume there is a ositive benefit, denoted by CO2, for each ton of CO 2 emission avoided by each unit sold of the green roduct. By introducing a monetary value to emissions, one can consider a combined government objective of minimizing the subsidy rogram cost, minus the benefit of emission reduction, i.e., r E [ min(q, D(z, ] CO2 E [ min(q, D(z, ]. (7 Similarly to Proosition 1, we can show that if there is an adotion target larger than a certain threshold Γ, then the objectives in (5 and (7 are equivalent and yield the same outcomes. In articular, if Γ Γ, the otimal subsidy olicy will be defined by the tightness of the adotion target constraint. Alternatively, if the subsidy level r required to reach Γ is significantly larger than the rice of carbon, CO2, the otimal subsidy is defined by the target constraint. Given a certain

13 Cohen, Lobel and Perakis: Consumer Subsidies with Industry Resonse for Green Technology Adotion 13 adotion target level Γ, there is a threshold level CO2 such that for any rice of carbon below this level, CO2 CO2, the otimal subsidy olicy is defined by the adotion target constraint. We next show that for an EV such as the Chevy Volt, a conservative estimate for the rice of carbon emission mitigated for each EV sold is much lower than this threshold level. As we mentioned, the ositive externalities of EVs corresond to the reductions in CO 2 emissions throughout their lifetime, converted to US dollars. Following the analysis in Arar (2010, the emission rate er unit of energy amounts to 755 [Kg CO 2 MWh 1 ]. By using the calculations in Cohen et al. (2013b, an estimate for an EV gas emission reduction is about 50.3 [Ton CO 2 ]. In order to convert this number to US dollars, we use the value assigned by the United States Environmental Protection Agency (EPA to a Ton of CO 2. The value for 2014 is 23.3 [$ (Ton CO 2 1 ] so that the monetary ositive externality of an EV is equal to $1, 172. One can see that the ositive externality for an EV is smaller than the consumer subsidies (equal to $7, 500. This imlies that it is sufficient to minimize exenditures, as described in (5. With the formal definitions of the otimization roblems faced by the sulier (1 and the government (2, in the next section we analyze of the otimal decisions of each arty and the imact of demand uncertainty. 4. The Price Setter Model For roducts such as electric vehicles, where there are only a few suliers in the market, it is reasonable to assume that the selling rice (MSRP of the roduct is endogenous. In other words, is a decision variable chosen by the sulier in addition to the roduction quantity q. In this case, the sulier s otimization roblem can be viewed as a rice setting newsvendor roblem (see e.g., Petruzzi and Dada (1999. Note though that in our roblem the solution also deends on the government subsidy. In articular, both q and are decision variables that should be otimally chosen by the sulier for each value of the subsidy r set by the government. To kee the analysis simle and to be consistent with the literature, we consider searately the cases of a stochastic demand with additive or multilicative uncertainty. In each case, we first consider general demand functions and then secialize to linear and iso-elastic demand models that are common in the literature. Finally, we comare our results to the case where demand is aroximated by a deterministic average value and draw conclusions about the cost of ignoring demand uncertainty. In ractice, comanies very often ignore demand uncertainty and consider average values when taking decisions such as rice and roduction quantities. As a result, we are interested in understanding how the otimal subsidy levels, rices and roduction quantities are affected when we exlicitly consider demand uncertainty relative to the case when demand is just aroximated by its deterministic average value. For examle, the comarison may be useful to quantify the value of investing some large efforts in develoing better demand forecasts.

14 14 Cohen, Lobel and Perakis: Consumer Subsidies with Industry Resonse for Green Technology Adotion We next resent the analysis for both additive and multilicative demand uncertainty Additive Noise Define additive demand uncertainty as follows: D(z, = y(z +. (8 Here, y(z = E [ D(z, ] is a function of the effective rice z = r and reresents the nominal deterministic art of demand and is a random variable with cumulative distribution function (CDF F. Assumtion 1. We imose the following conditions on demand: Demand deends only on the difference between and r denoted by z. The deterministic art of the demand function y(z is ositive, twice differentiable and a decreasing function of z and hence invertible. When = c and r = 0 the target level cannot be achieved, i.e., y(c < Γ. The noise is a random variable with zero mean: E[] = 0. We consider that the demand function reresents the aggregate demand for all the consumers in the market during the entire horizon. As a result, the assumtion that the target level cannot be achieved if the roduct is sold at cost and there are no subsidies translates to the fact that the total number of consumers will not reach the desired adotion target level without government subsidies. Under Assumtion 1, we characterize the solution of roblems (1 and (2 sequentially. First, we solve the otimal quantity q (, r and rice (r offered by the sulier as a function of the subsidy r. By substituting the otimal solutions of the sulier roblem, we can solve the government roblem defined in (2. Note that roblem (2 is not necessarily convex, even for very simle instances, because the government needs to account for the sulier s best resonse (r and q (, r. Nevertheless, one can still solve this using the tightness of the target adotion constraint. Because of the non-convexity of the roblem, the tightness of the constraint cannot be trivially assumed. We formally rove the constraint is tight at otimality in Theorem 1. Using this result, we obtain the otimal subsidy of the stochastic roblem (2, denoted by r sto. The resulting otimal decisions of rice and quantity are denoted by sto = (r sto and q sto = q ( sto, r sto. From roblems (1 and (2, the otimal rofit of the sulier is denoted by Π sto and government exenditures by Ex sto. We consider roblems (1 and (2, where demand is equal to its exected value, that is: E [ D(z, ] = y(z. We denote this deterministic case with the subscrit det, with otimal values: r det, det, q det, z det, Π det, Ex det. We next comare these metrics in the deterministic versus stochastic case.

15 Cohen, Lobel and Perakis: Consumer Subsidies with Industry Resonse for Green Technology Adotion 15 Theorem 1. Assume that the following condition is satisfied: The following holds: 2y (z + ( c y (z + c2 3 f (F 1 1 ( < 0. (9 c 1. The otimal rice of roblem (1 as a function of r is the solution of the following non-linear equation: y( r + E [ min(f 1 ( c, ] + y ( r ( c = 0. (10 In addition, using the solution from (10, one can comute the otimal roduction quantity: q (, r = y( r + F 1 ( c. (11 2. The otimal solution of the government roblem is obtained when the target adotion level is exactly met. 3. The otimal exressions follow the following relations: z sto = y 1( Γ K( sto z det = y 1( Γ ( q sto = Γ + F 1 sto c K( sto q det = Γ, sto where K( sto is defined as: K( sto = E [ ( min(f 1 sto c, ]. (12 sto If, in addition, the function y(z is convex: Π sto = sto = c + Γ y (z sto det = c + Γ y (z det Γ 2 y (z sto c (q Γ 2 sto Γ Π det = y (z det Remark 1. Note that for a general function y( r, one cannot derive a closed form solution of (10 for (r. Consequently, one cannot find a closed form exression for the otimal rice sto for a general additive demand. This is consistent with the fact that there does not exist a closed form solution for the rice-setting newsvendor. However, one can use (10 to characterize the otimal solution and even numerically comute the otimal rice by using a binary search method (see more details in the Aendix. Similarly, one cannot generally exress K( sto in closed form. Instead, K( sto reresents a measure of the magnitude of the noise that deends on the rice sto and the noise distribution.

16 16 Cohen, Lobel and Perakis: Consumer Subsidies with Industry Resonse for Green Technology Adotion K( sto is mainly used to draw insights on the imact of demand uncertainty on the otimal decision variables. Assumtion (9 guarantees the uniqueness of the otimal rice as a function of the subsidies, as it imlies the strict concavity of the rofit function with resect to. In case this condition does not hold, roblem (1 is still numerically tractable (see Petruzzi and Dada For the remainder of this aer, we will assume condition (9 is satisfied. For the case of linear demand we can use relation (15 instead, which is a sufficient condition for (9 that is satisfied in many reasonable settings. Note that the otimal ordering quantity in (11 is exressed as the exected demand lus the otimal newsvendor quantile ( c/ related to the demand uncertainty. The government can ensure that the exected sales achieve the desired target adotion level Γ by controlling the effective rice z. When demand is stochastic, in order to achieve an exected sales of Γ, the government must encourage the sulier to roduce a higher quantity than Γ to comensate for the demand scenarios where stock-outs occur. The additional roduction level is catured by K( sto. The otimal rice is characterized by the otimality condition written in (10 that deends on the cost and on the rice elasticity evaluated at that otimal rice, denoted by E d (. This ( can be rewritten so that the marginal cost equals the marginal revenue, i.e., c = 1 1/E d (. Even without knowing a closed form exression for the otimal rice, we can still show that the otimal rice decreases in the resence of demand uncertainty and so does the firm rofit. Remark 2. The results of Theorem 1 can be generalized to describe how the otimal variables (i.e., z, q, and Π change as demand uncertainty increases. Instead of comaring the stochastic case to the deterministic case (i.e., where there is no demand uncertainty, one can instead consider how the otimal variables vary in terms of the magnitude of the noise (for more details, see the roof of Theorems 1 and 5 in the Aendix. In articular, the quantity that catures the effect of demand uncertainty is K( sto. Since the noise has zero-mean, the quantity K( sto in (12 is always non-ositive. In addition, when there is no noise (i.e., = 0 with robability 1, K( sto = 0 and the deterministic scenario is obtained as a secial case. For any intermediate case, K( sto is negative and non-increasing with resect to the magnitude of the noise. For examle, if the noise is uniformly distributed, the inverse CDF function can be written as a linear function of the standard deviation σ as follows: F 1 ( c = σ ( 3 2 c 1. (13 Therefore, K( sto scales monotonically with the standard deviation for uniform demand uncertainty. In other words, all the comarisons of the otimal variables (e.g., effective rice, roduction

17 Cohen, Lobel and Perakis: Consumer Subsidies with Industry Resonse for Green Technology Adotion 17 quantities etc are monotonic functions of the standard deviation of the noise. For other distributions, the relationshi with the standard deviation is not as simle but the key quantity is K( sto. As a result, one can extend our insights in a continuous fashion with resect to the magnitude of the noise. For examle, the inequality of the effective rice is given by: z sto = y (Γ 1 K( sto. This equation is non-increasing with resect to the magnitude of K( sto and is maximized when there is no noise (deterministic demand so that z det = z sto = y 1( Γ. In general, as the magnitude of the noise increases, the gas between the otimal decision variables increase (see lots of otimal decisions as functions of the standard deviation of demand uncertainty in Figure 4 of Section 7. Remark 3. The solution of the otimal quantity q and the effective rice z rovide another interesting insight. Theorem 1 states that when demand is uncertain, the consumers are better off in terms of effective rice and roduction quantities (this is true for any decreasing demand function. Furthermore, the selling rice and the rofit of the sulier are lower in the resence of uncertainty, assuming demand is convex. These results imly the consumers are in general better off when demand is uncertain. Nevertheless, as we will show in Section 4.3, this is not always the case when we use the aggregate consumer surlus as a metric. In various settings, the selling rice is exogenously given (i.e., rice taker model. This setting is relevant for roducts where the market is fairly saturated with suliers. We resent the results for the rice taker model in the Aendix. We observe that for the rice setter model, the results deend on the structure of the demand function unlike in the rice taker case, where the results of Theorem 5 (see Aendix are robust with resect to the tye of demand. By focusing on a few demand functions, we can rovide additional insights. We will first consider the linear demand case, which is the most common in the literature. The simlicity of this demand form enables us to derive closed-form solutions and a deeer analysis of the imact of demand uncertainty. Note that the insights can be quite different for non-linear demand functions. The results resented in Theorem 1 justify the need for considering non-linear functions as well. For this reason, we later consider the iso-elastic demand case and comare it to the linear case. The imact of demand uncertainty on the subsidy level r and the overall government exenditure is harder to observe for a general demand form. In order to exlore this further, we focus on the cases of linear and iso-elastic demands. For both cases, we show that the subsidy increases with the added inventory risk catured by K( sto. Linear Demand In what follows, we quantify the effect of demand uncertainty on the subsidy level and the exected government exenditures. We can obtain such results for secific demand models, among them the linear demand model. Define the linear demand function as: D(z, = d α z +, (14

18 18 Cohen, Lobel and Perakis: Consumer Subsidies with Industry Resonse for Green Technology Adotion where d and α are given ositive arameters that reresent the maximal market share and the rice elasticity resectively. Note that for this model, a sufficient condition for assumtion (9 to hold is given by: α > 1 2c inf x f (x. For examle, if the additive noise is uniformly distributed, i.e., U[ a 2, a 2 ]; a 2 > 0, (note that since the noise is uniform with zero mean, it has to be symmetric we obtain: α > a 2 c. (15 One can see that by fixing the cost c, condition in (15 is satisfied if the rice elasticity α is large relative to the standard deviation of the noise. Next, we derive closed form exressions for the otimal rice, roduction quantities, subsidies, rofit and exenditures for both deterministic and stochastic demand models and comare the two settings. Theorem 2. The closed form exressions and comarisons for the linear demand model in (14 are given by: sto = c + Γ α = det ( q sto = Γ + F 1 sto c K( sto q det = Γ sto r sto = 2Γ α + c d α 1 α K( sto r det = 2Γ α + c d α Π sto = Γ2 α c (q sto Γ Π det = Γ2 α Ex sto = Γ r sto Ex det = Γ r det We note that the results of Theorem 2 can be resented in a more general continuous fashion as exlained in Remark 2. Surrisingly, the otimal rice is the same for both the deterministic and stochastic models. In other words, the otimal selling rice is not affected by demand uncertainty for linear demand. On the other hand, with increased quantities, the exected rofit of the sulier is lower under demand uncertainty. At the same time, the otimal subsidy level and exenditures increase with uncertainty. Therefore, both sulier and government are worse off when demand is uncertain. Corollary 1 and the following discussion rovide further intuition in how this cost of demand uncertainty is shared between the sulier and the government. Corollary q sto q det decreases in c and increases in Γ. 2. r sto r det increases in c and decreases in Γ.