metadata: authors: 0: author: Hallegatte,Stephane
1: author: Rozenberg,Julie
2: author: Vogt-Schilb,Adrien Camille
count: World
docna: 0: docna: Transition to clean capital, irreversible
            investment and stranded assets
docty: Policy Research Working Paper
subtopic: Climate Change Mitigation and Green House Gases,Science of Climate Change,Climate Change and Health,Climate Change and Environment,Disability,Services & Transfers to Poor,Economic Assistance,Access of Poor to Social Services,Energy Demand,Energy and Environment,Energy and Mining
teratopic: Macroeconomics and Economic Growth,Environment
abstracts: cdata!: This paper uses a Ramsey model with two
            types of capital to analyze the optimal transition to clean
            capital when polluting investment is irreversible. The cost
            of climate mitigation decomposes as a technical cost of
            using clean instead of polluting capital and a transition
            cost from the irreversibility of pre-existing polluting
            capital. With a carbon price, the transition cost can be
            limited by underutilizing polluting capital, at the expense
            of a loss in the value of polluting assets (stranded assets)
            and a drop in income. In contrast, policy instruments that
            focus on redirecting investments -- such as feebates or
            environmental standards -- prevent underutilization of
            existing capital, avoid stranded assets, and reduce
            short-term losses; but they reduce emissions more slowly and
            increase the intertemporal cost of the transition. The paper
            investigates inter- and intra-generational distributional
            impacts and the political acceptability of climate change
            mitigation policy instruments.
repnme: repnme: Transition to clean capital, irreversible
            investment and stranded assets
display_title: Transition to clean capital,
            irreversible investment and stranded assets
listing_relative_url: /research/2014/05/19457792/transition-clean-capital-irreversible-investment-stranded-assets
topicv3: Carbon Tax m1326372 1845,Energy and Extractives m1326595 507,Climate Change m1326829 567,Climate Change Mitigation m1327079 579,Social Assistance m1327201 1466
publishtoextweb_dt: 2014-05-01T00:00:00Z
historic_topic: Macroeconomics and Economic Growth,Environment
no_of_pages: 29
author_keys: 0: author_key: 1103816
1: author_key: 1296276
2: author_key: 1272850
model_extractions: 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 to analyze the optimal transition to clean capital when polluting investment is irreversible. The cost of climate mitigation decomposes as a technical cost of using clean instead of polluting capital and a transition cost from the irreversibility of pre-existing polluting capital. With a carbon price, the transition cost can be limited by underutilizing polluting capital, at the expense of a loss in the value of polluting assets (stranded assets) and a drop in income. In contrast, policy instruments that focus on redirecting investments—such as feebates or environmental standards—prevent underutilization of existing capital, avoid stranded assets, and reduce short-term losses; but they reduce emissions more slowly and increase the intertemporal cost of the transition. The paper investigates inter- and intra-generational distributional impacts and the political acceptability of 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<sup>1, 2,</sup><sup>_∗_</sup> , Adrien Vogt-Schilb<sup>1, 2</sup> , Stephane Hallegatte<sup>2</sup>

> _1CIRED, 45bis avenue de la Belle Gabrielle, 94736 Nogent-sur-Marne, France._

> _2The 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 perspective, 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 implies 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 price<sup>1</sup> 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 capital has been studied before, mostly through the lens directed technical change, focusing on the interaction between pollution and knowledge spillovers externalities (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)

> 1Countries 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 VogtSchilb 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 capital 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 accumulation 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 exogenous policy objective such as the UNFCCC 2<sup>_◦_</sup> 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 shortterm 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.<sup>2</sup> Investmentbased 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 mitigation policy and its short-term impacts, which may influence political acceptability. 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 preferable 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 individual 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 production 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 household, who receives the economy’s production from firms _yt_ , saves by accumu-

> 2A 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 assets<sup>3</sup> , receives income on assets at interest _rt_ and purchases goods for consumption _ct_ . Their wealth thus evolves as: At time _t_ , consuming _ct_ provides consumers with a utility _u_ ( _ct_ ). The utility function is increasing with consumption, and strictly concave ( _u_<sup>_′_</sup> _>_ 0 and _u_<sup>_′′_</sup> _<_ 0).

The household maximizes intertemporal discounted utility _W_ , given by: where _ρ_ is the rate of time preference.

Firms produce one final good _yt_ , using two types of available capital: polluting capital _kp_ (e.g., coal power plants, thermal engine vehicles) and clean capital _kc_ (renewable or nuclear power, electric vehicles).<sup>4</sup> Production is used for consumption ( _ct_ ) and investment ( _ip,t_ and _ic,t_ ).

Investment _ip,t_ and _ic,t_ increase the stock of installed capital, which depreciates exponentially at rate _δ_ :<sup>5</sup> The doted variables represent temporal derivatives.

Investment is irreversible (Arrow and Kurz, 1970):<sup>6</sup> This means that for instance, a coal plant cannot be turned into a wind turbine, 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: 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.

> 3Assets are capital and loans to other households.

> 4 _kp_ and _kc_ may also be interpreted as intangible capital; for instance, clean capital encompasses 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.

> 5We used the same depreciation rate for polluting and clean capital to keep notations simple, but this assumption plays no particular role in the analysis.

> 6Following 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_ installed 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.<sup>7</sup> 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_ .<sup>8</sup> GHG atmospheric concentration increases with emissions, and decreases at a dissipation rate _ε_ :<sup>9</sup> 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 consumption over time (Appendix A): As the elasticity of substitution is positive (<sup>_−_</sup> _cu_<sup>_u′′′_</sup> (<sup><u>(</u></sup> _c_<sup>_c_</sup> )<sup><u>)</u></sup><sup>_>_0),consumptiongrowsif</sup> 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: 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: The classical equilibrium in capital markets in a Ramsey model applies:

**Proposition 1.** _In the laissez-faire equilibrium, households are indifferent between 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 :_

> 7In this paper, underutilization of clean capital is never optimal: _∀t, qc,t_ = _kc,t_ .

> 8In the remaining of the paper, “carbon” will refer to GHG.

> 9The 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 productivity 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_ ̄ : This threshold can be interpreted as a tipping point beyond which the environment (and output) can be highly damaged, or as an exogenous policy objective such as the UNFCCC “2<sup>_◦_</sup> C target” (Allen et al., 2009; Matthews et al., 2009).

We solve for the welfare maximization program, in which institutions internalize 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 decentralized 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: 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): 6
Where _τ_ is the price of carbon expressed in dollars per ton: Before the ceiling on atmospheric GHG is reached, a classical result (e.g., Goulder 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): The steady state is reached when _mt_ = _m_ ̄ . In the steady state, atmospheric emissions are stable, implying that polluting capital is constant at _kp,t_ = _mε/G_ ̄ ̇ ( _mt_ = 0, eq. 11) and the rest of the economy keeps growing on a balanced 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): 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 capital equals the rental rate of clean capital:_

_The marginal productivity of polluting capital is equal to the rental rate of polluting capital plus the marginal cost of carbon emissions:_ 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 between consumption and investment is given by:_

_The rental rate of polluting capital can be lower than that of clean capital:_ _Where the irreversibility cost pt is the monetary impact of the irreversibility constraint on the rental rate of polluting capital:_ 7
**Figure 1:** Polluting and clean installed capital, and utilized polluting capital in the first-best optimum. Before _t_ 0, the economy is on the _laissez-faire_ equilibrium. At _t_ 0 the carbon price is implemented and polluting capital depreciates until _ti_ ( _∀t ∈_ ( _t_ 0 _, 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 decomposing 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:<sup>10</sup> The next proposition shows that an irreversibility cost necessarily appears when the carbon tax is implemented (in _t_ 0).<sup>11</sup> Once polluting capital has adjusted through natural depreciation, the irreversibility cost is nill.

**Proposition 4.** _Two phases can be distinguished during the optimal transition to clean capital:_

> 10Combining equations 25, 26 and 28.

> 11Contrary 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:_

- _Then, a phase when the market price of polluting and clean capital are equal and polluting investment is strictly positive:_

_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 productivity of clean capital, as the maximum regret cost is the cost of not having invested in clean instead of polluting capital before _t_ 0. 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:_

Proof. Eq. 26 implies that the rental rate of polluting capital _Rp,t_ is the difference 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 multiplier 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,t_ 0 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 the concentration ceiling (  ̄ _m_ ), brown capital is underutilized or not in the first-best 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 irreversibility 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.<sup>12</sup> 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 investmentbased 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 economy 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-

> 12The 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 investment 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: 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:<sup>13</sup> Investment-based instruments however induce a different short-term transition. 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:<sup>14</sup>

**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 ( _t_ 0 _, t_<sup> ̃</sup> ). During this interval, utilized polluting capital, hence output, is higher in the second-best strategy. □ Similarly to the carbon price, investment-based instruments differentiate the marginal productivities of capital (Appendix D): 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)

> 14Analytically 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 investmentbased instruments than with a carbon price.

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: 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 secondbest 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 investmentbased 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 shortrun (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: 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.<sup>15</sup> 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 investment 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.

> 15Investment costs can also be differentiated using financial markets, as proposed by Rozenberg et al. (2013).

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**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.<sup>16</sup>

# **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 underutilization 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 polluting 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 accumulates 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 investmentbased instruments may induce a smooth and acceptable transition to a lowcarbon economy.

> 16In this model the capital lifetime is endogenous and therefore people cannot extend the lifetime of their polluting capital.

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**Figure 6:** Depending on initial emissions (i.e. initial polluting capital _kb,t_ 0 ) and on the carbon budget (  ̄ _m − mt_ 0 ), 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 _m_ 0 and initial polluting capital _k_ 0. The lower blue line in Fig. 6 is expressed analytically in Appendix G and can be approximated by: According to Davis et al. (2010), the level of existing polluting infrastructure in 2010 is still low enough to achieve the 2<sup>_◦_</sup> 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<sup>_◦_</sup> C. Yet, reaching the 2<sup>_◦_</sup> 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.

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# **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 discounted 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 under 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 myopic agents or limited ability to commit. Nevertheless, our results suggest that investment-based instruments respond to a political acceptability issue as regards to climate mitigation policies.

# **Acknowledgements**

The authors thank Patrice Dumas, Marianne Fay, Louis-Gaetan Giraudet, Baptiste Perissin-Fabert and the participants of the 2014 AEA conference for their useful comments on a previous version of this article. All remaining errors are the authors’. We acknowledge financial support from the Institut pour la 16