A stationary equilibrium model of green technology adoption with endogenous carbon price

Dammann, Felix; Ferrari, Giorgio Working Paper A stationary equilibrium model of green technology adoption with endogenous carbon price Center for Mathematical Economics Working Papers, No. 688 Provided in Cooperation with: Center for Mathematical Economics (IMW), Bielefeld University Suggested Citation: Dammann, Felix; Ferrari, Giorgio (2024) : A stationary equilibrium model of green technology adoption with endogenous carbon price, Center for Mathematical Economics Working Papers, No. 688, Bielefeld University, Center for Mathematical Economics (IMW), Bielefeld, https://nbn-resolving.de/urn:nbn:de:0070-pub-29874160 This Version is available at: https://hdl.handle.net/10419/286394 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ 688 February 2024 A Stationary Equilibrium Model of Green Technology Adoption with Endogenous Carbon Price Felix Dammann and Giorgio Ferrari Center for Mathematical Economics (IMW) Bielefeld University Universit¨atsstraße 25 D-33615 Bielefeld ·Germany e-mail: [email protected] uni-bielefeld.de/zwe/imw/research/working-papers ISSN: 0931-6558 Unless otherwise noted, this work is licensed under a Creative Commons Attribution 4.0 International (CC BY) license. Further information: https://creativecommons.org/licenses/by/4.0/deed.en https://creativecommons.org/licenses/by/4.0/legalcode.en A STATIONARY EQUILIBRIUM MODEL OF GREEN TECHNOLOGY ADOPTION WITH ENDOGENOUS CARBON PRICE FELIX DAMMANN AND GIORGIO FERRARI Abstract. This paper proposes and analyzes a stationary equilibrium model for a competitive industry which endogenously determines the carbon price necessary to achieve a given emission target. In the model, firms are identified by their level of technology and make production, entry, and abatement decisions. Polluting firms are subject to a carbon price and abatement is formulated as an irreversible investment, which entails a sunk cost and results in the firms switching to a carbon neutral technology. In equilibrium, we identify a carbon price and a stationary distribution of incumbent, polluting firms, that guarantee the compliance with a certain emission target. Our general theoretical framework is complemented with a case study with Brownian technology shocks, in which we discuss some implications of our model. We observe that a carbon pricing system alongside installation subsidies and tax benefits for green firms trigger earlier investment, while higher income taxes for polluting firms may be distorting. Moreover, we discuss the role of a welfare maximizing regulator, who, by optimally setting the emission target, may mitigate or revert some parameters’ effects observed in the model with fixed limit. 1. Introduction We develop a model of carbon pricing within the framework of a general equilibrium model with strategic green technology adoption. The analysis is conducted for a competitive economy that is in a steady state or stationary equilibrium, where equilibrium variables remain constant over time. Climate change is one of the major topics in today’s world and addressing the challenge of reducing global emissions, as a large contributor of global warming, is a broadly studied and thoroughly discussed topic across several disciplines. Within the economic literature, ever since the contributions of Nordhaus (1977), it is widely accepted that any meaningful and efficient climate change policy to decrease emissions has to impose a price on the emissions of carbon dioxide and other greenhouse gases (cf. Stern, 2007). Its underlying idea is that by attaching a price to emissions, it creates financial incentives for those regulated to reduce their emissions, encourage them to adopt cleaner technologies, or to invest in renewable energy sources. As of today, approximately 23% of the world’s Date: February 23, 2024. 1 2 FELIX DAMMANN AND GIORGIO FERRARI total carbon emissions are subject to a carbon pricing system designed with the explicit goal of constraining emissions (cf. World Bank, 2023). The study and development of (effective) mechanisms for attaching a price to emission of pollutants is an important topic in the literature on environmental economics, and essentially boils down to two fundamental concepts. In the market-based or quantity approach, a regulator determines the maximum allowable level of overall emissions. Often implemented via a so-called cap-and-trade mechanism, it is considered to be one of the most promising and cost-effective market mechanisms in the effort to reduce carbon emissions. A prominent example includes the European Union’s emission trading system, which was implemented in 2005 and includes around 10.000 installations that cover around 40% of the total emissions. By setting a legally binding target for the maximum allowed level of greenhouse gas (GHG) emissions, this approach allows regulators to commit to concrete emission reduction goals on international, national or even industry specific level. Emission allowances are distributed to firms, either through auctions or as free allocations, and firms buy and trade allowances among themselves in order to account for their emissions. Consequently, this market-driven approach naturally gives rise to a market price for emission permits. In contrast to this mechanism, where the quantity of overall emissions is limited, the price approach to carbon emissions directly links the emission of one unit of pollutant to a fixed cost. Typically imposed by a regulator through a tax or penalty, the resulting level of overall emissions is a priori unclear, as market participants are not regulated in the amount of emissions they are allowed to emit. However, the carbon price may be determined by estimates of the price required to limit emissions below some predetermined level. In both cases, a carbon price – also referred to as carbon tax or social cost of carbon – emerges. The price encapsulates the cost associated with the right to emit one unit of pollutant, thus serving the purpose of correcting market distortions that result from market participants not weighing in the external effects of their harmful activities. As emphasized by Nordhaus (2007), “the key economic issue is how to balance the benefits and costs of global emissions reductions.” Moreover, a transparent and comparable carbon price is essential to provide incentives to firms as well as stimulate research and development in carbon neutral technologies (cf. Stern, 2007; Zhao, 2003). In particular, it should transmit the social cost of carbon emissions to the decisions of firms as well as individuals. GREEN TECHNOLOGY ADOPTION WITH ENDOGENOUS CARBON PRICE 3 In the literature, there exist broad contributions addressing the topic of carbon pricing and the economic impact of climate change in various climate economics settings. Nordhaus (2014) discussed the concept of social cost of carbon in order to mitigate losses that are caused by carbon emissions. Golosov et al. (2014) study a dynamic stochastic general equilibrium model, establishing a flexible optimal tax formula compatible with the climate impacts considered in Nordhaus (2014). Acemoglu et al. (2012) and Acemoglu et al. (2016) analyse directed technical change as well as optimal carbon taxes and subsidies. Other studies focused on the price formation of emission allowances, including works by Gr¨ull and Taschini (2011) as well as Carmona et al. (2009,2010) and Carmona and Hinz (2011), who establish market equilibria as well as design optimal emission trading mechanisms. The effect of cap-and-trade mechanisms and carbon taxes on firms’ production and abatement decisions was studied in Anand and Giraud-Carrier (2020) and Fan et al. (2023). We also mention A¨ıd and Biagini (2023), proposing a Stackelberg-type equilibrium model with a regulator optimally offsetting shocks in the carbon price dynamics; Colla et al. (2012), who examine endogeneous prices of permits and study the social welfare optimizing policy of a regulator; De Angelis et al. (2023) studying the effect of green investors on firms’ abatement strategies; Barnett (2023), who analyses the impact of climate change uncertainty on economic and financial outcomes, as well as Hitzemann and UhrigHomburg (2018), studying trading rounds of a finite set of firms which maximize over abatement and trading. In our model, we seek to embed the issue of carbon price formation into a stationary equilibrium model with strategic abatement investment. Firms are characterized by their idiosyncratic technology shocks, which evolve according to a general Itˆo-diffusion process, and we assume that firms’ activities lead to carbon emissions. Since emissions are regulated and firms’ profits are lowered through the implementation of a carbon price, they have the incentive to invest in a carbon-neutral technology. In equilibrium, we determine a stationary distribution of incumbent, polluting firms, the entry rate of new firms, as well as an optimal investment rule that is triggered by an equilibrium carbon price. The latter is endogenously determined and precisely reflects the cost necessary to achieve a certain emission target. Under reasonable and quite general assumptions on the involved functions and diffusion processes we are able to prove existence and uniqueness of this equilibrium. For exposition, in Section 5, we establish a specific formulation of the problem. We consider the case of technology shock processes evolving according to arithmetic Brownian motions and an AKstructure for firms’ production functions. Additionally, we assume a damage function of emissions on 4 FELIX DAMMANN AND GIORGIO FERRARI production, that admits a similar structure as the one considered in Golosov et al. (2014). Within this particular model, we conduct a comparative statics analysis to explore some of the parameters’ effects on the equilibrium values. For instance, we study the effect of a shift in the corporate tax rates of polluting or carbon neutral firms. Potential regulatory strategies might involve installation subsidies or different tax rates for these installations to incentivise polluting firms to invest in a carbon-neutral technology (see, for example, Farzin and Kort, 2000). We find that a policy penalizing polluting firms through increased taxes may yield suboptimal results, since firms’ reduced profits discourage potential firms from entering and thus causes a decreased market competition. This, in turn, leads to a smaller force for firms to become carbon-neutral, and we observe a decreasing carbon price that results in delayed investment. On the other hand, tax benefits for carbon neutral firms as well as subsidies on the investment cost lead to an opposing effect. Indeed, we observe that improving market conditions for carbon neutral firms prompts earlier investment by polluting firms and leads to a slightly higher equilibrium carbon price. Furthermore, we observe that reducing the cumulative emissions available to firms intensifies competition among them, which leads to an increased equilibrium carbon price. This effect also prompts firms to adopt a carbon-neutral technology earlier. It is important to notice that cumulative emissions are not part of the equilibrium values at this point, but are exogenously given (for discussions on this, we refer to Nordhaus (2014) and Stern (2007)). Clearly, this limit may be determined based on environmental considerations, such as the maximal allowed amount of emissions that can be released into the atmosphere while achieving a certain carbon concentration or limiting the global temperature increase to a certain degree. As such, the question of optimally setting the emission cap is far from being trivial. In the considered case study, we explore the concept of a regulator aiming to maximize social welfare by optimally setting the emission cap. Inspired by related contributions (see, for example, A¨ıd and Biagini, 2023; Colla et al., 2012; Ulph, 1996), we investigate a regulator balancing the benefits from emission inducing production and their societal damages. Our comparative statics analysis also involves a comparison of the equilibrium values with exogenous and endogenous emission limits. We observe that the regulator’s actions potentially counteract or mitigate effects on carbon price and investment decisions observed in the analysis with fixed overall emissions. For instance, an increased carbon intensity of production or elevated emission induced damages on production lead the regulator to set a lower emission target, encouraging earlier investment in carbon-neutral technology. GREEN TECHNOLOGY ADOPTION WITH ENDOGENOUS CARBON PRICE 5 Let us now discuss in detail the methodology that we follow for the construction of equilibrium. We start by studying the decision-making process of a single firm that is subject to the carbon pricing system and faces an irreversible investment opportunity of real-option type. We assume that the firm’s technology shock process evolves according to a general Itˆo-diffusion (cf. Hopenhayn, 1992; Luttmer, 2007; Miao, 2005). Since their production induced emissions impose a negative externality on the society, a emission regulation is imposed that charges a cost – the carbon price – for each unit of pollutant emitted, thus decreasing firm’s profits. As a response to this regulation, firms can potentially react by production adjustments, payment of the carbon price or abatement. We assume that firms’ abatement strategies, with the aim of lowering emissions, takes the shape of a real option problem of irreversible investment. Investment involves a sunk cost and results in the firm reducing their carbon emissions to zero. Hence, the firm can either continue paying the carbon price to account for their emissions, adjust its production or decide to exercise the option. In accordance with the existing literature (see A¨ıd and Biagini, 2023; Flora and Vargiolu, 2020; Huang et al., 2021), such investment should be understood as a switch to a different underlying technology, and we account for this by allowing the technology shock process to follow different dynamics after the investment. The first part of the paper thus addresses an optimal stopping problem (real option problem) that is solved using the guess-and-verify approach. We employ techniques presented in Alvarez (2001), that are tailored for dealing with general diffusions, in order to determine the firm’s optimal investment time in a carbon neutral technology, thereby becoming independent of the imposed carbon price. Our general framework, in which we do not fix any explicit profit functions or particular underlying diffusions, is enriched by the including the characteristic of the firm leaving market due to low levels of technology (absorption in zero) or non-observable reasons (Poisson death). It is important to notice that this general formulation of the problem allows the implementation of features and parameters beyond those specified in Section 5. This enables the study of different instances of the problem, thus drawing the focus to other parameters not captured in our specific model. Under reasonable assumptions on the involved diffusions and functions, we are able to derive a complete characterization of the value function as well as the optimal investment time of the firm. The latter is given by the first time the technology shock process exceeds a threshold value b, reflecting the intuition that technologically advanced firms are those first to invest in carbon neutral technologies. Moreover, we show that the investment threshold is decreasing in the carbon price on the market: If a polluting firm faces an increased cost on their emissions, the cost-saving effect of installing an 6 FELIX DAMMANN AND GIORGIO FERRARI emission reduction technology becomes larger and leads to an incentive to become carbon neutral at an earlier stage (cf. Huang et al., 2021). The single-firm optimal irreversible investment problem relates to the literature of optimal timing decisions in environmental economics, with the seminal papers of Pindyck (2000,2002) studying irreversible policy adoptions to reduce emissions of a pollutant. Since then, the real option approach to environmental investments has received much attention (cf. Boomsma et al., 2012; Detemple and Kitapbayev, 2020; Falbo et al., 2021). Let us mention Abadie and Chamorro (2008), Brauneis et al. (2013), Flora and Vargiolu (2020), and Insley (2003) that study emission reduction investments in the presence of diffusive carbon prices, as well as Basei et al. (2023) and Flora and Tankov (2023) as recent contributions incorporating the feature of Bayesian learning. Closest to our formulation is the work of Huang et al. (2021), studying the problem of a company that chooses to invest in an emission reduction technology in the presence of a carbon tax. Here, the emission rate is assumed to be diffusive while the investment cost admits jumps, and the authors succeed in determining the optimal investment time to reduce its emissions. Nevertheless, it is crucial to notice that in all the aforementioned contributions of irreversible investment the carbon price is either nonexistent or exogenously given by an uncontrolled diffusion or a constant tax. In this paper, we aim to fill this theoretical gap. To this end, we move to an aggregate level and consider a continuum of firms that are subject to idiosyncratic shocks and solve the optimal investment problem laid out in the first part. We seek to derive a stationary equilibrium consisting of an equilibrium carbon price and a stationary distribution of incumbent, polluting firms. First, since all firms will eventually exit (due to Poisson death or their technology shock process falling below zero) or invest (to become carbon neutral), we introduce new firms to the market via the so-called entry-condition. Here, we equate the cost of entering and the expected benefit of entering. As usual for stationary equilibrium models, this condition guarantees the balance of the inflow and outflow of firms, such that the mass of incumbent firms is constant. Second, we introduce an equilibrium condition that equates net emissions, resulting from firms’ production activities, and a constant emission level. Formulated in the spirit of market-clearing condition, the latter (fixed) parameter may be interpreted as an emission target or emission limit that is either imposed by a regulator or collectively decided on by firms (see also the discussion in Remark 2.2). We highlight that a similar condition is specified in the recent Anderson and Duanmu (2023), studying a general equilibrium model of a government setting either quotas or taxes GREEN TECHNOLOGY ADOPTION WITH ENDOGENOUS CARBON PRICE 7 on emissions, and then refraining from further actions. In the case of quotas – and in the same spirit as in this paper – the equilibrium carbon price is determined by equating quota and total net emissions. The resulting carbon price should thus not be understood as a flat tax rate on emissions that is exogenously imposed by a legislative body. Instead, it is endogenously determined by the competitive actions of firms resulting from a fixed cumulative amount of emissions among them. Hence, it reflects the cost attached to carbon emissions that is necessary to achieve a given emission target and to remain in a steady state. It is interesting to notice that, although each firm is subject to considerable change due to its idiosyncratic noise and strategic investment decisions, the resulting equilibrium values are constant. In this regard, our work closely relates to competitive equilibrium theory, that originated in Lucas and Prescott (1971), while dynamic models with entry and exit were introduced by Brock (1972) and Smith (1974). While these models did not contain any firm specific stochastic elements, Jovanovic (1982) first introduced a model including idiosyncratic productivity shocks. The notion of stationary equilibrium with entry and exit was then developed in the seminal papers of Hopenhayn (1992) and Hopenhayn and Rogerson (1993). Their approach serves as a tool to analyse long run behaviour of dynamic industries, resulting in equilibria with constant aggregate values. Since then, their techniques received much attention, resulting in numerous contributions, including Dixit and Pindyck (1994), Chapter 8, as well as Luttmer (2007) and Miao (2005), who study firms’ investment choices and entry and exit behaviour in related frameworks. Even though the literature on optimal carbon price mechanisms is extensive (see, for example, Acemoglu et al., 2012,2016; Carmona et al., 2009; Golosov et al., 2014 ), our approach that combines the features of stationary equilibria, endogenous carbon pricing as well as abatement strategies presents, to our knowledge, a novelty. Our modeling of abatement as a real option problem is motivated by the irreversibility in these choices. As highlighted by Abadie and Chamorro (2008), Brauneis et al. (2013), and Chesney and Taschini (2012), among others, investments in pollution reduction are usually expensive, durable and irrevocable. Consequently, abatement strategies in order to reduce the emissions are not decisions that firms are able to revise at a continuous rate, with typical examples including the switch to a different energy source or a different underlying technology. Notice that we are able to capture the effect of the latter policy by allowing for different dynamics of the underlying technology shock process after the investment of the firm. Our main contribution is then the study of the resulting 14 FELIX DAMMANN AND GIORGIO FERRARI 3. The single-firm investment problem Problem (2.4) takes the shape of an optimal stopping problem in the field of real options theory. Dating back to the contributions of Myers (1977) and McDonald and Siegel (1986), the real options approach to irreversible investment decisions has received much attention in various problems arising in economics and finance. Here, we mention Dixit (1989) as well as Pindyck (1988,1991). In the case, where the underlying economic shock process is one-dimensional – as in our case – explicit solutions are often feasible (cf. Dixit and Pindyck, 1994). We build on their analysis and use the connection of optimal stopping and free boundary problems (cf. Peskir and Shiryaev, 2006) in order to identify the optimal investment rule of the single firm. In the following, we denote G(z) := Φ2(z)−Φ1(z)−c(z) and by L=1 2σ2 1(z)∂zz +µ1(z)∂zthe infinitesimal generator associated to the diffusion Zof (2.1). We state the following assumption. Assumption 3.1. There exists a unique point ez∈(0,∞), such that L − (r+η)G(z) = π1(z) + L − (r+η)[Φ2(z)−c(z)]              >0,0≤z < ez, = 0, z =ez, <0, z > ez. (3.1) Assumption 3.1 is a well known criterion that is typical in optimal stopping problems in order to derive the existence and uniqueness of a point b∈R+triggering the optimal stopping time (see, e.g., Alvarez (2001) and Falbo et al. (2021)). It is crucial to notice that, at this point, no qualitative statements are possible regarding sufficient conditions on the parameters µi, σithat imply the suggested shape of the function (3.1) and thus of the monotonicity of A(·), as derived in (B.5). We remark that, under the assumption that µ1=µ2,σ1=σ2and a constant investment cost c(z) = I, the condition (3.1) simplifies to the assumption π1(z)−π2(z) + (r−η)I < 0 for z > ez and π1(z)−π2(z)+(r−η)I≥0 on z≤ez. Furthermore, we mention that if Assumption 3.1 is not satisfied, it can be shown that firms do not invest into a carbon neutral technology. In the case in which 0 is natural for the diffusion Z(and firms do not exit the market when their technology level is low), our analysis may lead to a stationary equilibrium in which neither entry nor exit takes place. Since this is not the scope of this work, we refrain from studying this case and stick to Assumption 3.1. GREEN TECHNOLOGY ADOPTION WITH ENDOGENOUS CARBON PRICE 15 The following theorem summarizes our findings in the single-firm problem. Its proof can be found in Appendix B.1. Theorem 3.2. (i) The value function vof (2.4)takes the form v(z) :=              0, z ≤0, Φ1(z) + Φ2(b)−Φ1(b)−c(b)ψ(z,0) ψ(b,0) ,0< z < b, Φ2(z)−c(z), z ≥b, (3.2) where b=b(cp)∈(0,∞)denotes the investment threshold triggering the optimal stopping time τb= inf{t≥0 : Zz t≥b}. It is given by the unique solution to the nonlinear algebraic equation A(b)=0, with A(·)given by A(z) := G′(z)[ψ(0)φ(z)−φ(0)ψ(z)] −G(z)[ψ(0)φ′(z)−φ(0)ψ′(z)] S′(z).(3.3) Here, S′denotes the scale density of the diffusion Zand ψ, φ denote the increasing and decreasing, respectively, fundamental solutions to the ordinary differential equation (L − (r+η))u= 0. (ii) The investment threshold bas well as the value function vare (strictly) decreasing in the carbon price cp. Moreover, the limit b∞:= limcp→∞ b(cp)exists finite. 4. A Continuum of Firms and Market Equilibrium In the latter section, we discussed the irreversible investment problem of a single firm, posed as an optimal stopping problem (2.4). In this section, we move to an aggregate level and consider a continuum of emitting firms, that each face idiosyncratic shocks to their technology shock process and solve the investment problem introduced in Section 2. In our forthcoming analysis, we aim to derive the long-run stationary equilibrium of the regulated market. Hence, in order to guarantee the existence and uniqueness of a competitive equilibrium with entry and exit and a positive carbon price cp>0, we state the following assumption on the exogenous entry cost ce. Assumption 4.1. We state the following assumptions. (i) The entry cost cesatisfies ce≤Zz z v(z; 0, Emax)ξ(dz).(4.1) 16 FELIX DAMMANN AND GIORGIO FERRARI (ii) We distinguish two cases. Recall that b∞= limcp→∞ b(cp)>0exists due to Theorem 3.2 and that supp{ξ(dz)}= [z,z]. If b∞> z, we assume ce>lim cp→∞ Zz z v(z;cp, Emax)ξ(dz).(4.2) Otherwise, if b∞≤z, we then assume ce>Zz z v(z;cp, Emax)ξ(dz).(4.3) The value cp>0is the unique solution to b(cp) = z(due to Theorem 3.2). Assumption 4.1 is not only necessary when deriving the existence of an equilibrium in the competitive market, but also admits a clear economic interpretation. This is due to the fact that it enforces the idea that – in any meaningful model – the equilibrium carbon price should be such that cp∈(0,∞), i.e. firms do not get a positive reward for emitting pollutants. Moreover, it guarantees that a positive number of firms are entering the market that do not immediately exercise their option to become carbon-neutral. For more details we refer to the proof of Proposition 4.2, here we concentrate on the intuition behind Assumptions 4.1 (i) and (ii). Notice that condition (i) implies that the entry cost ceis lower than the expected profit of entering firms in the absence of a carbon price cp. If not fulfilled, the entry cost would thus exceed the highest possible expected benefit. This could either lead to a negative equilibrium carbon price (which is not desirable from neither a modelling nor a regulators perspective) or (if the carbon price is assumed to be positive) discourage firms from entering the market. On the other hand, condition (ii) implies that the entry cost is larger than the expected benefit from entering a market with the largest carbon price that still guarantees a distribution of incumbent firms. An entry cost violating condition (ii) would thus lead to an imbalance, since no carbon price cp∈(0,∞) could prevent new entrants from flooding the market. We notice that, if b∞< z, condition (4.3) is not necessarily needed when deriving the stationary distribution. However, when violated, the entry condition (2.8) could lead to an equilibrium carbon price cpsuch that b(cp)< z. This would imply that all entering firms are immediately switching to become carbon neutral. While this could be desirable from an environmental perspective, it implies a zero mass of emitting firms in the long run steady state. We exclude this trivial case here. GREEN TECHNOLOGY ADOPTION WITH ENDOGENOUS CARBON PRICE 17 Existence and Uniqueness of an Equilibrium. Prior to presenting our central theoretical finding, which establishes the existence and uniqueness of a stationary equilibrium in our model, we offer a concise overview of the underlying rationale and the proof’s methodology. Our proof follows similar steps as those developed in Dixit and Pindyck (1994), Hopenhayn and Rogerson (1993), and Miao (2005). First, we derive the equilibrium carbon price using the entry condition (2.8). Via Assumption 4.1 and the derived monotonicity of the value function vwith respect to cp(see Theorem 3.2) it is straightforward to show that there exists a unique carbon price c∗ pthat leads (2.8) to hold with equality. The equilibrium value c∗ pis then used to derive the technology level b∗=b(c∗ p) at which firms choose to invest into becoming carbon neutral. Next, we solve for the equilibrium distribution ν∗, which is, similarly as in related contributions, not a probability measure (see Hopenhayn, 1992; Luttmer, 2007; Miao, 2005). Instead, for a given Borel set Bon the real line, ν∗(B) reflects the number of polluting firms whose technology shock process falls within the set B. Its support is given by the interval (0, b∗), since polluting firms exit when their technology either falls below zero (since they are assumed to be inefficient) or exceeds the threshold b∗(since they invest to become carbon neutral). Furthermore, it is crucial to notice that we are able to scale the distribution ν∗by the entry rate N∗when solving for it. More precisely, one can show (see Dixit and Pindyck, 1994; Hopenhayn and Rogerson, 1993) that the stationary distribution is linearly homogeneous in the entry rate N∗, such that ν∗=N∗f(z) for some function fto be found. The latter function is readily found using the Kolmogorov-foward equation, which is associated to the diffusion Zof (2.1) and complemented to include the two features entry and Poisson deaths (see Luttmer, 2007; Miao, 2005). It follows that we can compute fvia ordinary differential equations, that are, depending on the position of the exit threshold b∗, satisfied on particular intervals. More precisely, they are given by −∂ ∂z µ(z)f(z)+∂2 ∂z2hσ2(z) 2f(z)i−ηf(z) = 0,for z∈(0, z)∪(min(z, b∗), b∗),(4.4) −∂ ∂z µ(z)f(z)+∂2 ∂z2hσ2(z) 2f(z)i−ηf(z) + g(z) = 0,for z∈(z, min(z, b∗)).(4.5) with boundary conditions f(0) = f(min(z, b∗)) = 0, and where gdenotes the density function of the entry distribution ξ. Notice that, since b∗is endogenously determined, it is a priori unclear whether b∗≥zor b∗< z. The resulting stationary distribution of firms admits qualitatively different properties in these cases (notice that a fraction of entering firms is immediately switching to become carbon neutral in the latter case) and we thus distinguish them when deriving its scaled density f 18 FELIX DAMMANN AND GIORGIO FERRARI in the proof of Proposition 4.2 below. In the last step, we use the equilibrium condition (2.9) to solve for the entry rate N∗. Here, it is crucial to notice that, due to the linearity of ν∗in the entry rate N∗, the overall emissions as determined in the integral on the right hand side of (2.9) are also linear in N∗. The equilibrium entry rate N∗as well as the stationary distribution ν∗=N∗fare thus derived using simple calculations. We now state our main result. Its proof can be found in Appendix C. Proposition 4.2. There exists a unique stationary equilibrium, that includes a unique equilibrium carbon price c∗ p>0, an exit threshold b∗=b(c∗ p), an entry rate N∗and a stationary distribution ν∗ such that the entry condition (2.8)and the equilibrium condition (2.9)are satisfied. Moreover, b∗is the threshold that triggers the optimal stopping time τb∗in the single firm problem (2.4). We remark that our result does not require a sufficiently large parameter ηgoverning Poisson death. This is due to the fact that the distribution of incumbent firms is supported on the bounded interval (0, b∗). Indeed, in models with unbounded support and a non-stationary state process, a too small value of ηcould lead to an exploding number of firms with large technology levels (see, for example, Dixit and Pindyck, 1994; Miao, 2005). 5. A Case Study with Arithmetic Brownian Motions as Technology Shocks In this section we discuss an illustrative equilibrium model of firms switching to become carbon neutral. We begin by fixing a specific model and show that the model fulfills the assumptions we imposed throughout Sections 2and 4. In Section 5.1 we perform a comparative statics analysis on the equilibrium parameters, analysing the sensitivity of the values on some of the underlying parameters. Furthermore, we discuss how a regulatory body could choose to constraint the overall emissions in a strategic way. Section 5.2 assumes a social welfare maximizing regulator, and studies the effect on the resulting equilibrium. To begin with, we assume that the dynamics of Zand Z, as in (2.1)-(2.2), are given by arithmetic Brownian motions dZt=µ1dt +σ1dWt, t ≥0, Z0=z, dZt+τ=µ2dt +σ2dWt, t ≥τ, Zτ=Zτ, GREEN TECHNOLOGY ADOPTION WITH ENDOGENOUS CARBON PRICE 19 where µi∈Rand σi>0, i= 1,2. We assume that each firm has the production function y(z, k) = D(Emax)θzk,(5.1) where θ > 0 is a scale parameter and kdenotes the capital stock of the firm, such that we are in a classical AK-model (see for example W¨alde, 2011). Moreover, we assume that the externality, i.e. the atmospheric changes due to climate change caused by global emissions, may have a negative effect on production of each firm. Following Nordhaus (2014) and Golosov et al. (2014), we assume that damages are multiplicative and can be summarized by the damage function given by D(Emax) := exp −ρ(Emax −E)).(5.2) Golosov et al. (2014) assume a similar structure that first translates emissions to carbon concentration, which is then translated to damages. Here, we simplify the structural form by assuming that damages are directly triggered by emissions, and Emax −Emeasures the distance between overall emissions and a benchmark level E. The latter may be chosen in such a way that the corresponding carbon concentration is that of pre-industrial times (cf. Golosov et al., 2014). Notice that ρ > 0 results in a reduced productivity of production if the level of overall emissions is above the benchmark level E. Similar to Miao (2005), we assume a simple capital structure where firms rent capital from riskneutral investors who discount future cash flows at a constant rate r > 0. Moreover, we assume that capital depreciates with rate δ > 0, such that the rental rate for firms is given by r+δ. As in Section 2, we assume that firms’ production leads to pollution, and firms are subject to a carbon price that has to be paid according to their emissions. For simplicity, we assume that firms’ emissions are proportional to their output (see for example Acemoglu et al., 2016; Farzin and Kort, 2000; Krass et al., 2013; Ulph, 1996), i.e. e(y(z, k)) = λy(z, k),(5.3) for some λ > 0. The investment cost by the firm is assumed to be constant and equal to I−κ, where κ≥0 denotes a subsidy of the regulator given to firms in order to give incentives for an investment to become carbon neutral. We assume that firms face a constant elasticity demand function (see, for 20 FELIX DAMMANN AND GIORGIO FERRARI example Bertola, 1998; Krass et al., 2013), such that the market price is given by p(y) = y(z, k)−ε, ε ∈(0,1). Clearly, our model could embed situations in which consumers’ willingness to pay differs between products produced with different technologies (see, for example, Krass et al., 2013). Additionally to the restructering, the firm could also benefit from possible tax incentives the legislative body will implement in order to encourage firms to become carbon neutral. Hence, we assume that polluting and carbon neutral firms are taxed with potentially different tax rates before and after their restructure and denote them by τ1≥τ2. We note that Moyer et al. (2014) as well as Acemoglu et al. (2016) model the “carbon tax” as a production tax which differs by type of technology, whereas we model a carbon price alongside capital income taxes, and we refer to Barrage (2020) for a discussion on this topic. All in all, the firms’ profit functions before and after their investment are given by π1(z, k) := max k(1 −τ1)p(y)y(z, k)−δk −cpe(y)] −rk, π2(z, k) := max k(1 −τ2)p(y)y(z, k)−δk−rk. We can optimize these functions and determine optimal capital levels k∗ 1=(1 −ε)(D(Emax)θz)1−ε δ+cpλD(Emax)θz +r/(1 −τ1)1/ε, k∗ 2=(1 −ε)(D(Emax)θz)1−ε δ+r/(1 −τ2)1/ε,(5.4) which, plugged into the value function, yield π1(z) := π1(z, k∗ 1) = (1 −τ1)ε(1 −ε)D(Emax)θz δ+cpλD(Emax)θz +r/(1 −τ1)1−ε ε, π2(z) := π2(z, k∗ 2) = (1 −τ2)ε(1 −ε)D(Emax)θz δ+r/(1 −τ2)1−ε ε, e(z) := e(y(z, k∗ 1) = λ(1 −ε)D(Emax)θz δ+cpλD(Emax)θz +r/(1 −τ1)1 ε, and we observe that firm’s profits as well as their emissions are increasing in their current technology shock (see Hopenhayn, 1992). Regarding the entry decision of firms, we assume that the firms’ initial technology values after entry are uniformly distributed, i.e. ξ∼ U([z,z]) (cf. Miao, 2005). It is straightforward to verify that the proposed model satisfies the modelling assumptions made throughout the previous sections regarding the involved functions, and we provide a proof in Appendix D. We remark that, even in the explicit formulation of our model, a general verification of GREEN TECHNOLOGY ADOPTION WITH ENDOGENOUS CARBON PRICE 21 Assumption 3.1 is not possible. Clearly, this would lead to imposing specific assumptions on the involved parameters in our model, which is neither a straightforward task nor leading to any qualitative insights. 5.1. Comparative Statics Analysis. In the following, we discuss some of the implications of our case study on the equilibrium values and, especially, their sensitivity with respect to changes in the model’s parameters. To begin with, we fix the base case parameter values, which are summarized in Table 1. Parameter Value Polluting firm’s shock drift µ10.02 Polluting firm’s shock volatility σ10.15 Carbon Neutral firm’s shock drift µ20.02 Carbon Neutral firm’s shock volatility σ20.15 Polluting firm’s tax rate τ10.3 Carbon Neutral firm’s tax rate τ20.3 Depreciation Rate δ0.1 Riskless rate r0.5 Poisson death η0.04 Entry Cost ce28.6 Entry Distribution Interval (z,z) (5,30) Price Elasticity ε0.5 Scale Parameter Output θ0.3 Scale Parameter Emissions λ0.05 Scale Parameter Damage ρ0.02 Investment Cost I100 Subsidy κ0 Benchmark Emission Level E100 Cumulative Emissions Emax 102 Table 1. Base Case Parameter Values We want to emphasize that for all the parameters we have selected, both in the base case model and in the upcoming sensitivity analysis, we have ensured that the condition (3.1) of Assumption 3.1 is consistently met. Moreover, it is essential to note that these parameter values, although chosen to align with the estimated data, serve merely as illustrative benchmarks. Most of the data has been chosen to suit those assumed in related contribution as Golosov et al. (2014) and Miao (2005). To allow for a clear interpretation, the entry cost as well as the interval bounds for the entry distribution in the base case model have been calibrated such that the carbon price is c∗ p= 1. In the first part of our comparative statics analysis, we maintain cumulative emissions as a fixed parameter. Here, we let E= 100 and Emax = 102, such that the model depicts a 2% breach of the 22 FELIX DAMMANN AND GIORGIO FERRARI benchmark level. Later, in the subsequent part (see Section 5.2), we optimally select this parameter using a welfare maximization criterion of a regulator. In the following, we separately discuss the effects of a change in the underlying parameters on the equilibrium values. We focus on the effect on the carbon price c∗ p, the investment threshold b∗, as well as the overall output Y(c∗ p, Emax) and the turnover rate T∗among firms. The former can be easily computed via Y(c∗ p, Emax) := Zb∗ 0 y(z;c∗ p, Emax)ν∗(dz) = Zb∗ 0 D(Emax)θzk∗ 1(z, c∗ p, Emax)ν∗(dz),(5.5) where k∗ 1denotes the optimal capital demand as in (5.4). The turnover rate should be understood as the ratio between the entry rate to the mass of incumbent firms in equilibrium. We notice that the latter can be computed via M∗:= Zb∗ 0 ν∗(dz) = N∗Zb∗ 0 f(z)dz, such that the turnover rate writes as T∗=N∗ M∗=1 Rb∗ 0f(z)dz . It is interesting to observe that (5.5) displays an output effect of pollution regulation. As a response to the imposed carbon pricing system, firms decrease their production output in order to comply with the given emission target. More stringent regulation, imposed through a lowered emission limit, thus leads to a decrease in the cumulative output of firms. Sensitivity with respect to the Diffusion Coefficients. We investigate the impact of a change in the coefficients governing the underlying diffusions. Our findings are summarized in Table 2. We note that when the technology growth parameter µ1for polluting firms decreases, the exit threshold decreases correspondingly. This is a logical outcome since the profit function π1for polluting firms increases as technology levels improve. A smaller drift in this context leads to reduced expected operating profits, leading firms to invest earlier in achieving carbon neutrality. Consequently, the exit threshold bis lowered. Moreover, we observe that in equilibrium an increased technology growth among polluting firms results in a rising carbon price. This can be attributed to the fact that as firms experience greater technological advancements, their output and emissions also increase. To counteract this effect, in order to keep the cumulative emissions in balance with the equilibrium condition, a higher equilibrium carbon price encourages firms to opt for lower output levels. GREEN TECHNOLOGY ADOPTION WITH ENDOGENOUS CARBON PRICE 23 Carbon Price Investment Threshold Turnover Rate Overall Output Base case 1.00 32.78 0.0403 2040 µ1= 0.01 0.99 32.72 0.0401 2040 µ1= 0.03 1.01 32.84 0.0406 2040 µ2= 0.01 1.00 32.93 0.0403 2040 µ2= 0.03 1.00 32.62 0.0403 2040 σ1= 0.2 1.00 32.91 0.0404 2040 σ1= 0.25 1.00 33.04 0.0406 2040 σ2= 0.2 1.00 32.77 0.0403 2040 σ2= 0.25 1.00 32.77 0.0403 2040 Table 2. Comparative Statics with respect to the diffusion coefficients We also observe that a higher technology growth rate increases the turnover rate. It is worth noting that this increased turnover rate may have different causes: firms may exit the market due to decreased efficiency, as indicated by their technology shock process falling below zero, or they may become more efficient, leading to a higher number of firms adopting carbon-neutral technology and leaving the market while still operational. In this scenario, we find that, although the effect is minor, the latter factor seems to dominate. Regarding changes in the technology growth parameter µ2for carbon-neutral firms, we find that this has a negligible effect on the carbon price, even though it influences the exit threshold as expected. An increase in the technology growth of carbon-neutral firms boosts the expected profit for firms following their investment. This incentive encourages firms to invest sooner, resulting in a decrease in the exit threshold. Regarding a shift in the volatility of the technology shock process, we observe only minor effects. While a larger volatility of polluting firms’ technology shock process increases the turnover rate and leads to delayed investment, we find no significant effect of a shift in the parameter σ2on the equilibrium values. Sensitivity with respect to the Tax Rates τ1and τ2.We now study the effect of a shift in the corporate tax rates of firms. A potential regulator or legislative body could install different tax rates for polluting and carbon neutral firms, in order to encourage firms to invest earlier into becoming carbon neutral. However, in contrast to direct prohibitions on the use of certain technologies, setting a certain corporate tax level (as well as the carbon price) is an indirect tool that tries to provide incentives for firms to switch to the “right” technology. We note that this regulation policy could be installed by either increasing taxes for polluting firms, or by installing tax benefits (and thus 30 FELIX DAMMANN AND GIORGIO FERRARI a larger expected profit for potential entrants. As seen above, this increases the turnover rate and competition on the market, which results in a larger carbon price and a lower investment threshold. 5.2. A Welfare Maximizing Regulator. Here, we come back to the concept introduced in Remark 2.2, which posits that the overall emissions may be determined by deliberate decisions made by legislative bodies or environmental regulatory agencies. This concept is closely related with the principles underlying a cap-and-trade market framework, wherein a regulatory authority makes strategic decisions regarding the allocation of emission allowances to the market. In the context of our model, the cumulative emissions can be viewed as a regulatory constraint imposed on the collective emissions of firms. Within the literature, a common approach entails the perspective of a “welfare-maximizing” regulator, as in Barrett (2001), Colla et al. (2012), Ulph (1996), and Germain et al. (2004). In this framework, the regulator, when setting a limit on overall emissions, aims to balance the societal benefits derived from emissions reduction and the economic losses incurred due to the constraints imposed. In the mentioned contributions, it is assumed that the regulator maximizes a function encompassing aggregate production, adjusted for capital consumption, and the “cost of pollutioninduced damages”, i.e. max Emax Y(Emax)−rK(Emax)−ΓE1+w max ,(5.6) where Γ, w > 0 weigh the cost of pollution-induced damages (and thus translate a unit of emissions into a monetary unit), Ydenotes the aggregate supply of firms as in (5.5) and K(Emax) = Zb∗(cp,Emax) 0 k∗ 1(z, Emax)ν∗(dz) = Zb∗(cp,Emax) 0 k∗ 1(z, Emax)N∗f∗(z)dz denotes their consumption of capital. The proof of Proposition 4.2 reveals that the equilibrium entry rate N∗is determined via N∗=Emax Rb∗(cp,Emax) 0e(y(z;cp, Emax))f∗(z)dz =Emax λRb∗(cp,Emax) 0y(z;cp, Emax)f∗(z)dz , and hence, in equilibrium, we obtain Y(Emax)−rK(Emax)−ΓE1+w max =Emax Rb∗(cp,Emax) 0y(z;cp, Emax)f∗(z)dz λRb∗(cp,Emax) 0y(z;cp, Emax)f∗(z)dz −EmaxrRb∗(cp,Emax ) 0k∗ 1(z;cp, Emax)f∗(z)dz λRb∗(cp,Emax) 0y(z;cp, Emax)f∗(z)dz −ΓE1+w max GREEN TECHNOLOGY ADOPTION WITH ENDOGENOUS CARBON PRICE 31 Emissions Carbon Price Investment Threshold Turnover Rate Output Base case 102.0 1.00 32.78 0.0403 2040 Γ = 0.0985 101.3 1.01 32.28 0.0404 2026 Γ = 0.0990 100.8 1.02 31.91 0.0406 2016 Table 7. Comparative Statics for the welfare maximizing equilibrium values with respect to the scale parameter Γ =Emax1 λ−rRb∗(cp,Emax) 0k∗ 1(z;cp, Emax)f∗(z)dz λRb∗(cp,Emax) 0y(z;cp, Emax)f∗(z)dz −ΓEw max. Due to the dependencies of the firm’s production y, their capital demand k∗ 1(as in (5.4)), and the investment threshold b∗on the overall emissions Emax, a closed form solution for the maximization problem (5.6) does not seem feasible. To this end, we focus solely on the numerical analysis here. In accordance with Anand and Giraud-Carrier (2020), Pindyck (2002), and Pommeret and Prieur (2013), and Colla et al. (2012), Section 4, we set w= 1 so to obtain a quadratic cost of damages. Our numerical analysis suggests that this indeed guarantees the existence of a maximum in the regulator’s optimization problem (5.6), irrespective of the chosen parameter Γ >0. Clearly, the goals of the regulator are to keep the output at a high level, while minimizing the capital consumption as well as economic damages. These goals, as observed earlier, stand necessarily in conflict as the cumulative output strictly decreases in the cap on overall emissions. We highlight that also other criterions than that of (5.6) are possible to be implemented within the current framework. Including firms cumulative profits, surplus from the net revenue of collecting the carbon price or a measure of consumer surplus are possible ways to extend the proposed objective (see, for example, Anand and Giraud-Carrier, 2020; Krass et al., 2013). Note that the value of overall emissions Emax is now chosen endogenously as well, and thus becomes part of the equilibrium variables. We refer to the equilibrium, that involves the emission target set by the regulator via the optimality criterion (5.6), as the welfare maximizing equilibrium. In order to allow for a more comprehensible study of the latter, and to compare it with the equilibrium values we obtained in the previous comparative statics analysis for fixed overall emissions, we calibrate Γ such that the benchmark welfare maximizing equilibrium is such that E∗ max = 102, as in our base case model. We do not repeat the extensive comparative statics analysis studied before, but mainly focus on the emission related parameters λ, ρ (as studied in Section 5.1) and the scale parameter Γ that measures the cost of damage induced by pollution. 32 FELIX DAMMANN AND GIORGIO FERRARI Emissions Carbon Price Investment Threshold Turnover Rate Output Base case 102.0 1.00 32.78 0.0403 2040 λ= 0.049 102.0 1.02 32.78 0.0403 2081 104.1 0.99 34.41 0.0401 2125 λ= 0.051 102.0 0.98 32.78 0.0403 2000 100.0 1.01 31.33 0.0409 1961 ρ= 0.01 102.0 1.01 32.04 0.0405 2040 102.1 1.01 32.08 0.0405 2042 ρ= 0.03 102.0 0.99 33.53 0.0402 2040 101.9 0.99 33.42 0.0402 2038 Table 8. Comparative Statics with respect to the scale parameter λand ρ. For each parameter, the upper row recalls the equilibrium values for fixed overall emissions, while the lower row states the values in the social welfare maximizing equilibrium. Increasing the pollution damage factor clearly leads to an increased cost for the regulator. In Table 7we observe that the regulator reacts by decreasing the emission target Emax, even though this has the consequence of a lower overall output level. As observed in Section 5.1, decreasing the emission target leads to an increased carbon price, a lower investment threshold and a higher turnover rate. We conclude that a larger (negative) effect of pollution should be addressed by setting a stricter emission limit on the regulated firms. Moreover, we find that the higher the marginal damage, the lower the social welfare, as observed in Figure 5. This intuitive result complements the findings in Colla et al. (2012) and Krass et al. (2013). Next, we discuss the effect of a shift in the scale parameter λ, which measures the carbon intensity of production. Table 8summarizes our findings, where we compare the equilibrium values from the previous sensitivity analysis (for fixed emission target) with those resulting from the welfare optimizing equilibrium. Recall that an increased carbon intensity lead to a decreased overall output level and a lower carbon price, where the latter results from firms endogenously balancing their cost of production. Concerning the welfare-maximizing equilibrium, we note that as carbon intensity rises, the regulator takes action by lowering the emission target. This action can be interpreted as a clear signal from the regulator to polluting firms, indicating consequences for inefficient production facilities. Consequently, the output of polluting firms drops, and as competition among firms intensifies, the regulator’s action reverts the carbon price effect observed in the earlier sensitivity analysis. More precisely, we observe an increasing carbon price in response to the increasing carbon intensity. As a result, firms choose to invest in a cleaner technology at an earlier stage, such that the investment threshold bdeclines and the turnover rate increases. GREEN TECHNOLOGY ADOPTION WITH ENDOGENOUS CARBON PRICE 33 Figure 5. The social welfare for different values of the scale parameters Γ and ρ, respectively. Regarding the scale parameter ρ, we observe the following. Recall that an increase in ρincreases the damage of emissions on the production on both polluting as well as carbon neutral firms. While this decreased expected profits and thus lead to less competition in the equilibrium model with fixed overall emissions, we observe that the regulator’s actions mitigates this effect by lowering the emission target. Hence, the regulator acknowledges that production becomes less efficient when increasing the carbon induced damages, and penalizes polluting firms by reducing their available emissions. We observe that the latter leads to a decreased overall output of polluting firms as well as, as seen in Figure 5, a lower social welfare. 6. Conclusion In this paper, we discuss a model of strategic green technology adoption within a stationary equilibrium framework of competing firms. We succeed in deriving the existence and uniqueness of an endogenous carbon price and a distribution of polluting firms, that achieve a given emission target and keep the economy within a steady state. To this end, our notion of equilibria includes a compliance/consistency condition, that equates net emissions with a predetermined emission target. Within this framework, each firm maximizes profits and chooses an abatement strategy, which is triggered by a level b∗at which it is cost beneficial to switch to a carbon neutral technology. Our general formulation allows the study of several instances of this problem, including different technology shock process, profit functions and installation costs. Hence, we expect that our framework can accommodate other specific problem formulations that allow for the study of parameters not treated in this paper. 34 FELIX DAMMANN AND GIORGIO FERRARI In our case study of Section 5we showcase some of the implications of our model for technology shocks given by arithmetic Brownian motions and a simple AK-model with damage function on production. We observe that pollution regulation via a specified quota (and thus, a carbon price on emissions) affects firms in multiple ways. First, we note that a more stringent regulation leads to a decreased industry output and thus to a larger competition on quantities. Even though firms are faced with a constant elasticity of demand function, which is not affected by the output of competing firms, the competition arises through the stipulated emission target that effectively limits output produced with dirty technology. As a result, the carbon price increases with less emissions available. Second, pollution regulation has a large effect on firms’ abatement strategy. We observe that a carbon pricing system gives incentives to firms to switch once and for all to a carbon neutral technology, which makes them independent from the imposed market price on carbon. Clearly, firms are willing to invest earlier (and thus, for lower values of the underlying technology shock process) when the carbon price increases. Regarding the fixed parameters of the model, we observe that a slowdown of technology growth rate (of dirty technology) or faster technology growth (of carbon neutral technology) leads firms to switch earlier. Moreover, tax benefits for green firms (in form of a decreased corporate tax) as well as subsidies on the installation cost for abatement technology create financial incentives for polluting firms to invest, as reflected in a decreased trigger threshold b∗. In contrast, a policy that increases corporate taxes on polluting firms may yield suboptimal results, as it leads to a decreased market size, less competition and lower force to invest in green technology. The paper also allows the presence of a potential regulator, that aims to optimize its own objective. In our case study we shortly discuss possible consequences arising through the introduction of a welfare maximizing regulator, who optimally sets the emission target in order to weigh off the benefits and damages of polluting production. We find that the regulator may set an optimal emission target that mitigates or reverts the effect of some model’s parameters on the carbon price and the firm’s investment decision. For instance, a larger damage factor of pollution on production leads to less profits of market participants, which leads to less competition, a lower carbon price and a smaller force to become carbon neutral. The regulator mitigates this effect by lowering the emission target and forces firms to invest earlier. GREEN TECHNOLOGY ADOPTION WITH ENDOGENOUS CARBON PRICE 35 7. Acknowledgements The authors would like to thank Michael Barnett, Sara Biagini, Marta Leocata, Frank Riedel, and Luca Taschini for fruitful discussions and helpful suggestions. Moreover, the authors gratefully acknowledge financial support from Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 317210226 - SFB 1283. Appendix A. Technical Assumptions In this Section, we gather some of the assumptions needed for the analysis in the previous sections. Assumption A.1 is concerned with the coefficients in the SDEs (2.1)-(2.2), while Assumption A.3 lists the employed assumptions on the profit functions of the polluting and carbon-neutral firms, as well as the sunk cost and the emissions of the polluting firms. Assumption A.1 (Assumption on technology shock processes).We denote R+= (0,∞). We assume that the state space of the diffusions Zzand Zzare given by I= (x, ∞)and I= (y, ∞), where x, y ≤0. The coefficients µi:R→Rand σi:R→R+,i= 1,2are such that µi∈C1,σi∈C2 and |µi(x)−µi(y)| ≤ Ki|x−y|,|σi(x)−σi(y)| ≤ hi(|x−y|) for some Ki>0,hi:R+→R+strictly increasing, hi(0) = 0 and Z(0,ε) dx h2 i(x)=∞ for all ε > 0and x, y in I, or I, respectively. Remark A.2. We highlight that, a priori, we do not specify whether zero is attainable or unattainable for the diffusions Zand Z. In the forthcoming analysis, we will assume that the firms exit the market whenever their technology shock process falls below the level zero. Clearly, if zero is unattainable for the diffusion, no absorption takes place (consider, e.g., a geometric Brownian motion). On the other hand, it is straightforward to modify the assumption to include an absorption in a point ε > 0. Under the previous assumption, for all x∈ I, respectively I, there exists ε > 0 such that Zx+ε x−ε 1 + |µi(y)| σ2 i(y)dy < +∞, i = 1,2, such that (2.1)-(2.2) have a weak solution that is unique in the sense of probability law (see Karatzas and Shreve, 1991, Chapter 5.5). Since also those solutions are each pathwise unique by the YamadaWatanabe’s Theorem, it follows that (2.1)-(2.2) have unique strong solutions (see Karatzas and Shreve, 1991, Corollary 5.3.23) that are regular in the sense that any point of the interior of their respective state space can be reached in finite time with positive probability. We assume that the boundary points +∞are not attainable for neither of the two processes, i.e cannot be reached in finite time with positive probability. Assumption A.3 (Assumption on involved functions).We assume (i) π1(·;Emax,·)∈C1,1(R+×R+),π2(·;Emax)∈C1(R+),c(·)∈C1(R),e(·)∈C(R); (ii) 0 ≤π1(z;Emax, cp)≤K1(cp),0≤π2(z;Emax)≤K2for some K1(cp), K2>0and all (z, Emax)∈R2 +; (iii) ∂ ∂cpπ1(z;Emax, cp)<0for all z≥0; (iv) limcp→∞ π1(z, cp)=0for all z∈R+; (v) c(z), e(z)>0for all z∈R+, and |c′(0)|<∞. 36 FELIX DAMMANN AND GIORGIO FERRARI (vi) The distribution ξadmits a density function g(·)∈C1. While Assumptions A.3 (i), (v), (vi) deal with technical aspects, we can provide a theoretical foundation for assumptions (ii)-(iv). The second assumption posits that the profit function remains non-negative when considering positive values for technology z, overall emissions Emax, and carbon price cp. Furthermore, we assume that an increase in the carbon price, which firms must pay for each unit of emitted pollutants (which is indirectly linked to production output), adversely affects firm profits. Lastly, we presume that as the carbon price approaches infinity, firms’ profits tend to zero. In simple terms, emitting firms cease production entirely when faced with an exorbitant carbon price, resulting in zero profits. This assumption implies the absence of fixed production costs in our model, although it is straightforward to extent the model in this regard. Appendix B. Single-firm optimal investment problem In this section, we provide a solution to the optimal investment problem (2.4) of the single firm. Before we explain on how to derive a candidate value function and present the proof to Theorem 3.2, we first state the following technical remark. Remark B.1. Using arguments presented in Alvarez (1999), we are able to express the functions Φ1 and Φ2in a purely analytical way. In the following, we let ψ(z)and φ(z)denote the increasing and decreasing, respectively, fundamental solutions to the ordinary differential equation (L−(r+η))u(z) = 0, where Ldenotes the infinitesimal generator associated to the diffusion Z. We recall that the Greenkernel G:I × I → R+of the linear diffusion Zis given by Gr+η(z, y) = Z∞ 0 e−rtp(t;z, y)dt =(W−1ψ(z)φ(y), z < y W−1ψ(y)φ(z), z ≥y, where Wdenotes the constant Wronskian determinant of the fundamental solutions ψ(z)and φ(z), given by W=ψ′(z) S′(z)φ(z)−φ′(z) S′(z)ψ(z)>0, and where S′(z) = exp(−R(2µ1(z)/σ2 1(z))dz denotes the scale density of the diffusion Z. It can then be shown (see Alvarez, 1999) that Φ1(·)as in (2.6)can be rewritten as Φ1(z) = Z∞ 0 G(0,∞) r+η(z, y)π1(y)m′(y)dy, where m′(y)=2/(σ2 1(y)S′(y)) denotes the speed density of Zand G(0,∞) r+η(z, y)the Green-kernel of the constrained process Zthat is killed at 0, given by G(0,∞) r+η(z, y) = (W−1φ(y)ψ(z, 0), z < y, W−1φ(z)ψ(y, 0), z ≥y, where we let ψ(z, 0) = ψ(z)−(ψ(0)/φ(0))φ(z). We note that ψ(z, 0) = (ψ(z),if 0is natural or exit, ψ(z)−ψ(0) φ(0) φ(z),if 0is regular or entrance. For a classification of boundary points we refer to Borodin and Salminen (2015), Chapter 2. Analogously, we let ψ(z)and φ(z)denote the fundamental solutions to the ordinary differential equation (L − (r+η))u(z)=0, where Lis the infinitesimal generator associated to the diffusion Zof (2.2). We can proceed as above and obtain Φ2(z) = Z∞ 0 G(0,∞) r+η(z, y)π2(y)m′(y)dy, GREEN TECHNOLOGY ADOPTION WITH ENDOGENOUS CARBON PRICE 37 where W, S′(z), m′(z), G(0,∞) r+η(x, y)denote the Wronskian, scale density, speed density and Greenkernel of the constrained process Z, respectively. In the following, we derive a suitable candidate for the value function vof (2.4), which we then verify to be the true value function. To begin with, and via standard techniques, we associate the value function with a variational inequality of the form max (L − (r+η)w(z) + π1(z),Φ2(z)−c(z)−w(z)= 0,(B.1) with boundary condition w(0) = 0 and where L:= 1 2σ2 1(z)∂zz +µ1(z)∂zdenotes the infinitesimal generator associated to the diffusion Zof (2.1). Intuitively, equation (B.1) arises through the dynamic programming approach and reflects the two available strategies of the firm at any given point in time: If the firm chooses to invest in the carbon neutral technology, it receives a payoff of Φ2−c. Otherwise, the firm continues producing as a polluting firm, receives the running profit π1, and holds on to the option to invest. Equation (B.1) formalizes this heuristic argument and displays that the firm chooses the maximum between theses two rewards. As usual in optimal stopping problems, the state space then splits in two distinct regions: The waiting and stopping region. While it is optimal to invest immediately in the latter region, the firm should delay investment in the waiting region. In the following, we follow a guess-and-verify approach and conjecture that firms will invest (and thus become carbon neutral) whenever their technology level is sufficiently large. Hence, we guess that there exists a threshold b > 0 such that the stopping time τb:= inf{t≥0 : Zt≥b}(B.2) is optimal. Accordingly, we can relate the above variational inequality (B.1) to the following freeboundary problem                (L − (r+η)w(z) + π1(z)=0,0< z < b, (L − (r+η)w(z) + π1(z)≤0, b ≤z, w(z)=Φ2(z)−c(z), b ≤z, w(z)≥Φ2(z)−c(z),0< z < b, w(0) = 0, (B.3) where we separated the values of technology into the waiting region W= (0, b) and the stopping region S= [b, ∞). We solve the free-boundary problem (B.3) by first recalling that any solution to the ODE (L − (r+η))w(z) + π1(z) = 0 is given by w(z) = Aψ(z) + Bφ(z)+Φ1(z), where ψ(·) and φ(·) are as in Remark B.1. The boundary condition implies Aψ(0)+Bφ(0) = 0, such that w(z) = Aψ(z)φ(0) −φ(z)ψ(0) φ(0) + Φ1(z) =: Aψ(z, 0) + Φ1(z), where ψ(z, 0) is defined as in Remark B.1. For the ease of notation, we now let G(z) := Φ2(z)−Φ1(z)−c(z), and, by imposing smooth-fit and smooth-pasting conditions at the free boundary b, we obtain that the candidate value function takes the form w(z) :=      0, z ≤0, Φ1(z) + G(b) ψ(b,0) ψ(z, 0),0< z < b, Φ2(z)−c(z), z ≥b, (B.4) 38 FELIX DAMMANN AND GIORGIO FERRARI and the optimal stopping threshold bis given by the solution – provided that it exists – to the equation G′(b)[ψ(0)φ(b)−φ(0)ψ(b)] −G(b)[ψ(0)φ′(b)−φ(0)ψ′(b)] = 0. We define A(z) := G′(z)[ψ(0)φ(z)−φ(0)ψ(z)] −G(z)[ψ(0)φ′(z)−φ(0)ψ′(z)] S′(z), where S′(z) denotes the scale density of the diffusion Z. Furthermore, as in Remark B.1, we let m′(z)=2/(σ2(z)S′(z)) denote the speed measure density. We now search for a point bsuch that A(b) = 0. Since by direct computations one has d dz A(z) = m′(z)ψ(0)φ(z)−φ(0)ψ(z)L − (r+η)G(z),(B.5) we can write A(b) = A(0) + Zb 0 d dz A(z)dz, and notice that, under Assumption A.3, we have A(0) = G′(0)φ(0)ψ(0) −φ(0)ψ(0) S′(0) +G(0)φ(0)ψ′(0) −φ′(0)ψ(0) S′(0) =Φ2(0) −Φ1(0) −c(0)W=−c(0)W < 0, where the latter inequality follows from Assumption A.3. We are now ready to prove Theorem 3.2. B.1. Proof of Theorem 3.2.Proof of (i). We establish the result in three steps. Step 1. First, we prove that there exists a unique point b∈R+such that A(b)=0.To this end, we recall that A(b) = A(0) + Zb 0 d dz A(z)dz =−c(0)W+Zb 0 d dz A(z)dz. Since −c(0)W < 0 and A′(z)>0 for all z > ezunder Assumption 3.1, it is sufficient to prove that limb→∞ A(b)=+∞.Straightforward calculations, upon employing the mean value theorem for some point ξ∈(ez, b), yield lim b→∞ A(b) = −c(0)W+ lim b→∞ Zb 0 d dz A(z)dz =−c(0)W+Zez 0 d dz A(z)dz + lim b→∞ Zb ez m′(z)ψ(0)φ(z)−φ(0)ψ(z)L − (r+η)G(z)dz =−c(0)W+Zez 0 d dz A(z)dz + lim b→∞ L − (r+η)G(ξ) r+ηZb ez (r+η)m′(z)(ψ(0)φ(z)−φ(0)ψ(z))dz =−c(0)W+Zez 0 d dz A(z)dz + lim b→∞ (L − (r+η))G(ξ) r+ηψ(0)φ′(b) S′(b)−φ′(ez) S′(ez)−φ(0)ψ′(b) S′(b)−ψ′(ez) S′(ez) →+∞, and the latter follows from Assumption A.3, (L − (r+η))G(ξ)<0 (since ez < ξ), and φ′(b)/S′(b)↓0 as well as ψ′(b)/S′(b)↑+∞(see for example Borodin and Salminen, 2015, Chapter 2). GREEN TECHNOLOGY ADOPTION WITH ENDOGENOUS CARBON PRICE 39 Step 2. Next, we show that the candidate value function w(·) of (B.4) solves the variational inequality (B.1). Let b∈R+denote the unique solution to A(·) = 0, as determined in Step 1. By construction, (L−(r+η))w(z)+π1(z) = 0 on (0, b), while w(z) = Φ2(z)−c(z) on (b, ∞). In order to show that wsolves the variational inequality on R+, it thus remains to show (i) (L − (r+η))w(z) + π1(z)≤0 on (b, ∞) and (ii) w(z)≥Φ2(z)−c(z) on (0, b). Regarding (i), we recall that w(z)=Φ2(z)−c(z) = G(z)+Φ1(z) on (b, ∞). It follows that (L − (r+η))w(z) + π1(z) = (L − (r+η))G(z)<0, where the latter inequality follows from Assumption 3.1 and by construction, since b > ez. Regarding (ii), we recall that w(z) = Φ1(z) + (G(b)/ψ(b, 0))ψ(z, 0) on (0, b). Simple calculations reveal that w(z)≥Φ2(z)−c(z) is equivalent to G(b) ψ(0)φ(b)−φ(0)ψ(b)≤G(z) ψ(0)φ(z)−φ(0)ψ(z), and consequently, it is sufficient to prove that bis a local minimum of the function z7→ G(z)/ (ψ(0)φ(z)−φ(0)ψ(z)) on (0, b).We compute G(z) ψ(0)φ(z)−φ(0)ψ(z)′ =A(z)S′(z) (ψ(0)φ(z)−φ(0)ψ(z))2, which is zero for z=bby construction. Furthermore, G(z) ψ(0)φ(z)−φ(0)ψ(z)′′z=b =A′(b)S′(b) (ψ(0)φ(b)−φ(0)ψ(b))2+A(b)S′(b) ψ(0)φ(b)−φ(0)ψ(b)′ =A′(b)S′(b) (ψ(0)φ(b)−φ(0)ψ(b))2>0, where the latter inequality follows from Assumption 3.1 and b > ez. Hence, bis a local minimum of the function z7→ G(z)/(ψ(0)φ(z)−φ(0)ψ(z)) on (0, b) and the claim follows. Step 3. Last, we verify that the candidate value function windeed coincides with the true value function vof (2.4) and that the threshold btriggers the optimal stopping time τbas in (B.2). Let n∈Nand define σn:= inf{t≥0 : Zz t≥n}.We let τn=τ∧σnfor any stopping time τof the Brownian filtration. Due to our construction of the function w, we can employ Itˆo’s formula to obtain e−(r+η)(τn∧γ1)w(Zz τn∧γ1) = w(z) + Zτn∧γ1 0 e−(r+η)s(L − (r+η))w(Zz s) 1 {Zz s=b}ds +Mτn∧γ1 where γ1is defined as in (2.3) and Mtdenotes the stochastic integral Mt=Zt 0 e−(r+η)sσ1(Zz s)w′(Zz s)dWs, t ≥0. 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