WPS6859 Policy Research Working Paper 6859 Transition to Clean Capital, Irreversible Investment and Stranded Assets Julie Rozenberg Adrien Vogt-Schilb Stephane Hallegatte The World Bank Climate Change Group Office of the Chief Economist May 2014 Policy Research Working Paper 6859 Abstract This paper uses a Ramsey model with two types of capital and a drop in income. In contrast, policy instruments to analyze the optimal transition to clean capital when that focus on redirecting investments—such as feebates polluting investment is irreversible. The cost of climate or environmental standards—prevent underutilization mitigation decomposes as a technical cost of using of existing capital, avoid stranded assets, and reduce clean instead of polluting capital and a transition cost short-term losses; but they reduce emissions more slowly from the irreversibility of pre-existing polluting capital. and increase the intertemporal cost of the transition. With a carbon price, the transition cost can be limited The paper investigates inter- and intra-generational by underutilizing polluting capital, at the expense of distributional impacts and the political acceptability of a loss in the value of polluting assets (stranded assets) climate change mitigation policy instruments. This paper is a product of the Office of the Chief Economist, Climate Change Group. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at jrozenberg@worldbank.org. The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent. Produced by the Research Support Team Transition to Clean Capital, Irreversible Investment and Stranded Assets Julie Rozenberg 1, 2,∗, Adrien Vogt-Schilb 1, 2 , Stephane Hallegatte 2 1 CIRED, 45bis avenue de la Belle Gabrielle, 94736 Nogent-sur-Marne, France. 2 The World Bank, Sustainable Development Network, Washington D.C., USA Keywords: intergenerational equity, energy efficiency standards, mothballing, social and political acceptability, climate mitigation policy JEL classification: L50, O44, Q52, Q54, Q58 For the past centuries, economic growth has involved the accumulation of fossil-fueled capital that releases greenhouse gases (GHG) in the atmosphere (e.g. coal power plants, gasoline-fueled cars). From a global welfare perspec- tive, this accumulation of polluting capital is sub-optimal because it does not internalize the future economic damages caused by climate change. Stabilizing climate change requires near zero emissions in the long-run, and therefore im- plies stopping the accumulation of fossil-fueled capital. Future economic growth thus has to rely on clean capital. In theory, the optimal policy to trigger such a large-scale transition from polluting to clean capital is – in the absence of any other market failure – a Pigouvian price on GHG emissions, for instance a carbon tax (Nordhaus, 1991). However, governments have been timid about the carbon price1 and have relied instead on instruments that redirect investment towards clean capital, such as energy efficiency standards on new capital (such as the corporate average fuel economy (CAFE) standards in the automobile industry, efficiency standards for new buildings, or feebate programs that tax energy-inefficient equipment and subsidize energy-efficient equipment (IEA, 2014). In this paper, we investigate how the transition to clean capital is modified when using such investment-based instruments instead of a carbon price. We model the accumulation of productive clean capital to replace polluting capital, as suggested by Ploeg and Withagen (1991), and we focus on the effect of the irreversibility of past investments on the transition. The transition to clean cap- ital has been studied before, mostly through the lens directed technical change, focusing on the interaction between pollution and knowledge spillovers exter- nalities (e.g., Gerlagh et al., 2009; Grimaud et al., 2011; Acemoglu et al., 2012). ∗ Corresponding author Email addresses: rozenberg@centre-cired.fr (Julie Rozenberg ), vogt@centre-cired.fr (Adrien Vogt-Schilb), shallegatte@worldbank.org (Stephane Hallegatte) 1 Countries that price emissions at a national level are currently limited to the members of the European Union, Switzerland, Kazakhstan, Australia and New-Zealand. Combined with regional or sub-national initiatives, total priced emissions represent less than 10% of global emissions. Papers by Fischer et al. (2004), Williams (2011), Slechten (2013), and Vogt- Schilb et al. (2014), all study the optimal accumulation of clean capital but do not investigate formally implications for existing polluting capital. Here we discuss the pacing of abatement efforts over time and the impact of different policy instruments on the price of existing capital. Our analysis builds on a Ramsey model with two types of capital: “polluting” capital, which creates a negative externality (greenhouse gases emissions), and “clean” capital, which does not. Investment is irreversible, meaning that cap- ital can only disappear through depreciation. Firms may however underutilize existing polluting capital, so that abatement efforts can be divided out between two qualitatively different channels: (i) long-term abatement through accumu- lation of clean capital instead of polluting capital (e.g. people buy electric cars); (ii) immediate abatement through the underutilization (or early-retirement) of polluting capital (e.g. people drive less). We start from a laissez-faire economy, in which marginal productivities of polluting and clean capital are equal. We compare two strategies (carbon price and investment-based instruments) to maintain the concentration of greenhouse gases in the atmosphere below a certain threshold, corresponding to an exoge- nous policy objective such as the UNFCCC 2◦ C target. We find that mitigation costs decompose as a technical cost — using clean instead of polluting capital — and a transition cost due to the irreversibility of pre-existing polluting capital. This irreversibly cost quantifies the regret that society has because of excessive past investment in polluting capital (e.g. having built a coal power plant before the climate mitigation policy has been announced). In the long run both strategies lead to the same steady state, in which most installed capital is clean and GHG concentration is maintained at a constant level. The carbon price and investment-based instruments however induce different trajectories and costs over the short run. A carbon price maximizes inter-temporal welfare. It redirects all investment towards clean capital until polluting capital has depreciated to a level compatible with the long-term climate ceiling. The carbon price also induces a short- term decrease in the market price of existing polluting capital. This market price has three functions. It values polluting assets, it reflects the demand for new polluting capital and it signals the profitability of polluting capital to investors. If the climate constraint is stringent with regards to past polluting capital accumulation, the market price of polluting capital can even reach zero, and part of this capital is decommissioned. These assets that loose all their value because of the policy are often referred to as “stranded assets” (Goulder et al., 2010; Carbon Tracker, 2013). The underutilization or early retirement of polluting capital allows high short-term abatement but has significant impact on production. In contrast, investment-based instruments neither create stranded assets nor provide an incentive to underutilize polluting capital. Quite the contrary: by inducing a scarcity effect on polluting capital, these instruments increase the market price of existing polluting capital (i.e. generating “windfall profits” as in Goulder et al., 2010, when emission allowances are grandfathered instead of auctioned). Investment-based instruments yield a higher irreversibility cost than the carbon price as society keeps using obsolete polluting capital until the end of their lifetime instead of early-scrapping it – as if refusing to recognize that past accumulation of polluting capital was a mistake. Thereby, they are 2 less efficient (in inter-temporal welfare terms) than a carbon price.2 Investment- based instruments lead to a second-best pathway that reaches the same long-run objective as the optimal policy but delays efforts, with lower short-term impacts on output and higher efforts over the medium-run. Our results highlight a trade-off between the optimality of a climate miti- gation policy and its short-term impacts, which may influence political accept- ability. If we compare the instruments in terms of welfare maximization, the carbon tax alone is always the best policy. When looking at criteria such as short-term impacts, however, investment-based instruments may appear prefer- able to some decision-makers and voters. In particular, the impact of the carbon price on asset prices would primarily affect the owners of polluting capital and the workers who depend on them, transforming them into strong opponents to the mitigation policy (Jenkins, In press). Theoretically, lump-sum cash transfers can compensate the losers and tackle the equity issues faced when implementing a carbon tax (Arrow et al., 1996). In practice, however, it may not be feasible to monitor and compensate each indi- vidual loser of climate mitigation policies (e.g., Kanbur, 2010). Another option is to announce a carbon tax in advance to allow economic actors to anticipate it and avoid stranded assets (Williams, 2011), but doing so is made difficult by the governments’ limited ability to commit (Kydland and Prescott, 1977; Helm et al., 2003). Finally, one can use a cap-and-trade system where free emission allowances are distributed based on past emissions (grandfathering) or produc- tion capacity (Goulder et al., 2010). In this paper, we rethink investment-based instruments as a way to avoid stranded assets, therefore easing the political economy of climate mitigation. By spreading the costs over time and economic agents, investment-based instruments may reduce the number of opponents to mitigation policies and make the implementation of the carbon tax easier in the long-run. They however cannot curb emissions as fast as the carbon price can. If governments are not able or willing to implement a carbon tax in the near future and the transition has to be triggered by investment-based instruments for political reasons, their slowness makes their implementation (and enforcement) all the more urgent. The remainder of the paper is structured as follows. Section 1 presents the model and section 2 solves for the laissez-faire equilibrium. In section 3 we analyze the optimal growth path, that can be obtained with a carbon price, and we compare it with investment-based second-best instruments in section 4. In section 5, we study the timing issues of investment-based instruments and risks of lock-in. Section 6 discusses the results and concludes. 1. Model We consider a Ramsey framework with a representative infinitely-lived house- hold, who receives the economy’s production from firms yt , saves by accumu- 2 A large literature explores their drawbacks — such as the rebound effect if lower energy intensity leads to more extensive use of equipment (Goulder and Parry, 2008; Anderson et al., 2011) — and their rationale such as Tsvetanov and Segerson (2013). Parry et al. (In press) and Allcott (2013) find however that estimated mis-perceptions of energy savings are too low to justify CAFE standards in the automobile sector. 3 lating assets3 , receives income on assets at interest rt and purchases goods for consumption ct . Their wealth thus evolves as: ˙ t = rt · at + yt − ct a (1) At time t, consuming ct provides consumers with a utility u (ct ). The utility function is increasing with consumption, and strictly concave (u > 0 and u < 0). The household maximizes intertemporal discounted utility W , given by: ∞ W = e−ρt · u(ct ) dt (2) 0 where ρ is the rate of time preference. Firms produce one final good yt , using two types of available capital: pol- luting capital kp (e.g., coal power plants, thermal engine vehicles) and clean capital kc (renewable or nuclear power, electric vehicles).4 Production is used for consumption (ct ) and investment (ip,t and ic,t ). yt = ct + ip,t + ic,t (3) Investment ip,t and ic,t increase the stock of installed capital, which depreciates exponentially at rate δ :5 ˙ p,t = ip,t − δ kp,t k (4) ˙ c,t = ic,t − δ kc,t k (5) The doted variables represent temporal derivatives. Investment is irreversible (Arrow and Kurz, 1970):6 ip,t ≥ 0 (6) ic,t ≥ 0 (7) This means that for instance, a coal plant cannot be turned into a wind tur- bine, and only disappears through depreciation. However, firms may use only a portion qt of installed capital kt to produce the flow of output yt given by: yt = F (At , qp,t , qc,t ) (8) qp,t ≤ kp,t (9) qc,t ≤ kc,t (10) F is a classical production function, with decreasing marginal productivities, to which we add the assumption that capital can be underutilized. At is exogenous technical progress, and increases at an exponential rate over time. 3 Assets are capital and loans to other households. 4 kp and kc may also be interpreted as intangible capital; for instance, clean capital encom- passes existing clean technologies (e.g. clean electricity production and electrification of the economy) as well as patents, research and development expenses and human capital necessary to develop new clean technologies. 5 We used the same depreciation rate for polluting and clean capital to keep notations simple, but this assumption plays no particular role in the analysis. 6 Following the wording by Arltesou (1999) and Wei (2003) capital is putty-clay. 4 In the remaining of this paper, qt will be called utilized capital and kt in- stalled capital. Although it is never optimal in the laissez-faire equilibrium, the underutilization of installed polluting capital can be optimal when a carbon price is implemented.7 For instance, coal plants can be operated part-time and low-efficiency cars can be driven less if their utilization is conflicting with the climate objective. Polluting capital used a time t emits greenhouse gases (G × qp,t ) which accumulate in the atmosphere in a stock mt .8 GHG atmospheric concentration increases with emissions, and decreases at a dissipation rate ε:9 ˙ t = G · qp,t − εmt m (11) Note that since emissions are a function of polluting capital and capital has a decreasing marginal productivity, the carbon intensity of output increases with the polluting capital stock. 2. Laissez-faire equilibrium In the laissez-faire equilibrium, intertemporal utility maximization leads to a classical arbitrage equation which gives the basic condition for choosing con- sumption over time (Appendix A): c ˙ −u (c) = · (rt − ρ) (12) c c · u (c) As the elasticity of substitution is positive ( − u (c) cu (c) > 0), consumption grows if the rate of return to saving rt is higher than the rate of time preference ρ. Firms rent the services of polluting and clean capital from households at respective rental rates Rp,t and Rc,t . The flow of profit is given by: Πt = F (At , qp,t , qc,t ) − Rc,t · kc,t − Rp,t · kp,t (13) A competitive firm takes Rc,t and Rp,t as given and maximizes its profit by using all installed capital, equalizing at each time t the marginal productivity of polluting and clean capital to their respective rental rates: ∂qb F (qp,t , qc,t ) = Rp,t ∂qg F (qp,t , qc,t ) = Rc,t The classical equilibrium in capital markets in a Ramsey model applies: Proposition 1. In the laissez-faire equilibrium, households are indifferent be- tween investing in polluting or clean capital or lending to other households, so that the marginal productivities of clean and polluting capital net of depreciation are both equal to the interest rate : Rp,t = Rc,t = rt + δ (14) 7 Inthis paper, underutilization of clean capital is never optimal: ∀t, qc,t = kc,t . 8 Inthe remaining of the paper, “carbon” will refer to GHG. 9 The dissipation rate allows maintaining a small stock of polluting capital in the steady state. The linear relation between polluting capital and pollution emission is not a necessary assumption but simplifies the notations. 5 In the next section, we find that the carbon price forces the marginal produc- tivity of polluting capital to be higher than that of clean capital. Also, because investment is irreversible, the relative price of polluting capital decreases during the transition. We then discuss implications for the political economy of climate mitigation policies. 3. Discounted welfare maximization: carbon price In this section, we adopt a cost-effectiveness approach (Ambrosi et al., 2003) and analyze policies that allow maintaining atmospheric concentration mt below a given ceiling m¯: mt ≤ m ¯ (15) This threshold can be interpreted as a tipping point beyond which the environ- ment (and output) can be highly damaged, or as an exogenous policy objective such as the UNFCCC “2◦ C target” (Allen et al., 2009; Matthews et al., 2009). We solve for the welfare maximization program, in which institutions inter- nalize the GHG ceiling constraint. A social planner maximizes intertemporal utility given the constraints set by the economy budget, the capital motion law, investment irreversibility and the GHG ceiling. The same strategy can be decen- tralized by imposing the shadow price of emissions on producers and consumers through an optimal carbon tax or a universal cap-and-trade system (Appendix C). The social planner program is: ∞ max e−ρt · u(ct ) dt (16) c,i,k 0 subject to F (qp , kc ) − ct − ip,t − ic,t = 0 (λt ) ˙ p,t = ip,t − δkp,t k (νt ) ˙ c,t = ic,t − δkc,t k (χt ) ˙ t = G qp,t − εmt m (µt ) mt ≤ m ¯ (φt ) ip,t ≥ 0 (ψt ) qp,t ≤ kp,t (βt ) We indicated in parentheses the co-state variables and Lagrangian multipliers (chosen such that they are positive): λt is the value of income, νt and χt are the prices of polluting and clean capital, and µt is the price of carbon, expressed in terms of utility at time t. The present value Hamiltonian associated to the maximization of social welfare can be found in Appendix B. The main first-order conditions of our problem are (Appendix B.1): u (ct ) = λt = νt + ψt = χt (17) 1 ∂kc F = ((δ + ρ)χt − χ ˙ t) (18) λ 1 βt = ((δ + ρ)νt − ν ˙t) (19) λ ∂qp F = βt + τt · G (20) 6 Where τ is the price of carbon expressed in dollars per ton: µt τt = (21) λt Before the ceiling on atmospheric GHG is reached, a classical result (e.g., Goul- der and Mathai, 2000, footnote 11) is that the carbon price exponentially grows at the endogenous interest rate rt plus the dissipation rate of GHG (Appendix B.3): ¯ =⇒ τ˙t = τt (rt + ε) ∀t, mt < m (22) The steady state is reached when mt = m ¯ . In the steady state, atmospheric ¯ ε/G emissions are stable, implying that polluting capital is constant at kp,t = m ˙ t = 0, eq. 11) and the rest of the economy keeps growing on a balanced (m growth path, thanks to exogenous technical change At . In equations 18 and 19 we recognize the rental rates of clean and polluting capital Rc,t and Rp,t , as defined by Jorgenson (1967): 1 Rc,t := [(δ + ρ)χt − χ ˙ t] (23) λ 1 Rp,t := [(δ + ρ)νt − ν ˙t] (24) λ where χt and νt are respectively the clean and polluting capital shadow prices. The following proposition can be deduced from the first-order conditions: Proposition 2. Along the optimal path, the marginal productivity of clean cap- ital equals the rental rate of clean capital: ∂kc F = Rc,t (25) The marginal productivity of polluting capital is equal to the rental rate of pol- luting capital plus the marginal cost of carbon emissions: ∂qp F = Rp,t + τt G (26) Proof. Equation 25 derives from eq. 18 and 23. Equation 26 is obtained by substituting βt in eq. 20, using eq. 24. In the laissez-faire equilibrium, the marginal productivity of polluting capital was also equal to its rental rate. This is no longer the case when the pollution externality is internalized, as firms have to pay the carbon tax when they use polluting capital. Also, the rental rate of polluting capital Rp,t is no longer equal to that of clean capital, as it is now affected by an irreversibility cost: Proposition 3. Along the optimal path, the interest rate rt that arbitrates be- tween consumption and investment is given by: rt = Rc,t − δ (27) The rental rate of polluting capital can be lower than that of clean capital: Rp,t = Rc,t − pt (28) Where the irreversibility cost pt is the monetary impact of the irreversibility constraint on the rental rate of polluting capital: 1 ˙t ∈ [0, Rc,t ] pt = (ρ + δ )ψt − ψ (29) λt 7 Figure 1: Polluting and clean installed capital, and utilized polluting capital in the first-best optimum. Before t0 , the economy is on the laissez-faire equilibrium. At t0 the carbon price is implemented and polluting capital depreciates until ti (∀t ∈ (t0 , ti ), ib = 0). During this period, polluting capital may be underutilized (qp,t < kp,t ). Polluting investment then starts again, and the steady state is reached at tss . Proof. See Appendix B.2 for eq. 27. Equation eq. 28 is obtained by replacing νt by χt − ψt (eq. 17) in eq. 24. Since Rp,t = βt ≥ 0 (eq. 19), pt = Rc,t − Rp,t ≤ Rc,t . Because investment is irreversible, when the policy is implemented the stock of polluting capital cannot be instantaneously adjusted to its optimal level. Polluting capital therefore becomes overabundant and its rental rate decreases. The irreversibly cost pt quantifies the regret that society has because of excessive past investment in polluting capital (e.g. having built a coal power plant before the climate mitigation policy has been announced). It allows de- composing the shadow price of emissions τt as a “technical” abatement cost (e.g. renewable power plants are more expensive than coal power plants) plus an irreversibility cost:10 ∂qp F − ∂kc F p τt = + (30) G G economic cost technical cost irreversibility cost with p ∈ [0, ∂kc F ] The next proposition shows that an irreversibility cost necessarily appears when the carbon tax is implemented (in t0 ).11 Once polluting capital has ad- justed through natural depreciation, the irreversibility cost is nill. Proposition 4. Two phases can be distinguished during the optimal transition to clean capital: 10 Combining equations 25, 26 and 28. 11 Contrary to (Arrow and Kurz, 1970), who find that the irreversibility constraint is binding if the initial capital is higher than the steady-state level, here the irreversibility constraint is binding for any level of initial polluting capital because of the new constraint on emissions. 8 • First, a phase when the market price of polluting capital is lower than that of clean capital and no investment is made in polluting capital: 0 < pt ≤ Rc,t Rp,t < Rc,t ip,t = 0 • Then, a phase when the market price of polluting and clean capital are equal and polluting investment is strictly positive: pt = 0 Rp,t = Rc,t ip,t > 0 Note that during this phase net investment is negative (when accounting for depreciation) until it is equal to zero in the steady-state state. Proof. Appendix B.4. During the first phase, the irreversibility constraint prevents the economy from transforming polluting capital into either clean capital or consumption and the market price of polluting capital falls below the marginal utility of consumption (eq. 17). The maximum value of the irreversibility cost pt is ∂kc F , the marginal pro- ductivity of clean capital, as the maximum regret cost is the cost of not having invested in clean instead of polluting capital before t0 . Indeed, if pt = ∂kc F , the rental rate of polluting capital falls down to zero (eq. 28), reflecting that polluting capital is overabundant: Proposition 5. During the first phase (when polluting investment is nill) if the carbon price is higher than the marginal productivity of installed polluting capital, polluting capital is underutilized until its marginal productivity equals the carbon price:    pt = Rc,t   R =0 p,t τt G > ∂kp F (kp , kc ) =⇒ (31)    q p,t < kp,t ∂qp F (qp , kc ) = τt G  Proof. Eq. 26 implies that the rental rate of polluting capital Rp,t is the dif- ference between the marginal productivity of polluting capital and the carbon price. As the rental rate of polluting capital Rp,t is equal to the positive multi- plier associated to the capacity constraint βt (eq. 19 and 24), when the carbon price is higher than the marginal productivity of installed polluting capital the rental rate of polluting capital is nill and capital is underutilized. The underutilization of brown capital depends on the ceiling m ¯ , on the initial stock of brown capital kb,t0 and on other parameters of the model such as the functional forms of F and u, on the depreciation rate δ and the preference for the present ρ. As illustrated in Fig. 2, for a given set of functions and parameters 9 Figure 2: Depending on initial emissions (i.e. initial brown capital kb,0 ) and on ¯ ), brown capital is underutilized or not in the first-best the concentration ceiling (m optimum. the underutilization of brown capital happens if initial brown capital is high (right end of the x-axis) and/or if the ceiling is stringent (lower part of the y-axis). During the first phase when the irreversibility cost is positive, if the irre- versibility cost is too high, part of polluting capital can become obsolete and both the acquisition price (νt ) and rental rate (Rp,t ) of polluting capital fall down to zero.12 While all polluting capital looses value in the short-run, this obsolete capital becomes stranded as it is early-scrapped to increase short-term abatement. In this section, we have found that under irreversible investment, society has to live with past mistakes for a while, once it realizes it has been on a non-optimal growth path. A way to limit the associated irreversibility cost is to give up part of installed polluting capital in order to reduce emissions faster (thereby creating stranded assets). In the next section, we find that investment- based policies reduce emissions without affecting existing polluting capital, and therefore increase the social cost of GHG abatement. 4. Investment-based instruments Current climate mitigation policies rarely include carbon prices and rely instead on investment-based instruments such as energy efficiency standards or fiscal incentives for green investment (as feebates, which impose additional fees on polluting capital and rebates for clean capital). These instruments redirect investment towards clean capital but have no effect on the use of existing capital. In this section, we investigate the optimal transition to a clean-capital econ- omy using investment-based instruments. We find that (i) they are less efficient than the first-best carbon tax in terms of inter-temporal welfare maximization, (ii) they allow reaching the same steady state, and (iii) they induce a full utiliza- 12 The strictly positive marginal productivity of utilized polluting capital is transferred to households through the tax revenue τt G. 10 tion of polluting capital in the short run, thereby reducing short-term income losses. A way to trigger the transition to a clean economy is to differentiate invest- ment costs with feebate programs, i.e. fiscal incentives that include subsidies on clean investment (θc,t > 0) and taxes on polluting investment (θp,t > 0). With such a feebate program, πt , the flow of firms’ net receipt at time t is equal to: πt = F (qp,t , qc,t ) − (λt − θc,t ) ic,t − (λt + θp,t ) ip,t (32) Where λt is the price of investments. The optimal values of θc,t and θp,t can be obtained with a maximization of social welfare given the ceiling constraint. The same steady state as in the social optimum is reached (at a date tss,2 which is different than tss,1 in general). On the steady-state the optimal value of θc,t + θp,t is equal to the first-best carbon tax multiplied by the marginal emissions of polluting capital:13 ∀t ≥ tss,2 , θc,t + θp,t = τt,1 · G (33) Investment-based instruments however induce a different short-term transi- tion. Over the short-run, investment in polluting capital stops. However, as firms do not pay carbon emissions directly, it is never optimal to underutilize polluting capital (Appendix D). As a consequence, short-term output may be higher than in the first-best strategy:14 Proposition 6. With a feebate program, short-term output is equal or higher than in the first-best solution with a carbon price. Proof. The first-best carbon price may induce underutilization of polluting capital in the short-run (qp,1,t < kp,t ). In the second-best solution capital is not underused (qp,2,t = kp,t ). As both strategies start with a phase during which investment is nill, installed capital is identical with both policies in an interval ˜). During this interval, utilized polluting capital, hence output, is higher (t0 , t in the second-best strategy. Similarly to the carbon price, investment-based instruments differentiate the marginal productivities of capital (Appendix D): 1 ˙t + 1 (δ + ρ)(θc,t + θp,t ) − (θc,t ˙ ) ˙ + θp,t ∂qp F = ∂kc F − (ρ + δ )ψt − ψ λt λt pt τt,2 G (34) where pt is the irreversibility cost. In this second-best setting the shadow price of carbon τt,2 is still equal to a technical abatement cost plus the irreversibility 13 Note that the same outcome can be reached using taxes on polluting investment alone or subsidies to clean investment alone, since what matters is the sum of the tax plus the subsidy. A tax and a subsidy lead to different transfers in the society, which can play a key role on the acceptability of a particular environmental policy (e.g Sterner and H¨ oglund Isaksson, 2006; Fischer, 2008) 14 Analytically the effect on consumption is ambiguous because it involves the offsetting impacts from a substitution effect and an income effect: short-term output is higher, but investments in clean capital may also increase since the saving rate is endogenous. 11 Figure 3: The shadow price of emissions (or carbon price) is higher with investment- based instruments than with a carbon price. cost: ∂qp F − ∂kc F p τt,2 = + (35) G G economic cost technical cost irreversibility cost The irreversibility cost p is no longer bounded by the marginal productivity of clean capital but by the shadow price of carbon τt,2 (Appendix D). Indeed, preventing underutilization is like refusing to recognize that past accumulation of polluting capital was a mistake. When society keeps using obsolete polluting capital instead of early-scrapping it, the irreversibility cost can be as high as the cost of the carbon emissions that installed brown capital produces. The short-term utilization of obsolete polluting capital leads to a different shadow price of carbon than in the first-best case with a carbon tax: 1 ∀t ∈ Iu,1 , τt,2 − τt,1 = ∂qp F (qp,2 = kp ) − ∂qp F (qp,1 < kp ) + αt (36) G Where Iu,1 is the interval during which capital is underutilized on the first-best pathway and αt = pt − ∂kc F is the extra cost associated to the utilization of obsolete polluting capital on the second-best pathway. This extra-cost can be interpreted as a temporary subsidy on the utilization of polluting capital in the welfare-maximization framework (Appendix E). Figure 3 compares the shadow prices of carbon with the first and second- best policies. Investment-based policies generate a higher emissions shadow price than the carbon tax alone, however the dynamics of capital accumulation mean that the social cost of abatement cannot be translated into consumption losses in a trivial way (see also Vogt-Schilb et al., 2014). Even if investment- based instruments set a higher emissions shadow cost at each time t (Fig. 3), output is higher over the short-run (Prop. 6, Fig. 4). Investment-based policies only differ temporarily from the first-best pathway, in a way that smooths the transition costs: they decrease efforts in the short- run (Prop. 6), leave them unchanged in the long-run (eq. 33), and (therefore) increase efforts in the medium-run (Fig. 4). Moreover, feebate programs induce a different intra-generational distribution of abatement efforts from the carbon 12 Figure 4: On the left, output y in the two cases. In the short-run output is lower in the first-best case because of the adjustment of polluting capital utilization. On the right, consumption c is higher in the second-best case because of a higher output y . tss is the date at which the steady state is reached, it is reached sooner in the second-best case (tss,2 < tss,1 ). tax, since they avoid stranded assets. By preventing new investment in polluting capital, they even increase the value of existing polluting assets: Proposition 7. With a feebate program, the market price of polluting capital is initially higher than the price of clean capital, and than the marginal utility of consumption. Proof. First-order conditions for firms’ receipt maximization give: νt = χt + θc,t + θp,t − ψt (37) where νt is the price of polluting capital and χt is the price of clean capital. The policy creates a scarcity effect on polluting capital, that increases its price while the irreversibility constraint reduces its price in the short-run. Appendix D shows that θc,t + θp,t − ψt ≥ 0. Investment-based instruments are not limited to feebates and may include performance standards for new capital.15 Such performance standards include for instance existing energy-efficiency standards for new vehicles, buildings, and appliances. Proposition 8. In our model the optimal feebate program is equivalent to the optimal performance standard on new capital: (1) they induce the same invest- ment and output pathways and (2) they have the same impact on the price of polluting capital. Proof. Appendix F. As with feebates, performance standards induce a full utilization of existing polluting capital in the short-run and redirect investments towards clean capital. 15 Investment costs can also be differentiated using financial markets, as proposed by Rozen- berg et al. (2013). 13 Figure 5: GHG emissions in the two cases. The carbon price induces spare polluting capital and thus reduces carbon emissions faster in the short-run. They also create scarcity on existing polluting capital and therefore increase the price of polluting capital with regards to clean capital on secondary markets.16 5. Timing of action and carbon-intensive lock-in The utilization of investment-based instruments is limited by their slowness in reducing emissions. Indeed, since they maintain a full utilization of polluting capital in the short-run, investment-based policies result in higher short-term emissions than the carbon tax (Prop. 6 and Fig. 5) and might not be sufficient for stringent climate objectives as regards to past capital accumulation. Figure 6 proposes a visualization of this issue. Starting from low polluting capital stocks (thus low emissions), a carbon tax does not lead to underutiliza- tion of polluting capital and reaching the climate target is possible and optimal without a downward step in income. In this case, the carbon price consistent with the climate target leads to the exact same pathway as investment-based policies. This is a situation of “flexibility” in which a country can chose a pol- luting or a clean development path at low cost, using either a carbon price or investment-based instruments. But as long as climate policies are absent (or very lax), the economy accumu- lates polluting capital, making GHG emissions grow and reducing the residual carbon budget for a given climate target (the arrow “conventional growth”). At one point, the threshold when the marginal productivity of polluting capital is lower than the optimal carbon price is crossed (see eq. 31), meaning that polluting capital should be underutilized and output reduced. From there, a carbon price becomes more difficult to implement because of political-economy constraints. But the alternative option of using investment-based instruments is available, leading to higher inter-temporal costs but no immediate drop in income. There is a window of opportunity, during which alternative investment- based instruments may induce a smooth and acceptable transition to a low- carbon economy. 16 In this model the capital lifetime is endogenous and therefore people cannot extend the lifetime of their polluting capital. 14 Figure 6: Depending on initial emissions (i.e. initial polluting capital kb,t0 ) and on the carbon budget (m ¯ − mt0 ), the carbon tax and investment-based instruments can lead to different or similar outcomes (for a given set of parameters, and in particular ρ and δ ). If the carbon budget is too stringent, such that waiting for polluting capital depreciation is not sufficient, the investment-based instruments cannot be used. If the carbon budget is not stringent, there is no underutilization of polluting capital in the first-best optimum with the carbon tax and investment-based instruments are equivalent. While the economy is on the laissez-faire growth path (red arrow), polluting capital accumulates and the carbon budget is reduced for a given climate objective. If this occasion is missed (right hand side, Fig. 6), it becomes impossible to reach the climate target without underutilization of polluting capital and investment-based options are not available any more (if the climate objective is not revised). In this last area, not only the economic cost of reaching the climate target is higher, but the political economy also creates a carbon lock-in: the only option to reach the climate target requires early-scrapping and thus has a significant short-term cost, making it more difficult to implement successfully a climate policy consistent with the target. The zone in which polluting capital must be underutilized to remain below the ceiling depends on the capital depreciation rate δ , the GHG dissipation rate ε, initial GHG concentration m0 and initial polluting capital k0 . The lower blue line in Fig. 6 is expressed analytically in Appendix G and can be approximated by: G k0 m¯ < m0 + δ According to Davis et al. (2010), the level of existing polluting infrastructure in 2010 is still low enough to achieve the 2◦ C target without underutilizing polluting capital, suggesting that the global economy is not in this last region yet. They find that if existing energy infrastructure was used for its normal life span and no new polluting devices were built, future warming would be less than 0.7◦ C. Yet, reaching the 2◦ C target might imply to stop investing in polluting capital very soon, which depends on our ability to overcome infrastructural inertia and develop clean energy and transport services (Davis et al., 2010; Guivarch and Hallegatte, 2011). Note that Davis et al. (2010) do not discuss whether the least-cost policy would lead to underutilization, that is whether we are in the top or the middle triangle in Fig. 6. 15 6. Discussion Choosing the best instrument in terms of welfare results in choosing the lowest social cost of abatement but not the highest consumption at each time t. There is a trade-off between efficiency (maximum intertemporal welfare), intergenerational equity (distribution of efforts over time) and implementation obstacles (political economy). The carbon tax is the best tool to maximize dis- counted welfare, but public policy is especially difficult in contexts where costs are immediate, concentrated and visible, while benefits are invisible (avoided damages) and diffuse over time and over citizens (Olson, 1977). Policy-makers may use other criteria than social welfare maximization to choose the policy to implement (Beltratti et al., 1994; Chichilnisky et al., 1995). One possible reason why investment-based instruments are preferred by policy-makers is that they give the owners of existing polluting capital some time to adapt to the new economic conditions – without carrying a loss for past decisions – and prevent capital underutilization. Indeed, underutilization of capital may appear as a waste of resources, results in an output drop and creates unemployment. Also, the owners of obsolete polluting capital and the workers whose jobs depend on this capital can be strong opponents to climate policies. Governments may thus be captured by the owners of polluting capital (Laffont and Tirole, 1991) who claim compensations because they invested un- der pre-existing rules and will own stranded assets. Finally, governments may also be captured by individuals who have different time preferences from the social planner’s. Indeed, time preference heterogeneity makes it unappealing for some people to pay now for remote future benefits. This is even more so because future generations are likely to be richer and the ones benefiting from reduced climate change damages. Since investment-based strategies postpone mitigation efforts to the medium-run, they would be preferred by people with high discount rates. Investment-based instruments therefore ease the political economy of the transition to clean capital. While the outcome of such instruments is lower in terms of discounted intertemporal welfare, they have the potential to tackle both the effectiveness (they trigger a transition to clean capital) and the equity (they compensate losers) functions of a climate mitigation policy, as well as inter-generational distributional issues. Our analysis is incomplete and further analyses of the distributional impacts of mitigation instruments could model capital retrofit (an intermediary solution between investing in new clean capital and early-scrapping existing polluting capital) or learning-by-doing and knowledge spillovers (which would improve the productivity of clean capital). We also omitted to consider cases with my- opic agents or limited ability to commit. Nevertheless, our results suggest that investment-based instruments respond to a political acceptability issue as re- gards to climate mitigation policies. 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Appendix A. Maximization of the household’s utility The household maximizes their inter temporal utility (eq. 2) given the motion law of wealth (eq. 1). The present value Hamiltonian is: Hh (ct , at ) = e−ρt · {u(ct ) + λt [rt · at + yt − ct ]} (A.1) where λt is the shadow cost of investment in assets at time t. The first order conditions for a maximum of W are: ∀t, ∂c Hh = 0 ⇒ λt = u (ct ) (A.2) −ρt ∂ (e λt ) ˙ t = (ρ − rt )λt ∀t, ∂a Hh + =0⇒λ (A.3) ∂t The doted variables represent temporal derivatives. Differentiating eq. A.2 with respect to time and substitute for λ from eq. A.3, yields the Euler equation: c˙t −u (ct ) = · (rt − ρ) (A.4) ct ct · u (ct ) 19 Appendix B. Social optimum (section 3) The present value Hamiltonian associated to the maximization of social wel- fare (16) is: Ht = e−ρt · {u(ct ) + λt [F (qp , kc ) − ct − ip,t − ic,t ] + νt [ip,t − δkp,t ] ¯ − mt ] +χt [ic,t − δkc,t ] − µt · [G qp,t − εmt ] + φt · [m + ψt · ip,t + βt [kp,t − qp,t ]} (B.1) All multipliers are positive. The complementary slackness conditions are: ∀t, ψt ≥ 0 and ψt · ip,t = 0 (B.2) ∀t, βt ≥ 0 and βt · (kp,t − qp,t ) = 0 (B.3) ¯ − mt ) = 0 ∀t, φt ≥ 0 and φt · (m (B.4) Appendix B.1. First order conditions First order conditions give: ∂Ht =0⇒ u (ct ) = λt (B.5) ∂ct ∂Ht =0⇒ λt = νt + ψt ∂ip,t ∂Ht =0⇒ λt = χt ∂ic,t ∂Ht ∂ (e−ρt νt ) =− ⇒ −νt δ + βt = −ν ˙ t + ρνt ∂kp,t ∂t ∂Ht ∂ (e−ρt χt ) =− ⇒ λt ∂kc F (kp,t , kc,t ) − χt δ = −χ ˙ t + ρχt ∂kc,t ∂t ∂Ht =0⇒ λt ∂qp F (qp,t , kc,t ) − µt · G = βt ∂qp,t ∂Ht ∂ (e−ρt µt ) = ⇒ ˙ t − ρµt −φt + εµt = µ (B.6) ∂mt ∂t Appendix B.2. Equilibrium on the capital market and interest rate: proof of proposition 3 If we differentiate eq. B.5 with respect to time and substitute λt and λ˙t , we can write: ct · u (ct ) c˙t · = (ρ + δ − Rc,t ) (B.7) u (ct ) ct As in the laissez-faire equilibrium (eq. A.4), the interest rate rt that ensures households are indifferent between consumption and investment is thus given by: rt := Rc,t − δ (B.8) 20 Appendix B.3. Carbon price Eq. B.6 gives the evolution of µt . Using µ˙t = (λ˙t τt + λt τ˙t ) (from eq. 21), eq. B.5, eq. B.7 and eq. B.8 yields: φt τ˙t = τt [ε + rt ] − λt We call tss the date at which GHG concentration reaches the ceiling: ∀t ≥ tss , mt = m ¯ During the steady state, m ˙ t = 0 =⇒ Gqp,t = ε m ¯ (eq. 11). On the long run, installed capital is not underused, polluting installed capital is thus constant at kp,t = m¯ ε/G during the steady state. Before tss , φt = 0 (B.4). The carbon price thus exponentially grows at the endogenous interest rate plus the dissipation rate of GHG until the ceiling is reached: τ˙t = τt [ε + rt ] (B.9) These dynamics may be interpreted as a generalized Hotelling rule applied to clean air: along the optimal pathway, and before the ceiling is reached, the discounted abatement costs are constant over time. The appropriate discount rate is rt + ε, to take into account the natural decay of GHG in the atmosphere. Appendix B.4. The irreversibility constraint is binding in the short run : proof of proposition 4 A binding GHG ceiling is imposed at t0 . Before that, the economy was in the competitive equilibrium, such that clean and polluting capital have the same marginal productivity and installed capital is fully used (Proposition 1): lim qp,t = kp,t (B.10) t→t− 0 lim ∂qp F (qp,t , qc,t ) = ∂kc F (qp,t , qc,t ) (B.11) t→t− 0 We use a proof by contradiction to show that at t+ 0 (when the constraint is in- ternalized) the irreversibility condition is necessarily binding. Suppose that the transition starts with a phase when the irreversibility constraint is not binding, i.e. ψt = 0. This would lead to (Propositions 2 and 3): lim ∂qp F (qp,t , qc,t ) = ∂kc F (qp,t , qc,t ) + τt0 · G (B.12) t→t+ 0 Besides, investment means that capital is a continuous function of time: lim qp,t = kp,t (B.13) t→t+ 0 If the GHG ceiling is binding then τt0 > 0 (eq. B.9). So from eq. B.11 and eq. B.12: lim+ ∂qp F (qp,t , qc,t ) = lim+ ∂qp F (qp,t , qc,t ) (B.14) t→t0 t→t0 ∂qp F is a continuous function of qp,t so eq. B.14 implies that limt→t+ qp,t = 0 limt→t+ qp,t , which is incompatible with eq. B.10 and eq. B.13. 0 21 Appendix C. Decentralized equilibrium with a tax on emissions In a decentralized economy, it is possible to trigger the same outcome as in the social optimum with a lump-sum tax applied to carbon emissions. In this case, the firm’s flow of profit at time t is given by: Πt = F (qp,t , kc,t ) − Rc,t · kc,t − Rp,t · kp,t − τt G qp,t (C.1) With Rp,t and Rc,t the rental prices of polluting and clean capacities respec- tively, and τt the carbon tax. The tax is redistributed through the assets equa- tion: ˙ t = rt · at + yt − ct + τt G qp,t a (C.2) The Lagrangian corresponding to the firm’s maximization program is: L(t) = Πt + βt (kp,t − qp,t ) + γt (kc,t − qc,t ) (C.3) First order conditions are: ∂qg L = 0 ⇒ ∂qc F (qp,t , qc,t ) = γt (C.4) ∂qb L = 0 ⇒ ∂qp F (qp,t , qc,t ) = βt + τt · G (C.5) ∂kg L = 0 ⇒ γt = Rc,t (C.6) ∂kb L = 0 ⇒ βt = Rp,t (C.7) For all t, γt ≥ 0 and γt · (kc,t − qc,t ) = 0 βt ≥ 0 and βt · (kp,t − qp,t ) = 0 (complementary slackness conditions). With eq. C.4 we have γt = ∂qc F (qp,t , qc,t ) > 0, so qc,t = kc,t for all t. The combination of eq. C.4 and eq. C.6 gives ∂kc F (qp,t , kc,t ) = Rc,t Combining eq. C.5 and eq. C.7, we find ∂qp F (qp,t , kc,t ) = Rp,t + τt · G (C.8) In the equilibrium, the rental price of clean capacities is equal to the interest rate (plus delta): Rc,t = rt + δ , because clean capacities and loans are perfect substitutes as assets for households. When the irreversibility constraint is not binding (see eq. 6), and in particular on the balanced growth path, the rental rate of polluting capacities is equal to the interest rate as well and Rp,t = Rc,t = rt + δ . However, when the carbon price in implemented at t0 , the irreversibility con- straint is binding (4). In this case, since the use of polluting capacities suddenly becomes too expensive, the rental rate of polluting capacities is endogenously reduced. As a consequence of a lower rate of return for owners of polluting cap- ital, households stop investing in polluting capacities. If the carbon tax is very high, the rental rate of polluting capacities can even become nil and polluting capacities may be under-utilized. 22 Appendix D. Firms’ maximization problem with differentiation of in- vestment costs The present value Hamiltonian associated to the firm’s maximization pro- gram is: Ht = e−ρt · {F (qp,t , qc,t ) − (λt − θc,t ) ic,t − (λt + θp,t ) ip,t +νt [ip,t − δkp,t ] + χt [ic,t − δkc,t ] + ψt · ip,t + βt [kp,t − qp,t ]} First order conditions give: ∂Ht =0⇒ λt + θp,t = νt + ψt ∂ip,t ∂Ht =0⇒ λt − θc,t = χt ∂ic,t ∂Ht ∂ (e−ρt νt ) =− ⇒ − ν t δ + β t = −ν ˙ t + ρνt ∂kp,t ∂t ∂Ht ∂ (e−ρt χt ) =− ⇒ ˙ t = λt ∂kc F (kp,t , kc,t ) − χt δ ρχt − χ ∂kc,t ∂t ∂Ht =0⇒ λt ∂qp F (qp,t , kc,t ) = βt (D.1) ∂qp,t The complementary slackness condition ∀t, βt [kp,t − qp,t ] = 0 combined with equation D.1 gives that — if F satisfies the Inada conditions — capital is never underused with investment-based instruments ∀t, kp,t = qp,t . FOCs can be reduced to: νt + ψt = χt + θc,t + θp,t (D.2) 1 ∂kc F = ((δ + ρ)χt − χ ˙ t) (D.3) λ 1 ∂qp F = ((δ + ρ)νt − ν ˙t) (D.4) λ We thus obtain 1 ˙ ) − 1 (ρ + δ )ψt − ψ ˙ + θp,t ˙t ∂qp F = ∂kc F + (δ + ρ)(θc,t + θp,t ) − (θc,t λt λt θt pt (D.5) With pt the irreversibility cost and θt a positive term that depends on (θc,t +θp,t ). Equation D.5 is similar to eq. E.4 with θt = τt,2 G, where τt,2 is the shadow price of carbon. In the optimal pathway with a full-utilization of capital, θt is therefore equal to the shadow price of carbon (multiplied by G). In this setting under-utilizing polluting capital is never optimal because firms do not pay carbon emissions directly. Instead, investment in polluting capital is more expensive that investment in clean capital and over the short-run, as in the social optimum the economy does not invest in new polluting capital. Once polluting capital has depreciated to a level compatible with the GHG ceiling, polluting investments become profitable and start again. 23 The policy creates a scarcity effect on polluting capital, that increases its price (θc,t + θp,t , eq. D.2) while the irreversibility constraint reduces its price in the short-run (ψt , eq. D.2). Along the optimal transition to the new long-term steady state, ∂kc F ≤ ∂qp F ⇔pt ≤ θt (= τt,2 ) (D.6) ⇔ψt ≤ θc,t + θp,t so that the price of pre-existing polluting capital is higher than that of clean capital is the short-run. In the steady state, the irreversibility cost is nill (p = 0) and the marginal productivity of polluting capital is equal to that of clean capital plus θt . The same steady state as in the social optimum is reached and the optimal value of θt is equal to the first-best carbon tax multiplied by the marginal emissions of polluting capital: ∀t ≥ tss , θt = τt · G with tss the date at which the steady state is reached. With investment-based instruments, the shadow price of emissions τt,2 is still equal to a technical abatement cost plus the irreversibility cost: ∂qp F − ∂kc F p τt,2 = + (D.7) G G economic cost technical cost irreversibility cost with p ∈ [0, τt,2 ] The irreversibility cost p is now bounded by the shadow carbon price τt,2 (eq. D.6). One interpretation is that preventing under-utilization is like re- fusing to recognize that past accumulation of polluting capital may have been a mistake. By doing so, the irreversibility cost can be has high as the cost of the GHG emissions that installed brown capital produces. Appendix E. Maximization of social welfare with full utilization con- straint: temporary subsidy on existing polluting capital The same outcome as with feebates or standards can be reached with the same social planner program as in Appendix B and a full-utilization constraint: ∞ max e−ρt · u(ct ) dt (E.1) c,i,k 0 subject to F (qp , kc ) − ct − ip,t − ic,t = 0 (λt ) ˙ p,t = ip,t − δkp,t k (νt ) ˙ c,t = ic,t − δkc,t k (χt ) ˙ t = G qp,t − εmt m (µt ) mt ≤ m ¯ ( φt ) ip,t ≥ 0 (ψt ) qp,t ≤ kp,t ( βt ) qp,t = kp,t (αt ) 24 The present value Hamiltonian associated to the maximization of social wel- fare is: Ht = e−ρt · {u(ct ) + λt [F (qp , kc ) − ct − ip,t − ic,t ] + νt [ip,t − δkp,t ] ¯ − mt ] +χt [ic,t − δkc,t ] − µt · [G qp,t − εmt ] + φt · [m + ψt · ip,t + βt [kp,t − qp,t ] + αt [qp,t − kp,t ]} All multipliers are positive. Equations 19 and 20 become: 1 βt − αt = ((δ + ρ)νt − ν˙t) λ ∂qp F = βt − αt + τt · G The rental price of polluting capital is therefore equal to βt − αt . The condition on the marginal productivity of polluting capital becomes: ∂qp F = βt − αt + τt · G (E.2) Note that due to complementary slackness conditions, if βt > 0 then αt = 0 and if αt > 0 then βt = 0. In the first phase when polluting investment is nil, if the carbon tax is higher than the marginal productivity of the last unit of polluting capital, the value of polluting capital is nil, βt = 0 and the equation becomes: ∂qp F = −αt + τt · G (E.3) αt is a subsidy to the utilization of polluting capital. Similarly to the first-best pathway, the marginal productivities are differentiated as follows: ∂qp F = ∂kc F − pt + τt G (E.4) 0 < pt < τt G With the irreversibility cost pt > 0 during the first phase and pt = 0 when polluting capital reaches a sustainable level. In the long run when ib > 0 the equilibrium is equivalent to the social optimum. In the short run when ib = 0, ψt > 0 and Rp,t < Rc,t , except that in this case Rp,t becomes negative if the carbon price is higher than the marginal productivity of the last unit of polluting capital (expressed in output per emissions). Thus polluting capital is always fully-utilized. This instrument leads to the same investments and output as the differen- tiation of investment costs or standards, however it is not perfectly equivalent. Indeed, the carbon tax also affects polluting capital on the secondary markets, thus the price of polluting capital decreases in the short-run. Conversely, with taxes on investments or standards on investments, polluting capital becomes scarce and so its price increase on the secondary market. An instrument perfectly equivalent to the tax plus subsidy would be to differentiate capital costs, that is to tax both polluting investment and exchanges on the secondary market. 25 Appendix F. Investment regulation (performance standards) Another equivalent possibility is to regulate polluting investment through efficiency standards. In particular, the most polluting investments can be for- bidden. Here, we crudely impose polluting investments to be nil until polluting capital has depreciated to a level allowing to reach the carbon ceiling without under-utilizing polluting capital. We come back to the social planner’s program (beginning of section 3) and remove the concentration and ceiling constraints (eq. 11 and eq. 15). We can also remove the irreversibility constraint (eq. 6) which will not be binding in this case. Instead, we add a polluting investment constraint that forces ip,t to be equal to a standard at each point in time, and we call σt its Lagrangian multiplier: ∀t, ip,t = sdt ( σt ) (F.1) The standard sdt can be optimally set to equal polluting investments found in the previous section and the next section. Basically, sdt = 0 until polluting capacities have depreciated to a level compatible with the ceiling. The present value Hamiltonian associated to the maximization of social welfare is: Ht = e−ρt · {u(ct ) + λt [F (qp , kc ) − ct − ip,t − ic,t ] + νt [ip,t − δkp,t ] +χt [ic,t − δkc,t ] + σt · (sdt − ip,t ) + βt [kp,t − qp,t ]} (F.2) λt is the current value shadow price of income. νt and χt are the current shadow values of investments in polluting and clean capital. First order conditions can be reduced to the following equations: u (ct ) = λt = νt − σt = χt (F.3) λt ∂kc F = (δ + ρ)χt − χ ˙t (F.4) λt ∂qp F = βt (F.5) βt = (δ + ρ)νt − ν ˙t (F.6) Here, σt is equivalent to (θc,t + θp,t − ψt ) in the previous section. The maximization of intertemporal welfare results in the same equations as in the previous sections: Rp,t = Rc,t + nt (F.7) 1 with nt = ((ρ + δ )σt − σ˙t ) λt This equation is equivalent to Eq. D.5, with nt = θt − pt . The variable nt is positive, which means that the rental price of polluting capacities is higher than the interest rate. Indeed, as with the differentiation of investment costs the polluting investment standard creates a scarcity effect on polluting capital, which becomes more expensive than clean capital. This instrument must be thought of as temporary, since once polluting cap- ital has depreciated to a sustainable level, a carbon price can be implemented without inducing under-utilization of polluting capital, and thus becomes po- litically acceptable. Investment regulation can be compared with existing effi- ciency standards on cars or electric plants, that forbid the construction of the most polluting kinds of polluting capital. 26 Appendix G. Second-best infeasibility zone This zone defines the cases when the ceiling is reached before polluting ca- pacities have depreciated to a sustainable level. If no investment is made in polluting capacities, we have: kp,t = k0 e−δt Therefore, the stock of pollution follows this dynamic: ˙ = k0 e−δt − ε m m The solution to this differential equation is: G k0 −δt G k0 mt = − e + m0 + e−εt δ−ε δ−ε This function first increases to a maximum mmax = Gδk0 e−δt and then decreases. The maximum date is 1 mmax ε tmax = − ln( ) δ G k0 The expression of m at the maximum date gives the limit of the infeasibility ¯: zone if mmax = m G k0 ln( G m¯ ε k0 ) G k0 ε ¯ ε m ¯ =− m e + m0 + e δ ln( G k0 ) δ−ε δ−ε This can be rewritten: δ ε δ− G k0 ε δ δ−ε ¯ = m m0 + δ−ε G k0 δ The “clean incentives infeasibility zone” depends on the capital depreciation rate, the GHG dissipation rate, initial GHG concentration and initial polluting capacities. 27