WPS6609 Policy Research Working Paper 6609 How Capital-Based Instruments Facilitate the Transition Toward a Low-Carbon Economy A Tradeoff between Optimality and Acceptability Julie Rozenberg Adrien Vogt-Schilb Stephane Hallegatte The World Bank Sustainable Development Network Office of the Chief Economist of the September 2013 Policy Research Working Paper 6609 Abstract This paper compares the temporal profile of efforts to differ during the transition phase. The carbon price curb greenhouse gas emissions induced by two mitigation maximizes social welfare but may cause temporary strategies: a regulation of all emissions with a carbon under-utilization of brown capital, hurting the owners price and a regulation of emissions embedded in new of brown capital and the workers who depend on it. capital only, using capital-based instruments such as Capital-based instruments cause larger intertemporal investment regulation, differentiation of capital costs, or welfare loss, but they maintain the full utilization of a carbon tax with temporary subsidies on brown capital. brown capital, smooth efforts over time, and cause lower A Ramsey model is built with two types of capital: brown immediate utility loss. Green industrial policies including capital that produces a negative externality and green such capital-based instruments may thus be used to capital that does not. Abatement is obtained through increase the political acceptability of a carbon price. More structural change (green capital accumulation) and generally, the carbon price informs on the policy effect possibly through under-utilization of brown capital. on intertemporal welfare but is not a good indicator Capital-based instruments and the carbon price lead to estimate the impact of the policy on instantaneous to the same long-term balanced growth path, but they output, consumption, and utility. This paper is a product of the Office of the Chief Economist, Sustainable Development Network. 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 rozenberg@centre-cired.fr or shallegatte@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 How capital-based instruments facilitate the transition toward a low-carbon economy A tradeoff between optimality and acceptability Julie Rozenberg 1,∗, Adrien Vogt-Schilb 1 , Stephane Hallegatte 2 1 CIRED, Nogent-sur-Marne, France. 2 The World Bank, Sustainable Development Network, Washington D.C., USA Keywords: climate mitigation, intergenerational equity, energy efficiency standards JEL classification: L50, O44, Q52, Q54, Q58 Introduction For the past centuries, the global economy has been on a sub-optimal growth path in the sense that it did not internalize future damages caused by the release of greenhouse gases (GHG) into the atmosphere. These emissions are embedded in installed capital — such as fossil-fueled power plants, internal combustion engines in passenger vehicles, heat production — and infrastructure patterns such as transport networks or city density. In order to limit climate change damages, the international community has committed through the UNFCCC to maintain global temperature increase below two degrees compared to the pre-industrial climate, and this requires a drastic reduction of GHG emissions globally (IPCC, 2007). Doing so in a welfare-maximizing way requires internalizing the external- ity (damages from greenhouse gases emissions) with a global carbon price, e.g. through a carbon tax or a cap-and-trade system. A carbon price can, in par- ticular, induce a switch from carbon-intensive capital to clean capital such as renewable electricity, electric vehicles, rail transportation and insulated build- ings. It can also induce people to reduce the utilization of their polluting capital (Schwerin, 2013). However, the carbon price does not seem to be society’s preferred instrument, and up to now governments have been implementing incentives in favor of green capital1 , such as energy efficiency standards on new capital (e.g. CAFE stan- dards in the US or direct regulation for new buildings and home appliances) or fiscal incentives (see OECD, 2009, for vehicles). This alternative strategy gives firms and households the opportunity to make investments consistent with the turnover of their capital stock, that is to keep using existing capital until it ∗ Corresponding author Email addresses: rozenberg@centre-cired.fr (Julie Rozenberg ), vogt@centre-cired.fr (Adrien Vogt-Schilb), shallegatte@worldbank.org (Stephane Hallegatte) 1 At the exception of the EU-ETS and a few states that have implemented a carbon tax. — World Bank working paper — September 17, 2013 depreciates, while investing in cleaner new capital. Put differently, instead of regulating all GHG gases in the economy — as a carbon price would do — these policies focus on emissions embedded in new capital only. To give some insights on why such policies seem to be more politically accept- able than a carbon price, we compare analytically a carbon price with “capital- based” instruments, i.e. instruments that focus on capital instead of emissions (e.g. standards or subsidies on investments). Some studies compare the effi- ciency of such instruments (e.g, Fischer and Newell, 2008; Goulder and Parry, 2008), however, they do not explicitly consider the intertemporal distribution of abatement efforts nor model capital. Vogt-Schilb et al. (2012) model abatement through the deployment of green capital and find that abatement efforts are concentrated over the short term in response to a carbon tax. In this paper, we investigate how the intertemporal distribution of abatement efforts is modified when using alternative mitigation instruments. We use a simple Ramsey model with two types of capital, as proposed by Ploeg and Withagen (1991): “brown” capital, which creates a negative ex- ternality (greenhouse gases emissions), and “green” capital, which does not. Reducing emissions can be done through two channels. First, through a substi- tution between brown and green capital, i.e. structural change. This option is slow because it requires capital accumulation in the green sector. Moreover, as investment is supposed irreversible, brown capital can only disappear through depreciation. Second, it is possible to instantly reduce emissions through under- utilization of brown capital, i.e. through a contraction of the output volume. This option allows unlimited short-term abatements. Starting from a laissez-faire equilibrium in which capital is fully utilized and the marginal productivities of brown and green capital are equal, we model a social planner who decides to maintain the concentration of greenhouse gases in the atmosphere below a certain threshold. Two strategies are compared to com- ply with the ceiling. The first strategy uses a price on carbon emissions, that regulates all GHG emissions. The second strategy regulates emissions from new capital only, using three different capital-based instruments that are equivalent in terms of investments: (i) the carbon tax is completed by a temporary subsidy on brown capital; (ii) the capital cost of brown and green capital are differen- tiated, e.g. through subsidies for green investment; (iii) brown investments are temporarily regulated. In the long run the carbon price and the capital-based instruments lead to the same balanced growth path, in which the marginal productivity of brown capital is higher than that of green capital; this compensates for its higher cost, as using brown capital increases GHG atmospheric concentration. The two sets of strategies however induce different trajectories over the short run. A carbon price yields the first-best optimum pathway, that includes an ad- justment of brown capital utilization in the short-run. The partial utilization of brown capital has significant short-term impact on production and possibly con- sumption. Practically, this impact would primarily affect the owners of brown capital and the workers who depend on them. With capital-based instruments, total discounted welfare is lower than in the optimum, but output is higher over the short-run because brown capital is used at full capacity even during the transition. These capital-based instruments thus smooth out the transition toward a low-carbon economy, and can make more acceptable a carbon price in the longer run. 2 Our results do not depend on the social cost of carbon, as we model a social planner who imposes the optimal carbon ceiling. Comparing two sets of miti- gation strategies, they highlight a trade-off between the optimality of a climate mitigation policy and its short-term impacts, which influence implementation ease. If we compare the instruments in terms of welfare maximization, the car- bon tax alone is always the best policy. However, when looking at other criteria such as short-term impacts or ease of implementation of the policy, second-best strategies might appear preferable to many decision-makers. Indeed, strategies that focus on new capital or that subsidize temporarily polluting capital allow reaching the same long-run objective as the optimal policy but delay efforts, with lower short-term impacts on output and higher efforts over the medium- run. Since capital-based instruments postpone mitigation efforts compared with the first-best strategy, they induce higher marginal abatement costs, but they would be preferred by individuals with higher discount rates than the social planner. Capital-based instruments, since they stimulate the new green sector or tem- porarily accompany the brown declining sector, are industrial policies. For in- stance, a temporary subsidy to obsolete brown capital is comparable to Japanese “sunset” industrial policies, that supported declining traditional sectors in the middle of the 20th century (Beason and Weinstein, 1996). On the other hand, a subsidy to green investment is similar to industrial policies that help “sun- rise” green industries become competitive. Even though the usual reasons for implementing industrial policies — market failures, increasing returns in the green sector — are not modeled here, our paper brings an additional argument in favor of complementary industrial policies to smooth the transition towards a low-carbon economy. 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 capital-based second-best instruments in section 4. Section 5 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- lating assets2 , receives income on assets at interest rt and purchases goods for consumption ct . At time t, consuming ct provides consumers with a utility u (ct ). The utility function is increasing with consumption, and strictly concave (u (c) > 0 and u (c) < 0). The household maximizes their intertemporal discounted utility W , given by: ∞ W = e−ρt · u(ct ) dt (1) 0 where ρ is the rate of time preference. Firms produce one final good, using two types of available capital: brown capital 2 Assets are capital and loans to other households. 3 kb and green capital kg . Green capital encompasses existing green technologies as well as patents, research and development expenses and human capital nec- essary to develop new green technologies. Firms may use only a portion qt of installed capital kt to produce the flow of output yt given by: yt = F (At , qb,t , qg,t ) (2) qb,t ≤ kb,t (3) qg,t ≤ kg,t (4) F is a classical production function, with decreasing marginal productivities,3 to which we add the assumption that capital can be under-utilized. At is ex- ogenous technical progress, and increases at an exponential rate over time. In the remaining of this paper, qt will be called utilized capital and kt installed capital. Although it is never optimal in the laissez-faire equilibrium, the under- utilization of installed capital can be optimal when a carbon price is imple- mented4 . 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. Production is used for consumption (ct ) and investments (ib,t and ig,t ). yt = ct + ib,t + ig,t (5) Investment ib,t and ig,t increase the stock of installed capital, which depreciates exponentially at rate δ : ˙ b,t = ib,t − δ kb,t k (6) ˙ g,t = ig,t − δ kg,t k (7) The doted variables represent temporal derivatives. Investment is irreversible (Arrow and Kurz, 1970): ib,t ≥ 0 (8) ig,t ≥ 0 (9) This means that for instance, a coal plant cannot be turned into a wind turbine, and only disappears through depreciation. Brown capital used a time t emits greenhouse gases (G × qb,t ) which accumulate in the atmosphere in a stock mt . GHG atmospheric concentration increases with emissions, and decreases at a dissipation rate5 ε: m˙ t = G · qb,t − εmt (10) In the following section, we solve for the laissez-faire equilibrium. In the last two sections, we adopt a cost-effectiveness approach (Ambrosi et al., 2003) and 3 We assume decreasing returns to scale even in the green sector but a further extension of this work will be to assume increasing returns to scale in the short-run. 4 In this paper, under-utilization of green capital is never optimal so q g,t = kg,t . 5 The dissipation rate allows maintaining a small stock of brown capital in the balanced growth path. 4 analyze policies that allow maintaining atmospheric concentration mt below a given ceiling m¯ , a proxy for the increase in global temperature (Meinshausen et al., 2009): mt ≤ m ¯ (11) This threshold can be interpreted as a tipping point beyond which the environ- ment (and output) can be highly damaged. It can also be interpreted as an exogenous policy objective such as the UNFCCC “2C target”. 2. Laissez-faire equilibrium The laissez-faire equilibrium leads to classical results of a Ramsey model with two types of capital. Proposition 1. In the laissez-faire equilibrium, the marginal productivities of green and brown capital are equal. Consumption grows as long as the marginal productivity of capital — net of depreciation — is higher than the rate of time preference. Proof. Firms rent the services of capital from households, who own it. We denote Rb,t and Rg,t the rental prices of a unit of brown and green capital respectively. A firm’s total cost for capital is Rg,t · kg,t + Rb,t · kb,t . The firm’s flow of profit at time t is given by: Πt = F (At , qb,t , qg,t ) − Rg,t · kg,t − Rb,t · kb,t (12) A competitive firm, which takes Rg,t and Rb,t as given, maximizes its profit by using all installed capital and by equalizing at each time t the marginal productivity of brown and green capital to their respective rental prices: ∂qb F (qb,t , qg,t ) = Rb,t ∂qg F (qb,t , qg,t ) = Rg,t Since capital depreciates at the constant rate δ , the net rate of return to the owner of a unit of brown or green capital is respectively Rb,t − δ and Rg,t − δ .6 We model a closed economy, thus the assets owned by the households are installed capital, or loans to other households at rate rt . At equilibrium, households should be indifferent between investing in brown or green capital, or lending to other households, so that Rb,t = Rg,t = rt + δ (13) When solving for households’ utility maximization, we find the Euler equation, that gives the basic condition for choosing consumption over time (see Appendix A): ˙ c u (c) =− · (rt − ρ) (14) c c · u (c) 6 We implicitly assumed that the price of capital in units of consumables is 1, but this will not always be the case when the GHG ceiling is introduced. 5 u (c) The intertemporal elasticity of substitution is positive (− cu (c) > 0) so con- sumption grows if the rate of return to saving rt (i.e. the marginal productivity of capital, net of depreciation) is higher than the rate of time preference. The interest rate rt is the rate that converts future consumption into a current con- sumption that is equivalent in terms of social welfare, and equals the rate of return to consumption. If the interest rate equals the rate of time preference, consumption is constant over time. As a consequence of Proposition 1, if the output elasticity of brown capital is higher than that of green capital, the ratio of brown capital over green capital is higher than one. In other words, if using brown capital is more productive than using green capital, firms will invest more in brown capital. 3. Discounted welfare maximization: Carbon price In this section, we solve for the welfare maximization program, in which institutions impose the social cost of emissions on producers and consumers (e.g. an optimal carbon tax, a universal cap-and-trade system) in order to internalize the GHG ceiling constraint. A social planner maximizes intertemporal utility7 given the economy budget constraint (eq. 5), the capital motion law (eq. 6 and eq. 7), the irreversibility constraint (eq. 8) and the GHG ceiling constraint (eq. 10 and eq. 11). This latter constraint increases the social cost of brown capital, which is the source of the externality. Brown capital may thus be under-utilized in order to instantly reduce GHG emissions. It is however always nonoptimal to under-utilize green capital. Therefore, to keep the model simple, we model these two features (eq. 3 and eq. 8) for brown capital only. The social planner program is: ∞ max e−ρt · u(ct ) dt c,i,k 0 subject to F (qb , kg ) − ct − ib,t − ig,t = 0 (λt ) ˙ b,t = ib,t − δkb,t k (νt ) ˙ g,t = ig,t − δkg,t k (χt ) m˙ t = G qb,t − εmt (µt ) mt ≤ m ¯ (φt ) ib,t ≥ 0 (ψt ) qb,t ≤ kb,t (βt ) The variables in parentheses are the co-state variables and Lagrangian multi- pliers associated to each constraint. λt is the current value of income. νt and χt are the current values of brown and green capital. µt is the current cost of pollution in the atmosphere, expressed in terms of undiscounted utility at time t. We define τt as the current price of GHG, expressed in units of consumables. µt τt = (15) λt 7 The same optimal pathway can be obtained with a lump-sum carbon tax on GHG emis- sions, as it is shown in Appendix D. 6 Figure 1: Brown and green installed capital, and utilized brown capital in the first- best optimum. Before t0 , the economy is on the laissez-faire equilibrium. At t0 the carbon price is implemented and brown capital depreciates until ti (ib = 0). During this period, brown capital may be under-utilized (qb,t < kb,t ). Brown investments then start again, and the balanced growth path is reached at tb . We call tb the date at which GHG concentration reaches the ceiling8 : ∀t ≥ tb , mt = m ¯ . We show in Appendix B.1 that the carbon price exponentially grows at the endogenous interest rate plus the dissipation rate of GHG until the ceiling is reached. τ˙t = τt [ε + rt ] (16) When the ceiling is reached, mt = m ¯ , brown installed capital is constant at ¯ ε/G and the economy is on the balanced growth path. kb,t = m We find that three phases can be distinguished once the carbon price has been implemented (eq. B.7 and Fig. 1): a phase during which the irreversibility constraint is binding and brown investment is nil (between t0 and ti in Fig. 1), a phase during which brown investment is strictly positive (between ti and tb in Fig. 1), and the balanced growth path (after tb ). The phase when ib,t = 0 may be separated into two different phases (eq. B.8): a phase with full utilization of brown capital and a phase with under-utilization. In Fig. 1 brown capital is always under-utilized when ib,t = 0. The main first-order conditions of our problem are (Appendix B): u (ct ) = λt = νt + ψt = χt (17) 1 ∂kg F = (δ + ρ)χt − χ ˙t (18) λ 1 ∂qb F = (δ + ρ)νt − ν ˙ t + τt · G (19) λ As in Jorgenson (1967), the rental prices of green and brown capital Rg,t and 8 We assume that m = m ¯ on the interval [tb , +∞[, which is compatible with usual func- t tional forms, like the ones we use here for numerical illustrations. 7 Rb,t follow: 1 Rg,t = [(δ + ρ)χt − χ˙ t] (20) λ 1 Rb,t = [(δ + ρ)νt − ν˙t] (21) λ where χt and νt are respectively the marginal green and brown capital prices. We can thus deduce the following proposition from the first-order conditions. Proposition 2. Along the optimal path, the marginal productivity of green cap- ital is equal to the rental price of green capital, which is equal to the interest rate plus the depreciation rate. ∂kg F = Rg,t = rt + δ (22) Along the optimal path, the marginal productivity of brown capital must be equal to the rental price of brown capital plus the carbon price τt multiplied by the marginal emissions of production G. ∂qb F = Rb,t + τt G (23) Proof. See Appendix B.2. In the laissez-faire equilibrium, the interest rate was equal to both the net marginal productivity of green and brown capital, as shown in Proposition 1. Here, since the use of green capital does not create any externality, green capital is equivalent to capital in a simple Ramsey model and the interest rate is always equal to its rental price, net of depreciation. This is not true for brown capital when the pollution externality is internalized, as firms have to pay the carbon tax when they use brown capital. Also, the rental price of brown capital can be lower than the rental price of green capital when the irreversibility constraint is binding, as discussed in Proposition 3. Proposition 3. Two phases can be distinguished for the marginal cost of pro- ducing with brown capital: (i) When brown investment is nil, the value of brown capital is lower than the marginal utility of consumption, and the rental price of brown capital is lower than that of green capital. Rb,t = Rg,t − p (24) with 0 < p ≤ Rg,t In particular, when brown capital is overabundant, i.e. an additional unit of installed capital is not worth anything (νt = 0), the rental price of brown capital Rb,t decreases to zero (p = Rg,t ). In this case installed brown capital is optimally under-utilized such that the marginal productivity of utilized brown capital is equal to the carbon price (multiplied by G). ∂qb F = τt G with qb,t < kb,t (25) 8 (ii) When brown investments are strictly positive (and in particular on the balanced growth path), the rental price of brown capital Rb,t is equal to the rental price of green capital Rg,t = ∂kg F and: ∂qb F = ∂kg F + τt G (26) Firms must pay the carbon price when using brown capital for production, so the marginal productivity of brown capital should be higher than that of green capital, to adjust to its higher cost. Proof. See Appendix B.3. We show in the following proposition that the first phase necessarily happens first when a carbon price is implemented in the laissez-faire equilibrium. Proposition 4. When a GHG ceiling is enforced in the laissez-faire equilib- rium, the irreversibility constraint is binding (ib,t = 0). Proof. At t0 , the carbon price is implemented in a laissez-faire equilibrium, in which marginal productivities of brown and green capital are equal (Proposition 1). This condition on marginal productivities determines the ratio of installed brown capital over green capital. On the other hand, eq. 26 (∂qb F = ∂kg F + τt · G) implies that in the phase with ib,t > 0 and on the balanced growth path the ratio of installed brown capital over green capital is lower than in the laissez-faire equilibrium (because of decreasing marginal productivities). Since installed capital is necessarily the same at t− + 0 and t0 (just before, and just after the implementation of the carbon price), eq. 26 cannot be true at t+ 0 and the irreversibility constraint is necessarily binding in the beginning. A more complete proof can be found in Appendix C. During this first phase, the rental price of brown capital decreases below that of green capital and the economy stops investing in brown capital to invest only in green capital. The difference in rental prices appears because of the irreversibility constraint, that prevents firms from selling brown capital to either buy green capital or consume it. Since firms that use brown capital suddenly have to pay an additional production cost with the carbon tax, they can only keep producing if the rental price of brown capital decreases (otherwise, their marginal cost would be higher than their marginal benefit). This lower rental price is thus a transfer that “compensates” firms so that they can keep producing during the first phase. If the rental price of brown capital decreases down to zero, installed brown capital is optimally under-utilized to reduce carbon emissions.9 Indeed, if the marginal brown capital has no value anymore because τt is too high, it may be optimal to stop using it (in Fig. 1 installed brown capital is under-utilized until ti,1 ). In other words, since there is no fixed cost in using brown capital (Rb,t = 0), capital utilization may be reduced such as to equalize the marginal cost and marginal revenue (eq. 25). When the rental price is nil, the marginal 9 The decrease in R b,t (and optimal under-utilization of brown capital) is a function of the carbon price τt . 9 Figure 2: Depending on initial emissions (i.e. initial brown capital kb,0 ) and on ¯ ), brown capital is under-utilized or not in the first-best the concentration ceiling (m optimum. productivity of brown capital is transfered to households through the tax rev- enue τt G only. A direct consequence of eq. 25 is that during the first phase, installed brown capital is optimally under-utilized if the carbon price (multiplied by G) is higher than the marginal productivity of brown capital when all brown capital are used: τt G > ∂qb F |qb,t =kb,t . In particular, at t0 installed brown capital is under-utilized if: 1 τt+ ≥ ∂q F (t−0) 0 G b The under-utilization of brown capital depends on the ceiling m ¯ , on initial 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. Put more simply, as it is illustrated in Fig. 2, for a given set of functions and parameters the under-utilization of brown capital only 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). This can be interpreted in terms of time horizon: for a given level of initial emissions, the lower the ceiling the shorter the time before the ceiling is reached if all brown capital is utilized. If the time is short before the ceiling is reached, it is optimal to under-utilize brown capital in order to reduce emissions faster. Conversely, if the ceiling is to be reached in a long time, it is optimal to use all installed brown capital while it depreciates to a sustainable level. Because investment is irreversible, the society that we model has to live with past mistakes for a while, once it realizes it has been on a non-optimal growth path. A way to bypass this obstacle is to give up part of installed polluting capital in order to reduce emissions faster (Fig. 1, 2, and Prop. 4). Such a strategy reduces short-term output, but for stringent climate objectives (with regard to past accumulation of polluting capital), it is optimal. Under-utilization of existing capital may however be politically difficult. First, it appears as a waste of resources and creates unemployment (even though labor is not modeled here). Second, it affects primarily the owners of polluting capital and the workers whose jobs depend on this capital, transforming them into strong opponents to climate policies. 10 4. Capital-based policies It is possible to reduce carbon emissions through investment decisions — that is, to redirect investments towards green capital — without creating an incentive to reduce the utilization rate of brown capital, i.e. with no effect on production decisions. In practice, it can be done with capital-based instruments such as energy efficiency standards, fiscal incentives or differentiated interest rates depending on the carbon content of capital (Rozenberg et al., 2013). In this section, we consider the three following instruments: (i) the carbon tax is completed by a temporary subsidy on brown capital; (ii) the cost of brown and green capital are differentiated, e.g. with a subsidy on green investment; (iii) brown investments are regulated. All three instruments are equivalent if they are optimally designed to maxi- mize welfare given the ceiling constraint. They allow reaching the same balanced growth path as in the first-best optimum in the long-run, and they induce a full utilization of brown capital in the short-run. 4.1. Carbon tax plus temporary subsidy on brown capital Proposition 3 implies that when a carbon tax is implemented, if the rental price of brown capital falls down to zero in the short-run (because investment is irreversible) then it is optimal to under-utilize installed brown capital such that the marginal productivity of utilized brown capital is equal to the carbon price (eq. 24 and 25). As explained before, this might be unacceptable to the owners of brown capital, creating strong opposition against the measure. They may also consider the tax unfair, as they were not aware of the future carbon tax when they bought their capital. Under-utilization can however be prevented by subsidizing unprofitable brown capital in the short-run. Starting from the social optimum, eq. 23 becomes: ∂qb F = Rb,t + τt G − st (27) with st > 0 if Rb,t = 0 and st = 0 otherwise As explained in Appendix E, when the rental price of brown capital is nil, the subsidy is set as the difference between the carbon tax and the marginal productivity of brown capital when all brown capital is used.10 Firms thus have no incentive to under-utilize brown capital. This can only happen in the first phase, when brown investment is nil. Whenever Rb,t is strictly positive (and in particular in the long-run), brown capital is fully-utilized and the subsidy is equal to zero (so that the subsidy is only a temporary measure to smooth the transition). Note that since the complementary subsidy is a second-best strategy, the optimal value of the carbon tax is higher in the short-run than the one found in the first-best solution (section 3). However, the total cost borne by producers when they use brown capital is lower than in the first-best case. Here, the temporary subsidy is equivalent to a lower carbon tax in the beginning because we model only one kind of brown capital. If there was a continuum of brown 10 Note that the subsidy is always lower than the carbon tax. 11 capital with different carbon intensities, the subsidy would only go to carbon- intensive capital that would otherwise be discarded. In practice, a unique carbon price can be implemented to act as a signal for investments, and it can be completed by temporary subsidies to the most vulnerable firms or households, so that they can keep using their polluting capital. Of course, such policies might create regulatory capture (Laffont and Tirole, 1991), but could be a prerequisite for the implementation of the carbon tax. Such a subsidy to obsolete brown capital is comparable to Japanese indus- trial policies, that supported declining traditional sectors during the transition towards higher productivity sectors in the middle of the 20th century (Beason and Weinstein, 1996). 4.2. Differentiation of capital costs A second solution is to differentiate capital costs, for instance with fiscal incentives such as subsidies on green investment (θg,t < 0) or taxes on brown investment (θb,t > 0). Here we model lump-sum taxes on installed capital, as they are easier to model even though they are less realistic. In our model, since investment is irreversible, taxes on capital only have an impact on new capital, i.e. on investment decisions. They however induce transfers through rental prices that would not be necessary with taxes on investment (see Appendix F). The firm’s flow of profit at time t is given by: Πt = F (qb,t , qg,t ) − (Rg,t + θg,t ) kg,t − (Rb,t + θb,t ) kb,t (28) The optimal values of θg,t and θb,t can be obtained with a maximization of social welfare given the ceiling constraint. We solve the firm’s maximization problem in Appendix F and find that for all t it is optimal to have: qb,t = kb,t ∂qb F = Rb,t + θb,t ∂qg F = Rg,t + θg,t Under-utilizing brown capital is never optimal because firms do not pay carbon emissions directly. Instead, they pay a higher fixed price for brown capital than for green one, such that investment in brown capital is not profitable. Over the short-run, as in the social optimum (Prop. 4) the economy does not invest in new brown capital. Once brown capital has depreciated to a level compatible with the GHG ceiling, brown investments become profitable and start again. When brown and green investments are strictly positive, and in particular on the balanced growth path, the marginal productivity of brown capital is equal to that of green capital plus the sum of the tax and the subsidy (−θg,t is positive): ∂qb F (qb,t , qg,t ) = ∂qg F (qb,t , qg,t ) + (θb,t − θg,t ) To be on the same balanced growth path as in the social optimum, the optimal value of the tax plus the subsidy should be equal to the carbon tax multiplied by the marginal emissions of brown capital: ∀t ≥ tb , θb,t − θg,t = τt · G 12 with tb the date at which the balanced growth path is reached. This capital cost differentiation is similar to existing fiscal incentives, fee- bates programs or concessional loans for high efficiency homes or appliances. With a continuum of brown capital, the differentiation should be proportionate to the carbon content of each new investment. In practice, this tax on brown investment or subsidy to green investment can be done at the firm level but can also be subject to regulatory capture. Capital costs can also be differen- tiated using financial markets, as in Rozenberg et al. (2013). In this case, the differentiation would be calibrated on the carbon content of investments. 4.3. Investment regulation A third possibility to induce a shift from brown to green investment without reducing the utilization rate of brown capital is to regulate brown investment through efficiency standards. In particular, the most polluting brown invest- ments can be forbidden. Here, since we only model one kind of brown capital, we crudely impose brown investments to be nil until brown capital has depre- ciated to a level allowing to reach the carbon ceiling without using all brown capital. We come back to the social planner’s program (beginning of section 3) and remove the concentration and ceiling constraints (eq. 10 and eq. 11), as well as the irreversibility constraint (eq. 8). Instead, we add a brown investment constraint that forces ib,t to be nil, and we call σt its Lagrangian multiplier: ∀t, ib,t = 0 (σt ) (29) The maximization of intertemporal welfare results in the same equations as in the social optimum (Appendix G), except that the rental rate of brown capital is equal to that of green capital plus a positive term nt which depends on σt : ∂qb F = Rb,t (30) Rb,t = Rg,t + nt Therefore, the rental price of brown capital is higher than the interest rate. The brown investment regulation indeed creates a scarcity effect on brown capital, that becomes more expensive than green capital. Here again, this instrument must be thought of as temporary, since once brown capital has depreciated to a sustainable level, a carbon price can be imple- mented without inducing under-utilization of brown capital, and thus becomes politically acceptable. Investment regulation can be compared with existing efficiency standards on cars or electric plants, that forbid the sale of the most polluting kinds of brown capital. 4.4. Comparison with the social optimum All three capital-based instruments, if they are optimally designed given the ceiling constraint, lead to the same emissions and output pathways. As already noted, if the concentration ceiling is not stringent, these second- best instruments are equivalent to the carbon tax alone, because it is optimal to always use all brown capital in the short-run. On the other hand, if the ceiling is too stringent, such that waiting for brown capital depreciation is not sufficient to remain below the ceiling, these instruments cannot be used to reach 13 Figure 3: Depending on initial emissions (i.e. initial brown capital kb,0 ) and on the concentration ceiling (m ¯ ), the carbon tax and capital-based instruments can lead to different or similar outcomes (for a given set of parameters, and in particular ρ and δ ). If the ceiling is too stringent, such that waiting for brown capital depreciation is not sufficient, the capital-based instruments cannot be used. If the ceiling is not stringent, there is no under-utilization of brown capital in the first-best optimum with the carbon tax and capital-based instruments are equivalent. the target. This is illustrated in Figure 3. The “second-best infeasibility zone”, i.e. the zone in which brown capital must be under-utilized to remain below the ceiling, depends on the capital depreciation rate δ , the GHG dissipation rate ε, initial GHG concentration m0 and initial brown capital k0 . It is expressed analytically in Appendix H and if the carbon dissipation rate is small compared to the capital depreciation rate (ε δ ) it can be approximated by: G k0 ¯ < m0 + m δ According to Davis et al. (2010), existing polluting infrastructure in 2010 allowed staying above the zone in which under-utilization is necessary to stay below the ceiling, for a ceiling consistent with the 2 degrees target: they show 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 degrees Celsius. Yet, reaching the 2 degrees target might imply to stop investing in polluting capital tomorrow, which depends on our ability to overcome infras- tructural inertia and develop clean energy and transport services (Davis et al., 2010; Guivarch and Hallegatte, 2011). Davis and colleagues’ results do not give insights on whether the optimal approach to the 2 degrees target implies to under-utilize polluting capital, i.e. in which of the two upper zones we are. The interesting zone for our paper is the middle one, where it is optimal to under-utilize brown capital in the short-run, and thus where there is a difference between the carbon tax alone and second-best instruments. In this zone, mitigation efforts are increased in the short-run in the first-best optimum with a carbon tax, compared to the second-best alternatives. Proposition 5. With capital-based instruments that do not induce brown cap- ital under-utilization, output is higher in the short-run than in the first-best solution with a carbon price. 14 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 brown capital utilization. On the right, consumption c is higher in the second-best case because of a higher output y . tb is the date at which the balanced growth path is reached, it is reached sooner in the second-best case (tb,2 < tb,1 ). Proof. We showed with Proposition 4 that in the first-best optimum, when a carbon price is introduced, the utilization rate of brown capital may be dis- continuous. Therefore, in the short-run — that is, when the irreversibility constraint is binding — output is lower in the first-best optimum than in the second-best solution, in which the utilization rate of brown capital is continuous. If production is higher over the short-run in the second-best mitigation strat- egy, consumption can also be higher (we find so in the illustrative simulation of this paper). Analytically, however, the effect on consumption is ambiguous because it involves the offsetting impacts from a substitution effect and an in- come effect: short-term output is higher, but investments in green capital may also increase since the saving rate is endogenous. Eventually, all instruments lead to the same balanced growth path, but capital-based policies result in lower discounted welfare than in the social opti- mum, while they may increase the utility of current generations (Fig. 4). These policies generates higher short-term emissions (Fig. 6) than a carbon price, and because they are sub-optimal, they also generate higher marginal abate- ment costs (Fig. 5). The marginal abatement cost (MAC) is for instance equal to the carbon price τt in the first-best optimum, and to (θb,t − θg,t )/G with differentiated capital costs. It is interesting to note that in our model, and in particular because invest- ment is irreversible, the carbon tax (or tax plus subsidy) cannot be translated into consumption losses in a trivial way. At each point in time, the effect of the policy on output and consumption is disconnected from the MAC. Indeed, capital-based instruments are more expensive at each time t in terms of MAC (Fig. 5) while output (and possibly consumption) is higher over the short-run (Fig. 4). Put differently, in our framework the carbon price is not a good in- dicator to estimate the policy effect on instantaneous output and consumption (instead, it gives the impact on intertemporal welfare). On the other hand, the policy design influences the intertemporal distribution of mitigation efforts. Choosing the best instrument in terms of welfare thus results in choosing the lowest marginal abatement cost but not the highest consumption at each 15 Figure 5: The marginal abatement cost is higher with capital-based instruments than with a carbon price. In the optimal pathway, it is equal to the carbon price (τ ). Figure 6: GHG emissions in the two cases. The carbon price induces spare brown capital and thus reduces carbon emissions faster in the short-run. time t. There is however a trade-off between efficiency (intertemporal welfare), intergenerational equity (distribution of efforts over time) and implementation obstacles (political economy). Other criteria than social welfare maximization can be used to decide on the best policy to implement. For instance, Llavador et al. (2011) use the Intergenerational Maximin criterion, which maximizes the minimum utility over the whole trajectory. Using this criterion, capital-based policies would be preferred to the carbon tax alone. 5. Conclusion Current economies have expanded thanks to the installation and use of pol- luting capital (infrastructure, production processes, energy extraction) and have now come to realize that this accumulation is unsustainable. Reaching ambi- tious mitigation objectives such as the “2 degrees” target requires decreasing global emissions within the next one or two decades. Doing so with a carbon tax is likely to impose the early retirement of capital that could be operated for several more years (a process sometimes referred to as early-scrapping or moth- balling) while progressively accumulating green capital. Such a strategy can be 16 unacceptable if people have an aversion for capital under-utilization, as it cre- ates unemployment and may necessitate compensations for the owners of brown capital. Also, it reduces the income and consumption of current generations for the benefit of future ones, which may appear unattractive to individuals with high discount rates. We find that when all production capital is used — in our second-best strat- egy relying on capital-based instruments — the outcome in terms of discounted intertemporal welfare is lower but current generations have a higher income than when a carbon price is implemented alone. Such a transition towards a low-carbon economy can be triggered by green incentives that shift investments towards green capital without penalizing existing brown capital (e.g., efficiency standards on new cars or home appliances). It can also be done by subsidizing brown capital to ensure it is fully used until the end of its lifetime, despite the carbon tax. In those cases, current generations keep using their inefficient buildings and combustion engines, while redirecting their investment towards green capital. After some time, the only remaining “brown capital” is the one that does not need to be substituted by green capital, as in the optimal case. A carbon price can then be implemented more easily, since all instruments are then equivalent. Capital-based policies therefore only differ temporarily from the first-best path- way, in a way that smooths the transition costs: they decrease efforts in the short-run, increase them in the medium-run, and leave them unchanged in the long-run. These results are important for the political economy of climate change that has to deal, in particular, with the issues of sensitivity and preference hetero- geneity. Implementing a unique price on carbon emissions penalizes the owners of brown capital that would have to be compensated with interindividual trans- fers, and such transfers often face technical difficulties (Kanbur, 2010). Because capital-based instruments do not penalize (as much) the owners of brown capital and their employees, they mitigate these difficulties. Second, time preference heterogeneity (Greene, 2010; Heal and Millner, 2013) 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 while being the ones benefiting from reduced climate change damages. In this analysis, we have modeled a social planner who takes decisions given a set of parameters, and in particular given a discount rate (ρ) for welfare calculation. This discount rate is taken into account in the carbon ceiling, that increases social welfare compared to a baseline scenario with climate change damages. Given these preferences, the social planner sets the GHG ceiling and implements instruments to comply with the ceiling. But some individuals might have different time preferences from the social planner’s (Goulder and Williams, 2012). In particular, some individuals might have preferred a less restrictive GHG ceiling, i.e. lower short-term costs and higher long-term costs. Given the time profile of consumption with the two abatement strategies, it is clear that those individuals prefer the second option, because it shifts mitigation efforts to the future, compared to the first-best option. This suggests that the second- best strategy is more robust to preference heterogeneity than the first-best one, and supports the idea that policies focusing on new capital are more politically acceptable than a carbon price. The capital-based instruments that we modeled in this paper are industrial 17 policies: they subsidize the declining brown sector in order to avoid political opposition to the climate policy, or they trigger investment in the new green sector. As such, they create the same risks from capture and rent-seeking as most industrial policies (Laffont and Tirole, 1991). Applying them is there- fore challenging and requires strong institution settings and controls (Rodrik, 2008). However, we do not model here the usual factors that justify the use of industrial policies: learning-by-doing, imperfect appropriability of knowledge spillovers, increasing returns. Instead, we show that industrial policies can fa- cilitate the transition towards a low-carbon economy because they prevent the under-utilization of existing capital and reduce short-term output losses. In a further paper, we will consider increasing returns in the green sector to get a more exhaustive picture of the potential risks and benefits from green industrial policies. Acknowledgements The authors thank Patrice Dumas, Louis-Gaetan Giraudet, Marianne Fay and the participants of the 2013 EAERE and AFSE conferences for their use- ful comments on a previous version of this article. All remaining errors are the authors’. The views expressed in this paper are the sole responsibility of the authors. They do not necessarily reflect the views of the World Bank, its executive directors, or the countries they represent. References Ambrosi, P., Hourcade, J., Hallegatte, S., Lecocq, F., Dumas, P., Ha Duong, M., 2003. Optimal control models and elicitation of attitudes towards climate damages. Environmental Modeling and Assessment 8 (3), 133–147. Arrow, K. J., Kurz, M., Mar. 1970. Optimal growth with irreversible investment in a ramsey model. Econometrica 38 (2), 331–344, ArticleType: research- article / Full publication date: Mar., 1970 / Copyright c 1970 The Econo- metric Society. URL http://www.jstor.org/stable/1913014 Beason, R., Weinstein, D. E., 1996. Growth, economies of scale, and targeting in japan (1955-1990). The Review of Economics and Statistics, 286–295. URL http://www.jstor.org/stable/10.2307/2109930 Davis, S. J., Caldeira, K., Matthews, H. D., 2010. Future CO2 emissions and climate change from existing energy infrastructure. Science 329 (5997), 1330 –1333. Fischer, C., Newell, R. G., Mar. 2008. Environmental and technology policies for climate mitigation. Journal of Environmental Economics and Management 55 (2), 142–162. Goulder, L. H., Mathai, K., Jan. 2000. Optimal CO2 abatement in the presence of induced technological change. Journal of Environmental Economics and Management 39 (1), 1–38. 18 Goulder, L. H., Parry, I. W. H., Jul. 2008. Instrument choice in environmental policy. Review of Environmental Economics and Policy 2 (2), 152–174. Goulder, L. H., Williams, R. C., 2012. The choice of discount rate for climate change policy evaluation. Tech. rep., National Bureau of Economic Research. Greene, D. L., 2010. How consumers value fuel economy: A literature review. Assessment and standards division office of transportation and air quality U.S. environmental protection agency, EPA-420-R-10-008. Washington, DC: US Environmental Protection Agency. URL http://www.epa.gov/otaq/climate/regulations/420r10008.pdf Guivarch, C., Hallegatte, S., Oct. 2011. Existing infrastructure and the 2C target. Climatic Change 109 (3-4), 801–805. Heal, G., Millner, A., Apr. 2013. Discounting under disagreement. Working Paper 18999, National Bureau of Economic Research. IPCC, 2007. Summary for policymakers. In: Climate change 2007: Mitigation. Contribution of working group III to the fourth assessment report of the in- tergovernmental panel on climate change, b. metz, O.R. davidson, P.R. bosch, r. dave, L.A. meyer (eds) Edition. Cambridge University Press, Cambridge, UK and New York, USA. Jorgenson, D., 1967. The theory of investment behavior. In: Determinants of investment behavior. NBER. Kanbur, R., 2010. Macro crisis and targeting transfers to the poor. In: Glob- alization and Growth: Implications for a Post-Crisis World. Michael Spence, Danny Leipziger, Commission on Growth and Development, The World Bank, Washington DC, p. 342. Laffont, J.-J., Tirole, J., Jan. 1991. The politics of government decision-making: A theory of regulatory capture. The Quarterly Journal of Economics 106 (4), 1089–1127. URL http://qje.oxfordjournals.org/content/106/4/1089 Llavador, H., Roemer, J. E., Silvestre, J., 2011. A dynamic analysis of human welfare in a warming planet. Journal of Public Economics 95 (11), 1607–1620. URL http://www.sciencedirect.com/science/article/pii/ S0047272711000922 Meinshausen, M., Meinshausen, N., Hare, W., Raper, S. C. B., Frieler, K., Knutti, R., Frame, D. J., Allen, M. R., Apr. 2009. Greenhouse-gas emission targets for limiting global warming to 2[thinsp][deg]C. Nature 458 (7242), 1158–1162. URL http://dx.doi.org/10.1038/nature08017 OECD, Sep. 2009. Incentives for CO2 emission reductions in current motor vehicle taxes. Tech. rep., Organisation for Economic Co-operation and Devel- opment, Paris. Ploeg, F. V. D., Withagen, C., Jun. 1991. Pollution control and the ramsey problem. Environmental and Resource Economics 1 (2), 215–236. 19 Rodrik, D., 2008. Normalizing industrial policy. Working Paper 3, World Bank. Rozenberg, J., Hallegatte, S., Perrissin-Fabert, B., Hourcade, J.-C., 2013. Fund- ing low-carbon investments in the absence of a carbon tax. Climate Policy 13 (1), 134–141. Schwerin, H., 2013. Capacity utilization in pollution control. Vogt-Schilb, A., Meunier, G., Hallegatte, S., 2012. How inertia and limited potentials affect the timing of sectoral abatements in optimal climate policy. World Bank Policy Research (6154). Appendix A. Maximization of the household’s utility We consider a Ramsey framework with a representative infinitely-lived house- hold, who receives the economy’s production from firms yt , saves by accumulat- ing assets at , receives income on assets at interest rt and purchases goods for consumption ct . The assets dynamics are given by: ˙ t = rt · at + yt − ct a (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 (c) > 0 and u (c) < 0). The household maximizes their intertemporal utility, given by ∞ W = e−ρt · u(ct ) dt (A.2) 0 where ρ is the rate of time preference. The present value Hamiltonian is: Hh = e−ρt · {u(ct ) + λt [rt · at + yt − ct ]} (A.3) where λt is the shadow price of income at time t. The first order conditions for a maximum of W are: ∀t, ∂c Hh = 0 ⇒ λt = u (ct ) (A.4) −ρt ∂ (e λt ) ˙ t = (ρ − rt )λt ∀t, ∂a Hh = − ⇒λ (A.5) ∂t The doted variables represent temporal derivatives. If we differentiate eq. A.4 with respect to time and substitute for λ from this equation and λ ˙ from eq. A.5, we get the Euler equation, which gives the basic condition for choosing consumption over time: ˙ c u (c) =− · (rt − ρ) (A.6) c c · u (c) u (c) − cu (c) > 0 so consumption grows if the rate of return to saving is higher than the rate of time preference. If the interest rate equals the rate of time preference, consumption is constant over time. 20 Appendix B. First order conditions for the social optimum (section 3) The present value Hamiltonian associated to the maximization of social wel- fare is: Ht = e−ρt · {u(ct ) + λt [F (qb , kg ) − ct − ib,t − ig,t ] + νt [ib,t − δkb,t ] +χt [ig,t − δkg,t ] − µt · [G qb,t − εmt ] + φt · [m¯ − mt ] + ψt · ib,t + βt [kb,t − qb,t ]} (B.1) λt is the current value shadow price of income. νt and χt are the current shadow values of investments in brown and green capacities. µt is the current-value shadow price of pollution in the atmosphere, expressed in terms of undiscounted utility at time t. First order conditions give: ∂Ht =0⇒ u (ct ) = λt ∂ct ∂Ht =0⇒ λt = νt + ψt ∂ib,t ∂Ht =0⇒ λt = χt ∂ig,t ∂Ht ∂ (e−ρt νt ) =− ⇒ −νt δ + βt = −ν ˙ t + ρνt ∂kb,t ∂t ∂Ht ∂ (e−ρt χt ) =− ⇒ λt ∂kg F (kb,t , kg,t ) − χt δ = −χ ˙ t + ρχt ∂kg,t ∂t ∂Ht =0⇒ λt ∂qb F (qb,t , kg,t ) − µt · G = βt ∂qb,t ∂Ht ∂ (e−ρt µt ) = ⇒ ˙ t − ρµt −φt + εµt = µ ∂mt ∂t They can be reduced to the following equations: u (ct ) = λt = νt + ψt = χt (B.2) λt ∂kg F = (δ + ρ)χt − χ ˙t (B.3) λt ∂qb F = βt + µt · G (B.4) βt = (δ + ρ)νt − ν ˙t (B.5) µ˙ t = (ρ + ε)µt − φt (B.6) A simple interpretation for eq. B.2 is that along the optimal path, the current value of income (λt ) is the marginal utility of consumption at time t. It is also equal to the value of investments in green capital χt (eq. B.2). The implicit 1 rental value of green capital expressed in monetary terms is λ [(δ + ρ)χt − χ ˙ t] according to the definition given by Jorgenson (1967), where χt is the value of a unit of investment acquired at time t. It is always equal to the marginal productivity of green capital (eq. B.3). The complementary slackness conditions associated to the irreversibility constraint are: ∀t, ψt ≥ 0 and ψt · ib,t = 0 (B.7) 21 ψt is such that 0 ≤ ψt ≤ λ If ψt > 0, the value of brown capital is lower than the marginal utility of consumption (νt < λt ) and thus there is no investment in brown capital. If ψt = λ then the value of brown capital is zero. In this case βt = 0 (eq. B.5), and according to the complementary slackness condition associated to the “under-utilization” possibility, it means that qb,t = kb,t is not necessarily optimal anymore: ∀t, βt ≥ 0 and βt · (kb,t − qb,t ) = 0 (B.8) Still according to the definition given by Jorgenson (1967), βt is equal to the rental value of brown capacities, defined as (δ + ρ)νt − ν ˙ t (eq. B.5). The marginal productivity of brown capital is thus equal to the rental value of brown capital plus µ µt λt G (eq. B.4), where λt is the carbon price expressed in unit of consum- t ables. Appendix B.1. Carbon price We call τt the current price of GHG, expressed in units of consumables. µt τt = λt with µt the current-value price of pollution in the atmosphere and λt the current value income, both expressed in utility terms. eq. B.6 gives the evolution of µt . Using µ˙t = (λ˙t τt + λt τ˙t ) and λ˙t = (ρ − rt ), λt it can be written as the evolution of τt (the carbon price): φt τ˙t = τt [ε + rt ] − λt Before m reaches the ceiling, it is not binding and φt = 0. In that case the carbon price follows the following motion rule: τ˙t = τt [ε + rt ] Once the constraint is reached, ∀t mt = m ¯ , and φt > 0. 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 r + ε, to take into account the natural decay of GHG in the atmosphere (see for instance Goulder and Mathai, 2000, footnote 11, p6). Appendix B.2. Marginal productivities As in Jorgenson (1967), the rental prices of green and brown capacities are defined as follows: 1 Rg,t = [(δ + ρ)χt − χ˙ t] (B.9) λ 1 Rb,t = [(δ + ρ)νt − ν˙t] (B.10) λ If we differentiate eq. B.2 with respect to time and substitute λt and λ˙t , we can write: ct · u (ct ) c˙t · = (ρ + δ − Rg,t ) (B.11) u (ct ) ct 22 From eq. A.6 we can thus write: Rg,t = rt + δ (B.12) Combining eq. B.5 and eq. B.4 we find: ∂qb F = Rb,t + τt · G (qb,t ) Appendix B.3. Phases in brown capital Two cases can be distinguished: • If the irreversibility constraint is not binding, ψt = 0 (eq. B.7) and eq. B.2 gives νt = χt . Combining eq. 23 and eq. B.3 we find ∂qb F = ∂kg F + τt · G (qb,t ) • When the irreversibility constraint is binding (ψt > 0), we can differentiate eq. B.2 (νt = χt − ψt ) and substitute νt and ν ˙t in eq. B.5 to obtain βt 1 ˙t − (ρ + δ )ψt = (δ + ρ)χt − χ ˙t + ψ λt λt and using eq. B.3 we get βt 1 ˙t + (ρ + δ )ψt = ∂kg F − −ψ λt λt We call p = 1 (ρ + δ )ψt − ψ˙t and get λt ∂qb F = ∂kg F − p + τt · G (qb,t ) 0 < ψt ≤ λt ⇒ 0 ≤ λ 1 (ρ + δ )ψt − ψ˙t ≤ ρ + δ − λ˙t , t λt so that 0 ≤ p ≤ ∂kg F . When p = ∂kg F , the rental rate of brown capacities is nil and brown capacities may be under-utilized (slackness condition, eq. B.8) in order to adjust the marginal productivity of brown capital to the carbon price: ∂qb F = τt · G (qb,t ) Appendix C. Irreversibility constraint: proof of proposition 4 The GHG ceiling is imposed at t = t0 . Before that, the economy is in the competitive equilibrium so green and brown capacities have the same marginal productivity and capacities are fully used (Proposition 1). At t− 0 , i.e. just before 23 the ceiling is internalized (t < t0 ), we thus have the following limits for qb,t and ∂qb F : lim qb,t = kb,t (C.1) t→t− 0 lim ∂qb F (qb,t , qg,t ) = ∂kg F (qb,t , qg,t ) (C.2) t→t− 0 We use a proof by contradiction to show that at t+ 0 (when the constraint is internalized) the irreversibility condition is necessarily binding. Suppose that when t > t0 , the irreversibility condition is not binding, i.e. ψt = 0 (eq. B.7). According to proposition 3, it leads to: lim qb,t = kb,t (C.3) t→t+ 0 lim ∂qb F (qb,t , qg,t ) = ∂kg F (qb,t , qg,t ) + τt · G (C.4) t→t+ 0 So from eq. C.2 and eq. C.4: lim ∂qb F = lim ∂qb F (C.5) t→t+ 0 + t→t0 ∂qb F is a continuous function of qb,t so eq. C.5 implies that limt→t+ qb,t = 0 limt→t+ qb,t , and that is incompatible with eq. C.1 and eq. C.3. 0 Therefore, the irreversibility condition is necessarily binding at t = t+ 0 , i.e. ψt > 0. Two cases then need to be distinguished, whether brown capacities are fully used or not. If brown capacities are under-utilized (qb,t < kb,t and βt = 0), there is a discontinuity in output. Appendix D. 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 (qb,t , kg,t ) − Rg,t · kg,t − Rb,t · kb,t − τt G qb,t (D.1) With Rb,t and Rg,t the rental prices of brown and green capacities respectively, and τt the carbon tax. The tax is redistributed through the assets equation: ˙ t = rt · at + yt − ct + τt G qb,t a (D.2) As in the centralized equilibrium, the marginal revenue of brown capital is equal to ∂qb F = Rb,t + τt G while the marginal revenue of green capital is ∂kg F = Rg,t . The Lagrangian corresponding to the firm’s maximization program is: L(t) = Πt + βt (kb,t − qb,t ) + γt (kg,t − qg,t ) (D.3) First order conditions are: ∂qg L = 0 ⇒ ∂qg F (qb,t , qg,t ) = γt (D.4) ∂qb L = 0 ⇒ ∂qb F (qb,t , qg,t ) = βt + τt · G (D.5) ∂kg L = 0 ⇒ γt = Rg,t (D.6) ∂kb L = 0 ⇒ βt = Rb,t (D.7) 24 For all t, γt ≥ 0 and γt · (kg,t − qg,t ) = 0 βt ≥ 0 and βt · (kb,t − qb,t ) = 0 (complementary slackness conditions). With eq. D.4 we have γt = ∂qg F (qb,t , qg,t ) > 0, so qg,t = kg,t for all t. The combination of eq. D.4 and eq. D.6 gives ∂kg F (qb,t , kg,t ) = Rg,t Combining eq. D.5 and eq. D.7, we find ∂qb F (qb,t , kg,t ) = Rb,t + τt · G (D.8) In the equilibrium, the rental price of green capacities is equal to the interest rate (plus delta): Rg,t = rt + δ , because green capacities and loans are perfect substitutes as assets for households. When the irreversibility constraint is not binding (see eq. 8), and in particular on the balanced growth path, the rental rate of brown capacities is equal to the interest rate as well and Rb,t = Rg,t = rt + δ . However, when the carbon price in implemented at t0 , the irreversibility con- straint is binding (Appendix C). In this case, since the use of brown capacities suddenly becomes too expensive, the rental rate of brown capacities is endoge- nously reduced. As a consequence of a lower rate of return for owners of brown capital, households stop investing in brown capacities. If the carbon tax is very high, the rental rate of brown capacities can even become nil and brown capaci- ties may be under-utilized. It is possible to determine the rental value of brown − capital at t+ + + 0 thanks of the continuity of capacities between t0 and t0 : at t0 , the + + marginal productivity of brown capital is equal to ∂qb F (t0 ) = Rb (t0 ) + τt+ G 0 (eq. D.8). • If brown capacities are fully-utilized at t+ 0 (that is, qb,t+ = kb,t0 ), the 0 marginal productivity of brown capacities is necessarily equal to that of green capacities (∂qb F (t+ + 0 ) = ∂kg F (t0 )) because capacities are continuous − − and ∂qb F (t0 ) = ∂kg F (t0 ). − In this case Rb (t+ 0 ) = Rb (t0 ) − τt+ G. 0 • If brown capacities are under-utilized at t+ 0 , qb,t+ < qb,t− , Rb (t+ 0) = 0 0 0 (eq. D.7) and ∂kb F (kb,t , kg,t ) = τt · G. Brown capacities are thus under-utilized at t0 if τt+ G ≥ Rb,t− . In other words, 0 0 if the carbon tax (multiplied by the carbon intensity of capital G) is higher than the rental value of brown capacities in the laissez-faire balanced growth path, brown capacities are under-utilized in the short-run. Appendix E. Social optimum with a carbon tax and a temporary subsidy We start from the first order conditions found in the first-best optimum (section 3) and we modify 23 as follows: ∂qb F = Rb,t + τt G − st (E.1) with st calculated such that it is equal toτt G − ∂qb F |qb,t =kb,t + Rb,t 25 When the rental price of brown capacities is nil, the subsidy is set as the dif- ference between the carbon tax and the marginal productivity of brown capital when all brown capacities are used. In this case, firms have no incentive to under-utilize brown capacities. Whenever Rb,t is strictly positive (and in par- ticular in the long-run), brown capacities are fully-utilized and the subsidy is equal to zero. In a decentralized equilibrium, the subsidy would appear in the profit equa- tion as: Πt = F (qb,t , qg,t ) − Rg,t kg,t − Rb,t kb,t − (τt − st ) G qb,t (E.2) And it would be deducted from the households’ budget equation: ˙ t = rt · at + yt − ct + τt G qb,t − st qb,t a (E.3) Note that with the subsidy, in the short-run the optimal value of the carbon tax is different from the one found in the first-best solution (section 3). In the long-run, however, both instruments can lead to the same balanced growth path and the optimal carbon tax is the same. Appendix F. Firms’ maximization problem with differentiation of cap- ital costs Capital costs can be differentiated with fiscal incentives, e.g. subsidies on new green capacities (θg,t < 0) or taxes on new brown capacities (θb,t > 0). Here we model lump-sum taxes on all capacities, but they only have an impact on new investment decisions and are thus equivalent to taxes on new capacities. The optimal values of θg,t and θb,t can be obtained with a maximization of social welfare given the ceiling constraint. The firm’s flow of profit at time t is given by: Πt = F (qb,t , qg,t ) − (Rg,t + θg,t ) kg,t − (Rb,t + θb,t ) kb,t (F.1) The Lagrangian corresponding to the firm’s maximization program is: L(t) = Πt + βt (kb,t − qb,t ) + γt (kg,t − qg,t ) (F.2) First order conditions are: ∂qg L = 0 ⇒ ∂qg F (qb,t , qg,t ) = γt (F.3) ∂qb L = 0 ⇒ ∂qb F (qb,t , qg,t ) = βt (F.4) ∂kg L = 0 ⇒ γt = Rg,t − θg,t (F.5) ∂kb L = 0 ⇒ βt = Rb,t − θb,t (F.6) For all t, the complementary slackness conditions are γt ≥ 0 and γt · (kg,t − qg,t ) = 0 βt ≥ 0 and βt · (kb,t − qb,t ) = 0 Note that with the carbon tax, βt was equal to the rental value of brown capac- ities while here it is equal to the marginal revenue of brown capital (eq. F.6). 26 With eq. F.3 we have γt = ∂qb F (qb,t , qg,t ) > 0, so qg,t = kg,t for all t. Similarly, βt = ∂qb F (qb,t , qg,t ) > 0, so qb,t = kb,t for all t. The combination of eq. F.3 and eq. F.5, and eq. F.4 and eq. F.6 ∂qb F (qb,t , qg,t ) = Rb,t + θb,t ∂qg F (qb,t , qg,t ) = Rg,t + θg,t The irreversibility constraint is never binding for green investments, so as in the laissez-faire equilibrium, green capacities and loans are perfect substitutes as assets, and Rg,t = rt + δ , with rt the interest rate. Note that when the policies are implemented, the continuity of capacities imposes that Rg (t+ 0) + − θg (t+ 0 ) = R g ( t0 ). In other words, the rental price of green capacities suddenly increases when the subsidy is implemented. At the same time, the irreversibility constraint is binding for brown capacities and their rental price decreases below that of green capacities. For the same reason as for the green rental price, we − have Rb (t+ + 0 ) = Rb (t0 ) − θb (t0 ). Therefore, the economy does not invest in new brown capacities during this phase. As with the carbon price, these rental prices variations are transfers between households and firms that compensate for the tax or subsidy when investment is nil in brown capacities. Note that, contrary to the carbon tax, this policy may lead to a negative rental price for brown capacities when brown investments are nil, if θb,t is higher than the marginal productivity of brown capital. This negative rental price is equivalent to the subsidy we modelled in section 4.1. It is equal to Rb,t − st when st > Rb,t . On the balanced growth path, brown and green investments are positive so the irreversibility constraint is not binding and Rb,t = Rg,t . In this case the marginal productivity of brown capital is equal to that of green capital plus the sum of the tax and the subsidy (note that (−θg,t ) is positive): ∂qb F (qb,t , qg,t ) = ∂qg F (qb,t , qg,t ) + (θb,t − θg,t ) To be on the same balanced growth path as in the social optimum, the optimal value of the tax plus the subsidy should be equal to the carbon tax multiplied by the marginal emissions of brown capital: ∀t ≥ tb , θb,t − θg,t = τt · G with tb the date at which the balanced growth path is reached. Appendix G. Maximization of social welfare with standards on brown investments We come back to the social planner’s program (beginning of section 3) and remove the concentration and ceiling constraints (eq. 10 and eq. 11), as well as the irreversibility constraint (eq. 8). Instead, we add a brown investment constraint that forces ib,t to be equal to a standard at each point in time, and we call σt its Lagrangian multiplier: ∀t, ib,t = sdt (σt ) (G.1) The standard sdt can be optimally set to equal brown investments found in sec- tions 4.1 and 4.2. Basically, sdt = 0 until brown capacities have depreciated to a 27 level compatible with the ceiling, and then a carbon price can be implemented. The present value Hamiltonian associated to the maximization of social welfare is: Ht = e−ρt · {u(ct ) + λt [F (qb , kg ) − ct − ib,t − ig,t ] + νt [ib,t − δkb,t ] +χt [ig,t − δkg,t ] + σt · (sdt − ib,t ) + βt [kb,t − qb,t ]} (G.2) λt is the current value shadow price of income. νt and χt are the current shadow values of investments in brown and green capacities. First order conditions can be reduced to the following equations: u (ct ) = λt = νt − σt = χt (G.3) λt ∂kg F = (δ + ρ)χt − χ˙t (G.4) λt ∂qb F = βt (G.5) βt = (δ + ρ)νt − ν ˙t (G.6) The maximization of intertemporal welfare results in the same equations as in the social optimum, except for the marginal productivity of brown capital and the rental price of brown capacities: ∂qb F = Rb,t (G.7) Rb,t = Rg,t + nt (G.8) 1 with nt = ((ρ + δ )σt − σ˙t ) λt nt is positive if the standard imposes lower brown investment than in the laissez- faire equilibrium, and it is negative if it imposes higher investments. Here, since we want to force brown investments to be below that of the laissez-faire equilib- rium, nt is always positive, which means that the rental price of brown capacities is higher than the interest rate. Indeed, the brown investment standard creates a scarcity effect on brown capital, that becomes more expensive than green capital. Appendix H. Second-best infeasibility zone This zone defines the cases when the ceiling is reached before brown capac- ities have depreciated to a sustainable level. If no investment is made in brown capacities, we have: kb,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 28 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 “green incentives infeasibility zone” depends on the capital depreciation rate, the GHG dissipation rate, initial GHG concentration and initial brown capacities. 29