metadata: authors: 0: author: Bauer,Adam Michael
1: author: Hallegatte,Stephane
2: author: Mcisaac,Florent John
count: World
docna: 0: docna: The Timing versus Allocation Trade-off in
            Politically Constrained Climate Policies
docty: Policy Research Working Paper
abstracts: cdata!: When leaders face political economy
            constraints, is it best to delay all decarbonization
            initiatives until a sectorally coordinated strategy can be
            implemented, or is it preferable to implement an approach
            where sectors&#x2019; decarbonization strategies are
            uncoordinated? This question underscores a crucial trade-off
            &#x2013; here coined the &#x201c;timing versus
            allocation&#x201d; trade-off &#x2013; for
            politically constrained climate policymakers: (i) to
            sacrifice the optimal timing of climate policies to preserve
            the optimal allocation of emissions across economic sectors,
            or (ii) to preserve the optimal timing of abatement
            investment to the detriment of the allocation of emissions
            across sectors. This paper systematically explores this
            trade-off by presenting a modeling framework that elucidates
            the economic implications of various sub-optimal policy
            approaches to decarbonization that involve relaxing or
            delaying decarbonization efforts in a subset of sectors or
            economy-wide. The paper shows that the cost difference
            between an economy-wide, coordinated decarbonization
            strategy and an uncoordinated approach with heterogeneous
            carbon prices is smaller than the cost of delaying action
            and implementing a coordinated policy in the future. This
            implies that it is preferable to implement some policy in
            each sector, insofar as this is politically feasible, with
            less politically challenged sectors compensating with a
            marginal increase in policy ambition. The paper further
            elucidates how sectors with high annual emission rates, such
            as energy, are more costly to delay in comparison to their
            mid- to low-emission counterparts, such as industry, despite
            these sectors being nominally more costly sectors to decarbonize.
display_title: The Timing versus Allocation
            Trade-off in Politically Constrained Climate Policies
listing_relative_url: /research/2024/11/34419316/timing-versus-allocation-trade-off-politically-constrained-climate-policies
topicv3: Political Economy of Reform m1326263 836,Climate Change Mitigation m1327079 579
publishtoextweb_dt: 2024-11-07T00:00:00Z
dois: 10.1596/1813-9450-10971
origu: Off of Sr VP Dev Econ/Chief Econ (DECVP)
no_of_pages: 58
author_keys: 0: author_key: 2311156
1: author_key: 1103816
2: author_key: 2040293
model_extractions: Policy Research Working Paper 10971 The Timing versus Allocation Trade-off in Politically Constrained Climate Policies _Adam Michael Bauer Stephane Hallegatte Florent McIsaac_ Climate Change Group & Macroeconomics, Trade and Investment Global Practice November 2024 A verified reproducibility package for this paper is available at http://reproducibility.worldbank.org, click **here** for direct access.
Policy Research Working Paper 10971

# **Abstract**

When leaders face political economy constraints, is it best to delay all decarbonization initiatives until a sectorally coordinated strategy can be implemented, or is it preferable to implement an approach where sectors’ decarbonization strategies are uncoordinated? This question underscores a crucial trade-off—here coined the “timing versus allocation” trade-off—for politically constrained climate policymakers: (i) to sacrifice the optimal timing of climate policies to preserve the optimal allocation of emissions across economic sectors, or (ii) to preserve the optimal timing of abatement investment to the detriment of the allocation of emissions across sectors. This paper systematically explores this tradeoff by presenting a modeling framework that elucidates the economic implications of various sub-optimal policy approaches to decarbonization that involve relaxing or delaying decarbonization efforts in a subset of sectors or economy-wide. The paper shows that the cost difference between an economy-wide, coordinated decarbonization strategy and an uncoordinated approach with heterogeneous carbon prices is smaller than the cost of delaying action and implementing a coordinated policy in the future. This implies that it is preferable to implement some policy in each sector, insofar as this is politically feasible, with less politically challenged sectors compensating with a marginal increase in policy ambition. The paper further elucidates how sectors with high annual emission rates, such as energy, are more costly to delay in comparison to their mid- to low-emission counterparts, such as industry, despite these sectors being nominally more costly sectors to decarbonize.

This paper is a product of the Climate Change Group and the Macroeconomics, Trade and Investment Global Practice. 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://www.worldbank. org/prwp. The authors may be contacted at adammb4@illinois.edu, fmcisaac@worldbank.org, and shallegatte@worldbank. org. A verified reproducibility package for this paper is available at http://reproducibility.worldbank.org, click **here** for direct access.

<!-- Start of picture text --> TRANSPARENT<br>ANALYSIS<br>G<br>R E<br>N<br>S<br>I P<br>Y E<br>K A<br>C A<br>R P<br>LI R<br>O E<br>O C<br>W R<br>H<br>S<br>P<br><!-- End of picture text --> _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
The Timing versus Allocation Trade-off in Politically Constrained Climate Policies<sup>∗</sup> Adam Michael Bauer<sup>†</sup> St ́ephane Hallegatte Florent McIsaac

**JEL:** P18; Q52; Q54; Q58

**Keywords:** green investment; political economy; climate policy; second-best policies; adjustment costs

> ∗The authors thank Carolyn Fischer, Joe Pryor, and seminar participants at the World Bank Group for helpful discussions related to this work. The authors acknowledge financial support from the Climate Support Facility of the World Bank. AMB acknowledges financial support from a National Science Foundation Graduate Research Fellowship grant No. DGE 21-46756. 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. All remaining errors are the sole responsibility of the authors.

> † Correspondences sent to: adammb4@illinois.edu. Department of Physics, University of Illinois Urbana-Champaign, 1110 W Green St Loomis Laboratory, Urbana, IL 61801
# 1 **I Introduction**

2 Upon signing the Paris Agreement in 2015, a multitude of nations committed to pursuing policies 3 that would limit global warming to “well-below” 2 °C and to put significant effort towards limiting 4 warming to 1.5 °C (United Nations Framework Convention on Climate Change, 2015). While challenges 5 to decarbonization abound, political economy constraints (PECs) are of a particular consequence to 6 meeting the Paris Agreement targets. PECs are broadly defined as political headwinds – such as 7 thin majorities in legislative bodies, disapproving voter blocs, inhibiting government bureaucracy, or 8 powerful lobby groups with de facto veto power – that prevent leaders from implementing the most 9 efficient policies to address a particular challenge (Jenkins, 2014).<sup>1</sup> For instance, they often struggle 10 to implement policies to promote the decarbonization of the passenger transport sector (because of 11 opposition to higher gasoline prices) or the agricultural sector. Countries with oil and gas reserves 12 also have energy subsidies that are difficult to reform; for example, while the Group of Twenty and 13 the Asia-Pacific Economic Cooperation committed to phasing out inefficient energy subsidies in 2009, 14 many nations still heavily subsidize fossil fuel use (Asian Development Bank, 2015; Black et al., 2023; 15 Damania et al., 2023; Ihsan et al., 2024).

16 Indeed, current evidence suggests that while many countries have climate policies, they often start 17 with sectoral policies before implementing economy-wide policies (Dolphin et al., 2019; World Bank, 18 2023; Stechemesser et al., 2024). However, even these economy-wide policies often have a number of 19 exceptions; for example, they may only cover a share of total emissions (as is the case for European 20 ETS), have a number of exemptions (e.g., on air transport fuels), or apply different prices to different 21 CO2-emitting fuels (such as higher tax levels on liquid fuels compared to coal) (World Bank, 2024). One 22 practical example of such a decarbonization policy is the bifurcated European Union carbon dioxide 23 (CO2) emissions trading systems (ETSs), which imposes different carbon prices on different sectors of 24 the economy and contains a number of exemptions (European Commission, 2005, 2023; World Bank, 25 2024). Each of these examples illustrates how policymakers are managing the idiosyncrasies of the 26 green transition for their particular nation and navigating differences in abatement costs, sectoral 27 characteristics, and, especially, PECs.

28 This evidence would suggest that, generally, leaders are taking two approaches to climate policies 29 when faced with PECs: they are either (i) implementing sub-optimal, heterogeneous policies in the

> 1In the case of climate change, such a policy package would include a Pigouvian carbon tax or emissions trading system (ETS) (Pigou, 1920), along with complementary policies aimed at fostering innovation and supporting vulnerable communities.

2
30 near-term (such as isolated sectoral policies, subsidies-based strategies, or command-and-control regu31 lations), therefore sacrificing optimal sectoral coordination in favor of near-term investment efforts in 32 abatement, or (ii) biding their time for a change of political fortune, after which they will implement a 33 close-to-optimal policy for limiting fossil fuel emissions; in other words, they are wielding the “policy 34 instrument” of delay to forgo the optimal timing of climate policy to preserve the optimal allocation 35 of emissions across sectors.<sup>2</sup> We coin this trade-off between the optimal timing of CO2 abatement 36 investment and the optimal allocation of emissions across sectors as the “timing versus allocation” 37 trade-off in climate policy facing PECs.

38 This paper systematically explores this “timing versus abatement” trade-off in climate policy and 39 how it influences carbon price schedules, the temporal dynamics of abatement investment effort, and 40 the economic cost of climate policies. We present a multi-sector modeling framework that allows 41 the policymaker to wield two climate policy instruments: (i) a carbon tax/ETS (that can be either 42 homogeneous or heterogeneous across economic sectors), and (ii) the targeted delay of climate policies 43 in either a set of sectors or economy-wide. Our multi-sector approach has the distinct advantage of 44 allowing the planner to delay policies impacting only a subset of economic sectors.<sup>3</sup> We then elucidate 45 the impact of sub-optimal policies including delay on economy-wide decarbonization schedules and 46 carbon prices, the temporal distribution of investment in abatement technologies, and the sectoral and 47 aggregate economic cost of delay.

48 Our model specifies an economy with multiple sectors with differing abatement investment costs, 49 abatement potentials, and capital depreciation rates. In this context, the planner may delay decar50 bonization in three ways to overcome a given set of PECs. The first option is to reduce the stringency 51 of action in a set of politically challenged sectors by imposing a carbon price below the theoretical op52 timal.<sup>4</sup> Another choice is to delay _all_ climate policies impacting the set of challenged sectors. Finally, 53 they can choose to delay _all_ climate policies _economy-wide_ while they wait for political changes that 54 would make the theoretically efficient policy palatable. Each of these three scenarios underscores the 55 fundamental trade-off between the optimal timing of abatement investment and the optimal allocation 56 of emissions across economic sectors facing politically constrained climate policymakers.

> 2Another possibility is that leaders may be waiting to implement decarbonization policies until the cost of abatement technologies decline owing to global knowledge spillovers from countries with climate policies, i.e., they are “free riders” (Nordhaus, 2021).

> 3The targeted delay of climate policies in individual sectors differs from other approaches which take a global view (Lecocq et al., 1998; Sanderson and O’Neill, 2020) or focus on individual regions as opposed to sectors within those regions (Andaloussi et al., 2022).

> 4This approach – to implement a sub-optimal carbon price – is common in a number of nations with existing carbon prices that face PECs (Dolphin et al., 2019).

3
57 Importantly, our model includes the effects of adjustment costs (Lucas, 1967) on the green transition, 58 building on the modeling framework outlined in Vogt-Schilb et al. (2018). This is especially important 59 when analyzing the economic consequences of delaying decarbonization, as presumably the transition 60 must occur more rapidly in a setting with delay than in the optimal case.<sup>5</sup> This allows us to capture 61 the propensity for delay to lead to increasing levels of economic costs, as adjustment costs penalize a 62 swift green transition owing to, for example, resource scarcity and labor scarcity.

63 Our analysis proceeds as follows. We outline the general theoretical model in Section II. Our 64 different policy scenarios are outlined and linked to model parameters in Section III. We then show 65 the theoretical implications of PECs on policy in Section IV, and carry out numerical experiments to 66 quantify these effects in Section V. We discuss the implications of our findings and directions for future 67 study in Section VI and close in Section VII.

68 Before we continue with our analysis, we highlight a few of our key findings, and relate our work 69 to previous studies on decarbonization policy, second-best policies, and the political economy of the 70 green transition.

# 71 **I.1 Summary of contributions and findings**

72 From a theoretical perspective, we make two contributions. Our first contribution is a multi-sectoral 73 model of abatement investment that incorporates PECs on a subset of economic sectors’ decarboniza74 tion schedules to compare the costs and benefits of various approaches to managing PECs. This 75 modeling framework allows us to analyze both theoretically and quantitatively the impact of PECs, 76 particularly those that are restricted to individual sectors that may, for example, have strong anti77 decarbonization lobbies in a given nation. We demonstrate how a change in the decarbonization dates 78 in a subset of economic sectors translates into higher carbon prices in the remaining sectors, different 79 decarbonization dates, and higher overall policy costs relative to our optimal policy.

80 Secondly, we provide a proof that delaying action towards abating CO2 emissions either in a subset 81 of sectors or economy-wide is more costly than a sub-optimal abatement strategy in which action in 82 each sector is lower (resp. higher) in sectors with higher (resp. low) political economy constraints. By 83 allowing us to establish a hierarchy of the costliness of sub-optimal policy responses involving delay, this 84 proof allows us to conclude that it is the _timing_ of abatement investment, not the allocation of emissions

> 5This is obvious if one considers a nation intent on reaching net-zero by some future date; any delay in starting policy will shrink the horizon for the transition to occur, assuming of course that the nationally determined decarbonization goal is attained on schedule.

4
85 across sectors, that should be prioritized by climate policymakers facing PECs. We further show how 86 our model can be generalized beyond the issue of climate change to influence policy discussions more 87 broadly; indeed, we argue that our modeling approach may be applied to any nonrivalrous good that 88 is made rivalrous via an economy-wide quota for net accumulated use.

89 We further undertake numerical experiments to quantify the differences between the policy scenarios 90 we explore, and provide two high-level takeaway points. The first is that the aggregate economic cost 91 of policy increases nonlinearly as decarbonization in the challenged sector(s) is delayed. This finding 92 can be primarily explained by the influence of adjustment costs on the green transition: delaying 93 decarbonization in challenged sector(s) leads to an inflated carbon price in non-challenged sectors, 94 encouraging a swifter decarbonization, which adjustment costs make more expensive. Compared with 95 the usual framing – in which the cost of a sub-optimal distribution of effort across sectors is driven 96 by inter-sector differences in marginal abatement costs (Baranzini et al., 2017; Stiglitz, 2019; Gugler 97 et al., 2021) – here the increase in cost is driven by the combination of adjustment costs and the need 98 to decarbonize faster if action is delayed in some sectors.

99 Indeed, we find that the increase in total cost caused by delayed action in a sector is mostly driven 100 by its annual emissions rate, not its marginal abatement cost. This follows from a sector’s annual 101 emissions rate controlling the degree to which delaying that sector’s decarbonization distorts the optimal 102 allocation of emissions across the economy, thus implying that adopting second-best policies (such as 103 a sub-optimal carbon price or delaying the implementation of a carbon price) in sectors with large 104 annual emissions rates is especially costly. This insight provides a “rule of thumb” for policymakers 105 navigating the political and economic landscape of decarbonization: it is cheap (expensive, resp.) to 106 delay action in sectors with small (high, resp.) emissions, regardless of whether abatement is cheap or 107 expensive in these sectors.

108 The second contribution from our numerical experiments is to quantify the hierarchy of delay-based, 109 second-best approaches to decarbonization established by our theoretical results. For example, in the 110 case where decarbonization in the energy sector is delayed by a decade because of a lower carbon price 111 imposed by PECs, overall aggregate policy costs only increase by 1.7% relative to our optimal policy; 112 in monetary terms, this implies a $410 billion increase in cost. While this increase is non-trivial, it is 113 small compared to the $11.7 trillion increase (a ≳ 50% increase, relative to our optimal policy) that 114 results from delaying all decarbonization initiatives economy-wide for a decade. This relatively small 115 increase in costs in the former case owes to the fact that the planner is able to smooth investment over 5
116 time to accommodate the PEC, which is not the case when all action in politically challenged sector(s) 117 is delayed entirely by some amount of time.

# 118 **I.2 Related literature**

## 119 **I.2.1 Optimal climate policies**

120 Many papers have studied the structure and costliness of first-best climate policies, pioneered by Nord121 haus (1992). Our optimal policy is akin to such approaches, where the optimal carbon tax is enacted 122 immediately, albeit in a least-cost setting as opposed to a welfare-maximization framework. A number 123 of papers have focused on the level and structure of the optimal carbon tax. For example, Golosov 124 et al. (2014) study optimal carbon taxes in general equilibrium, and Dietz and Venmans (2019) revise 125 Golosov et al.’s findings by leveraging a state-of-the-art climate emulator. Cai and Lontzek (2019) com126 pute the optimal carbon tax schedule subject to climate and macroeconomic risks, including climate 127 tipping points; the economic impact of climate tipping points is further explored in Dietz et al. (2021). 128 While these models have been useful for informing policy discussions, they have been criticized along a 129 number of lines (Stern, 2013), making the policy contributions of such models, at best, limited, and at 130 worst, moot (Pindyck, 2013). An alternative approach to computing the optimal carbon tax schedule 131 is to compute the net-present value of streams of future climate damages, as done in Rennert et al. 132 (2022), where they compute the social cost of carbon under a number of future emissions, climate, 133 damage, and discounting scenarios.

## 134 **I.2.2 Second-best climate policies**

135 A number of studies focus on the efficiency of second-best policies for promoting energy efficiency 136 and mitigating climate change, motivated by the obstacle PECs create for the enactment of first-best 137 policies. These studies often consider the effects of a combination of taxes, subsidies, and command138 and-control regulation to address an environmental externality.<sup>6</sup> Our approach can perhaps be best 139 viewed through this lens, where we study the trade-off not between implementing a second-best policy 140 that utilizes a sub-optimal policy instrument (such as R&D subsidies), but in relaxing or delaying the

> 6 Indeed, a number of nations are pursuing policies that utilize these second-best instruments, especially tariffs and industrial policies, as a part of their broader decarbonization strategy. For example, there have been a series of tariff hikes for imported Chinese electric vehicles (EVs) in the United States (The White House, 2024; Dadush, 2024) and the European Union (Dadush and McCaffrey, 2024), while India has placed domestic investment requirements on all EV imports (Mehta and Shah, 2024). The United States also implemented a massive subsidy-based green industrial policy experiment via the Inflation Reduction Act of 2022 (Yarmuth, 2022). Each of these policies are designed to promote domestic job creation as a result of the green transition, while complying with the political constraints facing each polity.

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141 first-best policy. For example, Fischer and Newell (2008) explore the relative efficiency of a number of 142 sub-optimal policy approaches, such as R&D subsidies, emissions performance standards, and a fossil 143 power tax. One finding of note is that policies to promote innovation in abatement technologies are 144 only justified if an emissions price is in place already (Fischer, 2008), something that is lacking in 145 many nations pursuing subsidy-based or other second-best policies; Del R ́ıo Gonz ́alez (2008) explores 146 the interaction between subsidy- and tax-based environmental policies. However, some argue that first 147 implementing subsidy-based policies can be politically beneficial through coalition building, which can 148 spur carbon pricing in the future (Wagner et al., 2015; Meckling et al., 2017). More recently, Goulder 149 et al. (2024) explore the influence of pre-existing tax distortions on the efficiency of a carbon tax 150 affecting only a subset of economic sectors, finding that such “sectorally narrow” policies can be more 151 cost-effective than economy-wide policies if the carbon tax avoids sectors with a particularly high tax 152 interaction effect (Parry and Williams, 1999; Fullerton and Metcalf, 2001).

# 153 **I.2.3 The political economy of decarbonization**

154 While our work focuses on the possibility of PECs inhibiting the first-best climate policy, we do not 155 model the political economy explicitly. Besley and Persson (2023) study the PE of the green transition 156 in a dynamic setting where consumer values can change over time in response to new abatement 157 technologies, and policymakers are prevented from pursuing the optimal climate policy because of 158 regular elections.

159 While not quantitative from a modeling perspective, many scholars have explored the PE of a 160 transition away from fossil fuels. These studies range from examining carbon pricing and other climate 161 policies in particular nations (Crowley, 2013; Kurachi et al., 2022), to highlighting the effectiveness 162 of sub-national policy experiments (Bernstein and Hoffmann, 2018), to exploring the PE of a ‘just’ 163 green transition (Newell and Mulvaney, 2013). Hallegatte et al. (2023) propose a number of avenues to 164 address commonly faced PECs in decarbonization.

# 165 **I.2.4 Adjustment costs and socioeconomic inertia**

166 Our model includes a representation of adjustment costs through the use of a convex function for 167 abatement investments (following Vogt-Schilb et al. (2018) and Bauer et al. (2024a)). This approach 168 contrasts with a marginal abatement cost curve approach, where high levels of immediate abatement 169 incur no additional cost relative to low levels (this approach is common in many integrated assessment 7
170 models, _e.g._ , Barrage and Nordhaus, 2024). The theory of adjustment costs captures the opportunity 171 costs associated with the use of scarce resources, such as skilled labor (Lucas, 1967; Gould, 1968; 172 Mussa, 1977). Ha-Duong et al. (1997) develop a climate-economy model with adjustment costs and 173 demonstrate that adjustment costs and socioeconomic inertia act in concert to raise the optimal rate 174 of emissions abatement. More recently, Campiglio et al. (2022) include these effects in a dynamic 175 stochastic general equilibrium model of decarbonization, finding a net premium of 33% on the optimal 176 carbon price today relative to a model without adjustment costs.

## 177

# **II General theoretical model**

178 We consider a centralized planner that decarbonizes the economy for the least cost such that some 179 emissions budget, _B_ – referred to as the remaining carbon budget (RCB) – is not exceeded.<sup>7</sup> The 180 planner discounts the future at the social discount rate _r ≥_ 0. The economy has a set of _I_ sectors. 181 Investment and abatement in a given sector _i ∈ I_ are given by _xi_ ( _t_ ) _>_ 0 and _ai_ ( _t_ ) _>_ 0, respectively, 182 and the economy-wide cumulative emissions are given by _ψ_ ( _t_ ). Each sector can be described by: (i) 183 a convex cost of investment in abatement technologies, _ci_ ( _xi_ ( _t_ )); (ii) an emissions rate _a_ ̄ _i >_ 0; (iii) a 184 capital depreciation rate, 0 _< δi <_ 1; and (iv) a start time, before which no investment occurs, _t_ 0 _,i ≥_ 0. 185 The convexity of the cost function endows the system with adjustment costs (Lucas, 1967; Gould, 1968; 186 Mussa, 1977).

> 7The remaining carbon budget is a geophysical quantity that relates the amount of carbon dioxide emissions the atmosphere can withstand before a particular long-term temperature threshold is crossed (Intergovernmental Panel on Climate Change, 2021). While a global quantity from a geophysical perspective, the RCB parameter here can also be interpreted as the integrated flow of emissions a particular nation has committed to in compliance with their nationally determined climate target. Our model would then inform the cost of being compliant with such a nationally determined emissions budget.

8
# 187 **II.1 The optimal policy**

188 In this setup, the first-best policy<sup>8</sup> is the strategy where the planner immediately begins investing in 189 each economic sector, implying that for all _i ∈ I_ , _t_ 0 _,i_ = 0. The planner then solves, ̇ Subject to : _ai_ ( _t_ ) = _xi_ ( _t_ ) _− δiai_ ( _t_ ) _,_ 190 This model was studied extensively in Vogt-Schilb et al. (2018). Using the optimal path of abatement, 191 _a_<sup>_∗_</sup> _i_<sup>(</sup><sup>_t_),wecandeterminethe</sup><sup>_expost_optimalallocationofemissionstoasector</sup><sup>_i ∈I_as</sup>

> 192 which will be important for our discussion later. It is trivial to verify that � _Bi_<sup>_∗_=</sup><sup>_B_.</sup> _i∈I_

> 193 In solving (II.1), sectors can be further characterized by an optimal decarbonization date, _Ti_<sup>_∗>_0.</sup> 194 The decarbonization date is determined endogenously as the time at which _ai_ ( _t_ = _Ti_<sup>_∗_)=</sup><sup>_a_ ̄</sup><sup>_i_,binding</sup> 195 the abatement constraint at the upper bound; it is also inexorably linked to the endogenous carbon 196 price facing the sector, as higher carbon prices imply sooner decarbonization dates, and vice versa. 197 As we will make mathematically concrete below, a possible way to accommodate a PEC is to add a 198 constraint on the decarbonization date, assuming it has to be delayed relative to its optimal value, by 199 deflating the carbon price facing investors in a sector with political headwinds.<sup>9</sup>

> 8Throughout, we refer to the solution of (II.1) as the “optimal” or “first-best” policy; every other policy we analyze is referred to as “sub-optimal” or “second-best”. This does not imply that, from a political perspective, this policy is easily implementable; on the contrary, most carbon price schemes or ETSs have a number of exemptions and compromises (World Bank, 2024). However, for our purposes, the solution to (II.1) represents both a qualitative and quantitative benchmark against which we can measure the qualitative and quantitative implications of various sub-optimal policies later.

> 9This could be an implicit goal in the European Union’s ETS2. ETS2 follows different rules and has a lower starting price than ETS1, implying that the covered sectors ( _e.g._ , transportation) will be decarbonized later than if they were incorporated into ETS1, where carbon prices are often over 80€/tCO2 (European Commission, 2023).

9
# 200 **II.2 Incorporating political economy constraints**

201 To incorporate PECs, of the set of economic sectors _I_ , we denote some subset, _C ⊂ I_ , as being 202 politically challenged, while the remaining sectors, _N_ := _I \ C_ , are not (comparatively) politically 203 challenged. The planner accommodates the PEC by splitting the optimal ETS described by (II.1),<sup>10</sup> in 204 which all sectors face a unique and perfectly-credible emissions cap, into two separate, independent and 205 sub-optimal schemes. They then allocate a premium amount of emissions, _Bp >_ 0, to the politically 206 challenged sectors to the detriment of the non-challenged sectors. This procedure poses each set of 207 sectors with their own carbon price, _μj_ ( _t_ ), and emissions cap _Bj_ , and their stock of CO2 emissions are 208 tracked separately, _ψj_ ( _t_ ), for each _j ∈{C, N }_ . 209 It is worth mentioning that throughout, we consider a planner that always meets the constraint 210 that emissions are kept below _B_ . In reality, it may be possible that PECs force the planner to simply 211 adopt a higher carbon budget than is deemed globally optimal, or at a national level, for the leader to 212 weaken its NDC. This, of course, will come at a price, whether that be in terms of increased climate 213 damages or reputational costs for failing to achieve commitments made to the international community. 214 For simplicity, we do not attempt to quantify these costs in our analysis, and focus strictly on a how 215 to reallocate the emissions budget one has decided on to comply with the PECs one faces.

216 Given the above discussion, we can formulate the optimal control problem for the politically con217 strained policymaker as,

> 218 It is clear from our formulation that if _t_ 0 _,i_ = 0 and _Bp >_ 0 that we essentially have (II.1) with sub-

> 10We will refer to our policies throughout as emissions trading systems, but we note that the same results can, of course, be found for a carbon tax regime.

10
219 optimal emissions allocations, with _Bp_ controlling the degree to which the allocations are sub-optimal. 220 We solve this model in complete generality in Appendix A. 221 One theoretical result of importance is as follows.

222 **Lemma II.1.** _Along the optimal path of_ (II.3) _, the marginal investment costs c_<sup>_′_</sup> ( _xi_ ( _t_ )) _in a sector i ∈ I_ 223 _faced with the carbon price μj_ ( _t_ ) _for j ∈{C, N } can be decomposed into three parts, such that_ 224 _where c_<sup>_′_</sup> _i_<sup>(</sup><sup>_δia_ ̄</sup><sup>_i_)</sup><sup>_isthemarginalinvestmentcostinsectoriinthesteady-state,Ei_(</sup><sup>_t_)</sup><sup>_isthevalueof_</sup> 225 _emissions reductions from a marginal unit of investment at time t, Oi_ ( _t_ ) _is the “forgone opportunity”_ 226 _effect of a marginal investment in abatement, and Li_ ( _t_ ) _is the long-run value of abatement capital._ 227 _Proof._ See Appendix A.

228 This result is of significance for our discussion for two reasons. The first is that it is general for 229 any shape of the cost function inasmuch as the cost function is convex, therefore generalizing the 230 insights gleaned from our numerical simulations in Section V where we assume the cost function is 231 quadratic. The second is that PECs alter both the carbon price<sup>11</sup> and the decarbonization date of a 232 sector by changing its allocation of emissions. Therefore, this framework allows us to transparently 233 and intuitively interpret the influence of PECs on investment schedules under different policy suites, 234 tow which we now turn our attention.

## 235

# **III Policy suites**

236 Policymakers can relax or delay decarbonization initiatives in a set of economic sectors, or across the 237 entire economy, to accommodate PECs. These approaches generally take two forms, as either (i) homo238 geneous or (ii) heterogeneous policies across sectors. A homogeneous policy faces each economic sector

> 11In our framework, the optimal carbon price is determined endogenously as the shadow value of emissions reductions, see Appendix A.

11
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|---|---|---|---|---|
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|Problem<br>solved|(II.1)|(II.3)|(II.3)|(II.1)_a_<br> economy co|
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12
239 with the same, harmonized, perfectly-credible carbon price; in the face of PECs, the policymaker could 240 delay the onset of this carbon price schedule, therefore sacrificing the _efficient temporal distribution_ 241 _of abatement investment_ across sectors, to ensure the _efficient allocation of emissions_ to each sector 242 of the economy.<sup>12</sup> In contrast, a heterogeneous policy divides the economy into multiple carbon price 243 regimes, sacrificing the efficient allocation of emissions across each economic sector in favor of the op244 timal temporal allocation of investment subject to inefficient emissions budgets. The latter approach 245 can itself take two forms, in both relaxing decarbonization initiatives in the sectors facing PECs (i.e., 246 implementing a sub-optimal carbon price immediately) or by delaying the onset of a carbon price in 247 challenged sectors for some amount of time. See Table I for a summary; throughout, we will refer to 248 each “policy suite” by its number in the first column of Table I for convenience. 249 We formulate four political economy-constrained policy suites that allow us to model each of these 250 approaches to decarbonization. Each policy suite, to differing degrees, forces the planner make con251 cessions between the optimal timing of abatement investment and the optimal allocation of emissions 252 budgets across sectors because of PECs, therefore allowing us to probe the impact of the “timing versus 253 allocation” trade-off on politically constrained climate policies. Throughout our discussion, we refer to 254 a delay in either decarbonization or starting climate policy by _δT >_ 0. The first suite we consider is 255 trivial: the no PECs scenario. The planner simply enacts the first-best, optimal policy by solving (II.1). 256 Throughout, we will use this baseline as a way to quantify the impact of PECs in the other three suites 257 we consider.

258 The second policy suite involves a heterogeneous policy enacted immediately. The planner accom259 modates political headwinds by allocating a premium amount of emissions to the set of challenged sec260 tors, such that the decarbonization dates for the sectors _i ∈ C_ are shifted by _δT_ , making _Ti_ = _Ti_<sup>_∗_+</sup><sup>_δT_.</sup> 261 This approach could be attractive if the planner can implement _some_ policy immediately, but cannot 262 implement the first-best because of PECs. One interpretation of this policy suite is that the poli263 cymaker is pursuing a strategy of “ambition ramp-up” in the politically challenged sectors, where a 264 lower-than-optimal carbon price is enacted immediately and ambition is “ramped-up” in the future. 265 The required emissions premium to achieve this outcome is unique to the challenged sectors in question, 266 see Appendix B. Because policy is able to be implemented in each sector immediately, _t_ 0 _,i_ = 0 for all 267 _i ∈ I_ . This policy suite characterizes an approach that is focused on preserving the optimal timing of

> 12We note that we are considering a policy-driven transition away from greenhouse gas-emitting capital. In reality, some investment may persist in the absence of explicit government policy, though this does vary with the degree to which investors believe a climate policy will eventually be enacted (Brunner et al., 2012).

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268 abatement investment at the expense of the optimal allocation of emissions across sectors. 269 The third policy suite represents a heterogeneous carbon price policy that is delayed by some 270 number of years. For non-politically challenged sectors, the policymaker is able to implement policy 271 immediately; hence, _t_ 0 _,i_ = 0 for all _i ∈ N_ . However, in the politically challenged sectors, no policy 272 is able to be implemented for some number of years, implying _t_ 0 _,i_ = _δT_ for _i ∈ C_ . This sets the 273 emissions premium at _Bp_ = � _a_ ̄ _iδT_ and shifts all the decarbonization dates for _i ∈ C_ by _δT_ . This _i∈C_ 274 policy suite represents a “middle of the road” approach to managing the “timing versus allocation” 275 trade-off; non-challenged sectors experience investment immediately with a distorted emissions budget, 276 while abatement investment in politically challenged sectors is delayed for some amount of time, while 277 maintaining their optimal emissions allocation once the policy is introduced. 278 The final policy suite we consider a delayed homogeneous approach, where economy-wide decar279 bonization initiatives are delayed by some number of years. This is the case, for instance, in nations 280 where climate policy is “held hostage” by some set of economic sectors, or in nations where policy281 makers are waiting for all PECs to be removed to implement the first-best (albeit delayed) allocation 282 and sequencing of efforts across sectors. As an example of the latter case, a nation that depends on 283 personal cars for transportation may have strong opposition to climate policies in the transport sector, 284 and this opposition could be so strong that it impacts the entire nation’s climate agenda, therefore 285 halting economy-wide climate policy. From a modeling perspective, this approach is equivalent to 286 solving (II.1) with an emissions cap equal to _B −_ � _a_ ̄ _iδT_ , where the total costs are discounted by a _i∈I_ 287 factor of exp( _−rδT_ ). This policy suite prioritizes the optimal allocation of emissions across economic 288 sectors while sacrificing the optimal timing of abatement investment (by delaying the start of climate 289 policies).

290

# **IV Theoretical implications of political economy constraints**

291 Each of the political economy-constrained policy suites outlined above impacts policy in two ways: 292 via the carbon price facing each group of sectors and via the decarbonization date of each sector. 293 Non-challenged sectors face inflated carbon prices owing to their reduced carbon budgets, leading to 294 sooner-than-optimal decarbonization dates. The logic is reversed for the challenged sectors, whose emis295 sions allocations are higher-than-optimal, leading to deflated carbon prices and longer decarbonization 14
296 dates.<sup>13</sup> We can formalize these intuitions into the following proposition.

297 **Proposition IV.1.** _For a challenged sector k ∈ C and a non-challenged sector i ∈ N , that face carbon_ 298 _prices μC and μN , respectively, we have,_ 299 _implying non-challenged sectors’ decarbonization dates are sooner when decarbonization is delayed in_ 300 _the sectors k ∈ C;_ 301 _which means the carbon price in the non-challenged sectors increases; and_ 302 _dictating that the carbon price facing the challenged sectors decreases or stays the same._ 303 _Proof._ See Appendix A.

304 The changes in sectoral characteristics described by Proposition IV.1 impact each channel of the 305 sectoral marginal investment cost to differing extents. Using the framework established in Lemma II.1, 306 we can elucidate these impacts in the following.

307 **Lemma IV.2.** _Consider a sector k ∈ C and a sector i ∈ N . If the decarbonization of sector k ∈ C is_ 308 _marginally delayed such that its decarbonization date, Tk > Tk_<sup>_∗whereT ∗_</sup> _k_<sup>_istheoptimaldecarbonization_</sup> 309 _date, then the optimal stream of marginal investment costs in the sector i ∈ N is impacted in three_ 310 _ways:_

- 311 _1. The marginal value of emissions reductions in the sector is increased;_

- 312 _2. The marginal forgone opportunity of future investments in the sector is increased;_

- 313 _3. The marginal long-term value of abatement capital is increased._

314 _Proof._ Consider a sector _k ∈ C_ and a sector _i ∈ N_ , and allow the decarbonization date of sector _k ∈ C_ 315 to be increased such that _Tk > Tk_<sup>_∗_where</sup><sup>_T ∗_</sup> _k_<sup>is the optimal decarbonization date.Then using(II.4),we</sup>

> 13The only exception to this is when the optimal policy is delayed; this implies a larger decarbonization date, but the same carbon price.

15
316 verify the sign of the marginal change of each component of the stream of marginal investment costs 317 as follows: 318 where we used the Leibniz integral rule in to derive (IV.5) and used (IV.1)–(IV.3) throughout; note 319 especially that _∂Ti/∂Tk <_ 0 for every _i ∈ N_ .

320 Each of these theoretical results in (IV.4)–(IV.6) admit intuitive explanations. The value of emis321 sions reductions at a time _t_ in a non-challenged sector, as a result of delaying decarbonization in a 322 politically challenged sector, increases in direct proportion to the rate at which the carbon price facing 323 that sector increases with delay. This is natural in our framework, as the carbon price is the sole bench324 mark for the monetary value of emissions reductions along the optimal path (as it is the shadow price 325 of emissions). The marginal forgone opportunity effect is positive because: (i) emissions are valued at 326 a higher price (because of a higher carbon price), making forgone opportunities for future investment 327 more valuable; and (ii) the decarbonization date is sooner, implying longer time horizons over which 328 opportunities are forgone. Finally, the long-term value of abatement capital in non-challenged sectors 329 grows, because decarbonization happens sooner, implying that (i) the capital stock will be marginally 330 longer-lived and therefore more valuable, and (ii) we achieve full potential earlier and therefore the 331 value is discounted less.

331 332 Two additional theoretical corollaries can be immediately drawn from Lemma IV.2.

333 **Corollary IV.3.** _The influence of delaying the decarbonization of a set of politically challenged sectors_ 334 _on the politically challenged sector’s stream of marginal investment costs is opposite as its influence_ 335 _on the marginal investment costs for non-challenged sectors, as outlined in Lemma IV.2; that is, each_ 336 _inequality in_ (IV.4) _–_ (IV.6) _is negative. To see this, one need only insert μC as opposed to μN in_ (IV.4) _–_ 337 (IV.6) _, note that ∂μC/∂Tk <_ 0 _, and to set ∂Ti/∂Tk to unity (as i_ = _k)._ 338 **Corollary IV.4.** _The total policy cost of decarbonization for non-challenged sectors will be higher than_ 339 _the optimal case, and vice versa for the politically challenged sectors._ 16
340 Corollary IV.4 is especially relevant in the context of integrated assessment models with adjustment 341 costs, which make a swift transition away from emitting capital particularly costly, as considered here. 342 The changes in the stream of marginal investment costs for both the challenged and non-challenged 343 sectors established by Lemma IV.2 and Corollary IV.3 manifest at the aggregate level to change the 344 overall cost of policy. However, it is not immediately clear which policy response would be more costly 345 in the aggregate; indeed, each policy response results in varying levels of increased spending in non346 challenged sectors and a corresponding amount of decreased spending in the challenged sectors. This 347 ambiguity motivates our main theoretical result.

348 **Theorem IV.5.** _Consider_ (II.1) _and_ (II.3) _. Then for equivalent amounts of delay in the decarboniza-_ 349 _tion of the challenged sectors, policy suite 4 is the most expensive sub-optimal policy response to PECs,_ 350 _followed by policy suite 3, with policy suite 2 being the least expensive sub-optimal response for a fixed_ 351 _amount of delay._ 352 _Proof._ See Appendix C.

353 Theorem IV.5 allows us to establish a clear hierarchy of the costliness of each sub-optimal policy 354 response in Table I: policy suite 2 is the least costly sub-optimal response, followed by policy suite 355 3, with policy suite 4 being the most expensive. It follows from this theoretical result that it is the 356 _timing_ of abatement investment, rather than the allocation of emissions across sectors, that should be 357 prioritized when formulating politically constrained, second-best climate policies. This result provides 358 a useful benchmark for policymakers considering sub-optimal policies to combat climate change that 359 incorporate delays, and proves that doing “something” in the near-term is more cost-effective than 360 waiting to implement the theoretically optimal policy sometime in the future.

361 However, because the decarbonization dates in each sector and the carbon prices they face are 362 determined endogenously in the model (see Appendix A for details), we cannot compute closed-form 363 analytic expressions for their values and gradients to quantify effects outlined in Proposition IV.1 and 364 how these changes influence overall policy costs via Theorem IV.5. This is important for evaluating 365 the trade-offs between each sub-optimal policy response to PECs. For example, if, on the one hand, 366 the difference in cost between policy suite 2 and 4 is small, then the policymaker may be tempted to 367 implement policy suite 4, as it is the most lenient to political challenges. On the other hand, if the 368 cost difference between policy suite 2 and 4 is large, then a policymaker may be convinced to overcome 369 political constraints to implement some policy in the near term, rather than waiting to implement the 17
**TABLE II. Results of calibration scheme for global parameters and sectoral characteristics.**

|Global parameters:|_r_ = 2 % yr<sup>_−_1</sup>|_B_ = 625 GtCO2|||
|---|---|---|---|---|
|Sector| ̄_a_<br>�<br>GtCO2 yr<sup>_−_1�</sup>|_δ_ [% yr<sup>_−_1</sup>]| ̄_c_|�<br>$ tCO<sup>_−_1</sup><br>2<br>GtCO2 yr<sup>_−_3</sup><br>�|
|Waste|0.82|3.3||12954|
|Industry|5.47|4||5566|
|Forestry|8.25|0.8||2259|
|Agriculture|4.07|5||7567|
|Transport|3.74|6.7||1942|
|Energy|11.99|2.5||895|
|Buildings|3.2|1.7||4122|

370 theoretically optimal policy later on. 371 We now design a set of numerical experiments to quantify the effects of Proposition IV.1 and Theo372 rem IV.5. Using (II.1) and (II.3), we simulate each policy suite outlined in Table I. These experiments 373 allow us to quantify the influence of each policy response on: (i) the decarbonization dates of all sectors 374 and the carbon prices they face; (ii) how changes in the decarbonization date and carbon price affects 375 the temporal distribution of abatement investment across each sector of the economy; and (iii) the 376 degree to which changes in the investment path impacts overall sectoral and aggregate policy costs.

377

# **V Numerical experiments**

## 378 **V.1 Design**

379 We follow Vogt-Schilb et al. (2018) and Bauer et al. (2024a) to design a set of numerical experiments 380 that evaluate the economic costs of PECs in each political economy policy suite in Table I. We con381 sider an economy with seven economic sectors ( _i.e._ , _|I|_ = 7), following the framework outlined in the 382 Intergovernmental Panel on Climate Change’s (IPCC) Sixth Assessment Report (AR6) (Intergovern383 mental Panel on Climate Change, 2022b). The economic sectors we consider are: waste management, 384 buildings, forestry, industry, energy, agriculture, and transportation.<sup>14</sup> In all of the results we show 385 below, the planner aims to decarbonize the world economy such that global average temperatures do

> 14For more information on what specific emissions sources and abatement technologies are included in each of these sectors, we refer readers to Intergovernmental Panel on Climate Change (2022b); see the Technical Summary, Chapter TS5, Section 9, “Mitigation Potentials Across Sectors and Systems”, p. 123-125 and figures therein.

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386 not breach 1.7 °C in the long-run, in compliance with the Paris Agreement targets.<sup>15</sup> The planner 387 applies a discount rate of 2%, in line with empirical evidence on long-run real interest rates (Bauer and 388 Rudebusch, 2023) and the median estimate from a recent expert elicitation (Drupp et al., 2018). See 389 Appendix D for full details on our calibration of each model parameter. The results of our calibration 390 is summarized in Table II.

391 In our simulations of policy suites 2 and 3, we assume that two sectors may be politically chal392 lenged: energy and industry. Our motivation is that energy is the sector with the highest abatement 393 potential (and therefore annual emissions rate), implying that delaying this sector would require the 394 largest reallocation of emissions allocations in the ETS. Industry, on the other hand, has moderate 395 abatement potential (about half that of energy), but is over six times as expensive in terms of marginal 396 investment costs compared to energy. Therefore, these two sectors provide useful guideposts for our 397 discussion, where energy signifies delaying a cheap sector with substantive annual emissions, whereas 398 industry represents delaying a sector with moderate emissions but with expensive marginal investment 399 costs. Moreover, modeling delay in the energy sector could represent a situation in which a strong 400 fossil fuel lobby delays climate action, while delaying heavy industry could represent a nation that is 401 concerned about the competitiveness of domestic industries and therefore delays the decarbonization 402 of its industrial sector. We will explore the impact of delaying decarbonization for each sector listed in 403 Table II, and relate this to their sectoral characteristics, in Section VI. 404 To summarize, our experiments solve for the optimal abatement investment schedule in each policy 405 scenario described in Table I with either energy or industry serving as the politically challenged sector 406 (i.e., each set of simulations are carried out independently, with one politically challenged sector). This 407 means we solve either (II.1) or (II.3) with a specified _δT_ delay to the decarbonization date in our 408 politically challenged sector. We repeat this process for a number of _δT ∈_ (0 _,_ 10] with a discretization 409 of 0 _._ 1 years. Each model is solved in continuous time over an infinite horizon.

19
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# 440 **V.2.2 Impact on temporal distribution of abatement investment**

441 The impact of PECs on each sector’s decarbonization date and their carbon prices shown in Figures I 442 and II influences the optimal temporal distribution of investment via the mechanisms outlined in 443 Lemma IV.2. To this end, we compute each individual piece of the marginal investment path – that 444 is, the value of emissions reductions, _E_ , the forgone opportunity effect, _O_ , and the long-run value of 445 abatement capital, _L_ – for two sectors, buildings and energy, for each policy suite in Figure III. (In 446 policy suite 2 and 3, the decarbonization date of energy is a decade higher than in the optimal case.) 447 In the buildings sector (an example of a non-politically challenged sector in policy suites 2 and 448 3), we find that the value of emissions reductions (Panel IIIa)) starts higher for policy suites 2-4, 449 in-line with an inflated carbon price facing the sector. We further find that the growth rate of the 450 value of emissions reductions is roughly the same across policy suites; this can be explained by the 451 social discount rate and capital depreciation rates being the same across policy suites (see (IV.4)). 452 The opposite is the case for the forgone opportunity effect. We find that the starting level for the 453 forgone opportunity effect is roughly consistent for each policy suite (with the exception of policy suite 454 4, where economy-wide investments in decarbonization are delayed by a decade), and grows faster for 455 increasing degrees of distortion in the allocation of emissions (see Panel IIIb)). This is because the 456 extent of forgone investment opportunities depends on both the carbon price and the decarbonization 457 date (see (IV.5)), which change simultaneously when policies are distorted relative to the optimal case. 458 The combined impact of changes in the value of emissions and forgone opportunities is for a higher 459 starting level of investment that grows more rapidly and declines sooner relative to the optimal case 460 (Panel IIIc)). Finally, the long-run value of abatement capital exhibits behavior similar to the forgone 461 opportunity effect, see Panel IIId). Note that the impact on the marginal investment cost path for 462 energy in policy suite 4, where all sectors are delayed by a decade, is similar to the impact on the 463 agriculture marginal investment cost path for policy suite 2-4. 464 The influence of PECs on the marginal investment path of energy, the politically challenged sector 465 in our policy suite 2 and 3 experiments, is shown in Figure IIIe)-h). We find that the difference in 466 how the decarbonization of energy is delayed across policy suites 2 and 3 have important implications 467 for the temporal distribution of investment. In policy suite 2, the impact of PECs on the stream 468 of marginal investment costs is inverse to that of the buildings sector for the same policy suite; the 469 value of emissions reductions is depressed early on, and the forgone opportunity effect grows (that is, 470 becomes more negative) slower than in the optimal case. The combined effect is to start lower and 23
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522 where we find the marginal cost of delaying decarbonization to be increasing in delay. This result is 523 perhaps not surprising given the presence of adjustment costs: as decarbonization in the challenged 524 sector(s) is increasingly delayed, the more non-challenged sectors are squeezed to decarbonize sooner, 525 which adjustment costs make nonlinearly more expensive. The requisite decrease in spending in the 526 challenged sector cannot account for the increasing expenses in the non-challenged sectors, leading to 527 a higher overall cost of policy. Another interpretation of Figure VI is that when all policies are delayed 528 (as in policy suite 4), objectives becomes increasingly difficult to achieve without strong, costly action. 529 With a smaller delay, or a sub-optimal set of policies in politically challenged sectors ( _ ́a la_ policy suite 530 2 or 3), the increase in cost remains roughly linear and thus less costly than policy suite 4. 531 Despite their qualitative similarities, we find significant quantitative variation in the aggregate 532 policy cost across policy suites. For the delayed economy-wide policy suite, aggregate policy costs 533 can increase by as much as 50% (see Panel VIb)), and the marginal cost of delay (Panel VIc)) can 534 exceed $2.5 trillion per year of delay. In policy suites 2 and 3, we find that increases in the aggregate 535 policy cost are much less than in delayed economy-wide policy case. In our policy suites 2 and 3 536 experiments, we find that delaying energy is comparatively more expensive than industry; as energy is 537 more emissions intensive than industry (see Table II), the required emissions premium to delay energy 538 is higher than industry, leading to non-challenged sectors to be decarbonized sooner and increasing 539 costs (see Table III).

540 One final observation from Figure VI is that the quantitative implications of delay as outlined in 541 policy suite 2 are strikingly small in comparison to the other policy suites. For energy, we find that 542 delaying decarbonization by a decade in policy suite 2 results in just a 1.7% increase in overall policy 543 costs, and a 0.6% increase for delaying industry for a decade; in monetary terms, this implies increases 544 of a $410 billion and $140 billion in cost, which is dwarfed by the $11.7 trillion increase by a decade of 545 delay in policy suite 4. The sizeable difference between monetary impacts in policy suite 2 as opposed 546 to policy suite 3 or 4 results from _some_ degree of action being taken immediately. This allows the 547 policymaker to smooth investment over time, which is not possible in policy suites 3 and 4, where 548 all action in the challenged sectors is delayed. As a result, policies in policy suite 2 require far less 549 emissions premiums to accommodate delay, thus limiting their impact on aggregate policy costs.

27
<!-- Start of picture text --> Policy suite 2: Sector decarbonization effort relaxed<br>1.02 a) b) c) e Waste<br>° ° ° e Industry<br>2 1015 Forestry<br>> 3 e = Agriculture<br>36 Transport<br>£6oe= 1.01 ee = BuildingsEnergy<br>2 I 1.005 ° ° e ° ° e e 10 year delay<br>—H e e e + 5 year delay<br>+ * *<br>l * e+ + : tot + + :<br>Policy suite 3: Sector decarbonization effort delayed<br>1.125 e d) e e f).<br>E 1.100<br>a<br>3 1.075<br>oa!<br>ovr<br>2<br>1.050<br>3 ©<br>z ! * * *<br>= <1 095 ° ° e .°e we e<br>1.000 “ot 2 “7s 3<br>2 4 6 5 10 3.0 3.5 4.0<br>Captial depreciation rate, Abatement potential, Log marginal investment cost,<br>6; (% / yr) aj (GtCO / yr) logio(ci) ($ / (tCO2/yr’))<br><!-- End of picture text -->
564 stringent than the optimal policy (see the relative carbon prices in Figure II). Therefore, the political 565 environment after the delay period not only has to be amenable to the original optimal policy in the 566 delayed sectors or across the entire economy, but it may require an environment that is exceedingly 567 (perhaps unreasonably) favorable so that an even more stringent carbon price can be implemented. 568 A corollary of these results is that, in formulating climate policies under PECs, preserving the 569 optimal timing of abatement investment is a higher priority than the optimal allocation of emissions 570 across sectors; in other words, doing “something” is very much better than waiting to implement a 571 more theoretically efficient policy in the future. Comparing the overall cost increases between policy 572 suite 2 and 3 support this point: the difference in the additional policy cost induced by delaying 573 the decarbonization of energy by a decade between policy suites 2 and 3 is _∼_ $2 trillion, granted 574 this difference is smaller for industry. Of course, each of these options are better than delaying all 575 climate policies.<sup>21</sup> We conclude that, if a policymaker were facing political headwinds while still 576 wanting to achieve climate goals, implementing a sub-optimal carbon price or ETS that delays the full 577 decarbonization of challenged sectors may be an attractive policy approach. This conclusion would 578 support the idea of exemptions for politically sensitive sectors if and only if they make it possible to 579 implement the policy sooner.<sup>22</sup> 580 Throughout, we argued that energy is the more expensive sector to delay decarbonizing because of 581 its high abatement potential. We now make this notion concrete by computing the relative policy cost 582 of delaying each sector in our numerical experiments by five and ten years in Figure VII. We find that 583 our earlier arguments are well-supported: energy is consistently the most expensive sector to delay. 584 In computing the correlation of each sectoral characteristic and relative aggregate policy costs (see 585 Table IV), we find that the annual emissions rate is consistently the most correlated sectoral parameter 586 with changes in relative policy cost increases, followed by the marginal investment cost and the capital 587 depreciation rate.<sup>23</sup> These results would suggest that policymakers using delay to assuage PECs should

> 21One possible exception to this conclusion could be policies which are administratively burdensome and do not lead to tangible emissions reductions. In this case one would have the worst of both worlds: nontrivial costs and a reduced remaining carbon budget, which would demand an accelerated and costly policy schedule later to remain compliant with the emissions budget.

> 22This point would suggest that the approach taken by the EU in their establishment of ETS2 is prudent from an economic and political perspective.

> 23We note that there is significant anti-covariance between emissions rates and marginal investment costs in the IPCC AR6 data (Intergovernmental Panel on Climate Change, 2022b); energy and forestry are at once the cheapest sectors and the most emissions intensive, while waste has the lowest emissions rate and the highest marginal investment cost. This happenstantial attribute of the data does skew our results somewhat; for example, excluding energy, forestry, and waste results in unclear correlation structures between additional policy costs and sectoral characteristics. However it would be difficult to claim a relationship with only four sectors is at all statistically significant. We therefore rely on the intuition established above in our analysis: the higher the emissions rate, the more of a distortion in the optimal allocation of emissions across ETSs, causing higher additional policy costs relative to the optimal. Note this statement is

29
**TABLE IV.** _r_<sup>2</sup> **values for regressions between parameter values and relative policy cost after delay, shown in Figure VII.**

|Policy suite|Delay amount|(years)<br>_δ_| ̄_a_|log10( ̄_ci_)|
|---|---|---|---|---|
|Policy suite 2|5|0.28|0.95|0.61|
||10|0.26|0.97|0.69|
|Policy suite 3|5|0.32|0.89|0.69|
||10|0.28|0.92|0.69|

588 focus on delaying sectors with low levels of emissions – not those with high marginal abatement costs 589 – as these would induce the least amount of cost increases in aggregate. Put differently, while it 590 makes sense to be pragmatic and delay action in politically challenging sectors, this does not apply to 591 the largest emitting sectors; indeed, some political constraints can and should be navigated through 592 compromise and second-best policies, while others must be overcome, as the costs of the second-best 593 option significantly outweigh the benefit of bending to PECs.

# 594 **VI.2 The influence of technology change**

595 One aspect of decarbonization policies not explicitly addressed in our analysis is endogenous technology 596 change (Hogan and Jorgenson, 1991; Acemoglu et al., 2012; Armitage et al., 2023).<sup>24</sup> Endogenous 597 technological progress – or “learning by doing” (Schmidt and Sewerin, 2017) – is spurred by investment 598 in a given sector and can impact the allocation of investment over time (Kverndokk et al., 2004; 599 Gillingham and Stock, 2018). Nowhere have the effects of learning-by-doing been more pronounced 600 than in photovoltaic solar panels, the cost of which have declined substantially over the last decade 601 plus (Bollinger and Gillingham, 2019). Recent work has further shown that learning-by-doing and 602 knowledge spillovers substantially boost global welfare gains as a result of the Inflation Reduction 603 Act (Arkolakis and Walsh, 2023).

604 While not modeled explicitly, we can comment on how our results would interact with endoge605 nous technology change. The first clear implication is that, since the rate of endogenous technolog606 ical progress depends on the cumulative amount of investment in a given sector (akin to “Wright’s 607 law” (Wright, 1936)), delaying investment in sectors with high rates of endogenous learning would be

> made independent of the level of marginal investment costs.

> 24Although, in Appendix A, we do show that an economy-wide, exogenous rate of technological progress, 0 _≤ φ ≤_ 1, is equivalent to a shift in the social discount rate, _r → r_ + _φ_ .

30
608 more costly than delaying sectors with low rates of learning. The same argument applies to sectors 609 with high degrees of inter-sectoral knowledge spillovers, such as energy (Richels and Blanford, 2008). 610 Finally, having a credible carbon price in place, even if temporarily deflated, may spur additional pri611 vate sector investment in abatement technologies (Brunner et al., 2012) which could bring down costs. 612 This would reinforce our overall conclusion that preserving the optimal timing of abatement investment 613 is of a higher priority than preserving the optimal allocation of emissions across sectors.

# 614 **VI.3 Generalized insights**

615 While we focused our discussion on the issue of combating climate change, a generalization of our 616 model of carbon emissions abatement investment, given by either (II.1) or (II.3), to general abatement 617 investment is natural. In essence, our model takes atmospheric CO2, a previously nonrival good between 618 sectors of the economy, and makes it rivalrous between sectors via a quota on net accumulated use; 619 the quota is represented by _B_ . The remaining parameters can be generalized as follows: _a_ ̄ _i_ represents 620 the exogenous use-rate of the good in each sector; _xi_ ( _t_ ) represents the investment in a generalized 621 abatement technology, with convex investment costs _ci_ ( _xi_ ( _t_ )); _ai_ ( _t_ ) represents a generalized abatement 622 capital stock; the economy-wide use of the good is accumulated via _ψ_ ( _t_ ); and the shadow value of a 623 marginal unit of abatement is given by _μ_ . Given this “mapping”, our insights are made generalizable to 624 any nonrivalrous good that is made rivalrous via an economy-wide quota, as are the cost implications 625 of delay-based policies in transitioning away from the use of this good. In particular, our findings via 626 Theorem IV.5 still hold; that is, implementing some policy in the near-term is a more cost-effective 627 approach to complying with PECs than delaying implementation of the theoretically optimal response 628 in the future, regardless of the nature of the good itself, insofar as it can be reasonably modeled by our 629 framework.

# 630 **VI.4 The timing of damages**

631 While the policies we discuss in our setup are always compliant with the implied temperature target, 632 the rate at which the temperature target is reached varies across policies. This owes to the fact that 633 abatement schedules are different across policies, and cumulative emissions are constrained only to not 634 exceed the ETS cap. While this approach has the advantage of getting around thorny issues of defining 635 a damage function for climate change (that is highly uncertain (Intergovernmental Panel on Climate 636 Change, 2022a) with existing parameterizations having been widely criticized (Stern, 2013; Pindyck, 31
<!-- Start of picture text --> neeees Policy suite 1: 1.70 TITS TT SS TS es SS are<br>35 First-best. policy a) . b)<br>‘ __., Policy suite 2: Energy g -<br>~ 30 .: Policydecarbonizationsuite 3: Energy effort relaxed S* 1.65 a:<br>& : decarbonization effort delayed aan :<br>me<br>8 29 ** _..., dePo l icyay suite 4: Economy-wide Sae 2, 1.60 ::<br>% 20 : BE ‘<br>eS n :<br>5 15 3.8 j<br>a ce<br>e 10 \ & 1.50<br>ca . Ey<br>0 “, Se“ea SSS mx S145 ---= Temperature target<br>2025 2050 2075 2100 2025 2050 2075 2100<br>Year Year<br><!-- End of picture text -->
**TABLE V. Year that 1.6 °C and 1.7 °C are reached for different baselines.**

|Baseline|1.6 °C|1.7 °C|
|---|---|---|
|Baseline 1|2041|2096|
|Baseline 2 (Energy)|2040|2090|
|Baseline 3 (Energy)|2039|2080|
|Baseline 4|2036|2058|

654 to the extent that the optimal policy in each policy suite warms faster than the optimal policy, but 655 that our general ranking of the costliness of these policy suites given by Theorem IV.5 would still hold.

# 656 **VI.5 Other sources of risk**

657 Throughout, we have only considered the risk of PECs in formulating second-best policies to address 658 climate change. However, it is well-known that the climate response is uncertain (Sherwood et al., 659 2020; Intergovernmental Panel on Climate Change, 2021), and that this makes the issue of addressing 660 climate change one of “risk management” (Stern, 2013). Indeed, a number of papers have highlighted 661 how climate and economic uncertainty implies more stringent climate policies (see Lemoine and Rudik 662 (2017) for a review, and Lemoine (2021)). It stands to reason that the overall cost of PECs would only 663 increase in the presence of climate uncertainty, especially if the worst-case climate scenario arises<sup>27</sup> ; for 664 example, Daniel et al. (2019) show that risk from climate damages grows quadratically in delay.

# 665 **VI.6 Future directions**

666 We imagine many extensions of the present work. For one, future work could include the myriad other 667 risks present in the climate-economic system that are not discussed here, such as compounding climate 668 risk (Bauer et al., 2024b), transition risk (Campiglio et al., 2022; Barnett, 2023), and climate tipping 669 points (Dietz et al., 2021). One can easily imagine the prospect of delaying decarbonization interacting 670 with each of these risks to exacerbate the economic costs of delay and reinforce the central findings of 671 this paper. Secondly, here we represented climate policies as a carbon price (implemented through an 672 ETS), but other policy instruments could be considered as counterfactuals to the policy suites we dis673 cuss; for example, subsidies for green innovation, command-and-control technology standards, or green 674 portfolio standards could either increase or decrease the economic costs in each policy suite (Fischer

> 27Bauer et al. (2024a) amend the framework of (II.1) to include climate uncertainty by making the remaining carbon budget uncertain and showed that the overall urgency of climate policy is increased.

33
675 and Newell, 2008; Armitage et al., 2023). Future work could also quantify the additional economic cost 676 of missing technology and knowledge gains as a result of delaying climate policy. Finally, while our 677 numerical examples take a global view, it would be interesting and useful to apply this framework to 678 country-level nationally determined contributions to the Paris Agreement to determine how the con679 stitution of an individual nation’s economy either exacerbates or nullifies the increase in policy costs 680 described here; this is especially the case as the emissions intensity of each sector can vary substantially 681 across countries, meaning that the costliest sectors to delay are bound to change based on the specifics 682 of the economy being examined.

## 683

# **VII Conclusion**

684 In this paper, we elucidated how the “timing versus allocation” trade-off underscores climate policies 685 under PECs, and how these PECs impact economy-wide decarbonization dates, optimal carbon prices, 686 the temporal distribution of abatement spending, and aggregate policy costs. We presented four distinct 687 political economy-constrained policy suites to explore qualitative and quantitative changes in climate 688 policies to navigate political challenges. We close by reemphasizing two major themes of this work. 689 Firstly, we showed, both theoretically and in numerical experiments, that preserving the optimal 690 timing of abatement investment outweighs preserving the optimal allocation of emissions across sectors 691 for climate policies under PECs. We found that implementing an uncoordinated policy suite across 692 sectors is more cost-effective than delaying action to improve coordination, and can substantially lower 693 the cost burden of assuaging political headwinds. Implementing a sub-optimal carbon price in politically 694 challenged sectors,<sup>28</sup> with a modest increase in the carbon price for other sectors, can lead to both 695 delayed decarbonization in the politically challenged sector (thus complying to political constraints) 696 and only moderate overall policy cost increases; our numerical experiments estimate, at the maximum, 697 a 1.7% increase in aggregate policy costs as a result of delaying challenged sectors’ decarbonization by 698 a decade using this policy approach. On the other hand, doing nothing in a set of politically challenged 699 sectors or economy-wide leads to an increased cost of policy relative to doing something; indeed, our 700 numerical experiments suggest that overall policy costs could increase by as much as 50% if all policies 701 are delayed by a decade.

> 28Note that throughout we use a carbon price as a proxy for emission reduction in general, but in practice these reductions may be actualized through other policy instruments. For example, if the optimal building energy efficiency standard (that would create a similar marginal cost of abatement as other sectors) is politically infeasible, it is better to implement some improved standards rather than delaying standards altogether.

34
702 The second key point is that delaying decarbonization is most costly for sectors with high emissions 703 intensities. This is because the amount of emissions that need to be reallocated (in a sub-optimal 704 fashion) is largest when emissions intensive sectors are delayed, leading to the largest distortion in 705 carbon prices and decarbonization schedules, which adjustment costs make especially expensive.<sup>29</sup> 706 From this we can conclude that sectors such as energy, forestry, and industry would be the most 707 expensive to delay, whereas waste and buildings would be the least expensive in relative terms. (Of 708 course, delaying any sector is always more expensive than the optimal policy.) This conclusion can 709 provide guidance to policymakers facing difficult choices in domestic climate agendas as to what set of 710 second-best policies they may pursue, and their economic consequences.

## 711

# **Author affiliations**

712 Adam Michael Bauer

- 713 1. Department of Physics, University of Illinois Urbana-Champaign, Urbana, IL 61801

- 714 2. The World Bank Group, Washington DC

- 715 St ́ephane Hallegatte

- 716 1. The World Bank Group, Washington DC

- 717 Florent McIsaac

718

1. The World Bank Group, Washington DC

719

# **Data availability statement**

720 The code to reproduce the calculations in this paper is available at https://github.com/adam-bauer721 34/BHM-pol-econ-reprod.

> 29Throughout, we consider an ETS for modeling purposes, but similar conclusions would be found if we focused instead on a “carbon budget” approach (Department for Energy Security and Net Zero, 2023), regulations, and other policy instruments.

35
»»» »» »»» »»
733 date _Ti_ in each sector, as shown in the following two Lemmas.

734 **Lemma A.1.** _Consider_ (II.3) _. For all t < Ti with i ∈I, λi_ ( _t_ ) = 0 _._ 735 _Proof._ Recall that the decarbonization date _Ti_ is defined as _Ti_ := inf _{t ∈_ R<sup>+</sup> : _ai_ ( _t_ ) = _a_ ̄ _i}_ . It 736 immediately follows that, prior to the decarbonization date, _a_ ̄ _i − ai_ ( _t_ ) _>_ 0 by definition, implying that 737 _λi_ ( _t_ ) = 0 for all _t < Ti_ by complementary slackness.

738 **Lemma A.2.** _Consider_ (II.3) _. For all t < Tj_<sup>_∗where j∈{C, N} and T ∗_</sup> _j_<sup>:= max</sup><sup>_{Tk_:</sup><sup>_k∈j},φj_(</sup><sup>_t_) = 0</sup><sup>_._</sup> 739 _Proof._ Consider a set of sectors _j ∈{C, N }_ and the maximum decarbonization time for that set of 740 sectors _Tj_<sup>_∗_:=max</sup><sup>_{Tk_:</sup><sup>_k∈j}_.Giventhat</sup><sup>_ψj_(</sup><sup>_t_=0)=0and</sup><sup>_a_ ̄</sup><sup>_i−ai_(</sup><sup>_t_)</sup><sup>_>_0forall</sup><sup>_i∈j_and</sup><sup>_t<Ti_,</sup> 741 _ψ_ ̇ _j_ ( _t_ ) _>_ 0 for all _t < Tj_<sup>_∗_.The constraint</sup><sup>_ψj_(</sup><sup>_t_)</sup><sup>_≤Bj_,where</sup><sup>_Bj_is given by(II.2) for</sup><sup>_j∈{C, N}_binds at</sup> 742 the time _Tj_<sup>_∗_= max</sup><sup>_{Tj_:</sup><sup>_j∈{C, N}}_, i.e., when the last unit of emissions has been emitted before total</sup> 743 decarbonization. Hence, for all _t < Tj_<sup>_∗_,</sup><sup>_Bj−ψj_(</sup><sup>_t_)</sup><sup>_>_0,andbycomplementaryslackness,</sup><sup>_φj_(</sup><sup>_t_) = 0for</sup> 744 _t < Tj_<sup>_∗_.</sup> 745 Using Lemmas A.1 and A.2, the first order conditions for the costate variables become, 746 for all _t < Ti_ . Solving (A.7) implies the following path for the carbon price in the sector groupings 747 _j ∈{C, N }_ is given by 748 where we have defined _μj_ ( _t_ = 0) = _μj_ as the initial carbon price, which follows an exponentially in749 creasing path, in-line with the Hotelling rule (Hotelling, 1931).<sup>30</sup> Using (A.9) in (A.8) and rearranging, 750 we have 751 for each _j ∈{C, N }_ and _i ∈ j_ , where we have used the fact that _μj_ is interpreted as a tax to change

> 30In an abuse of notation, we have relabeled _t − t_ 0 _→ t_ , as the shift does not materially change the results if _t_ is always defined as the first period when investment begins.

37
752 its sign. We can solve (A.10) using variation of parameters, 753 where _Cx ∈_ R is a to-be-determined constant. Note we integrate from _t → Ti_ as our boundary condition ̄ 754 for investment – that in the steady state, _ci_ ( _xi_ ( _Ti_ )) = _ci_ ( _δiai_ ) – occurs at the decarbonization time _Ti_ . 755 Using the aforementioned boundary condition, we can solve for _Cx_ and write the final solution for the 756 optimal marginal cost of investment as 757 Eqn. (A.12) can be written in an equivalent form, which provides the proof for Lemma II.1.

758 _Proof of Lemma II.1._ Eqn. (A.12) can easily be rewritten as 759 where _Ei_ ( _t_ ) is the value of the emissions avoided by an investment at time _t_ in sector _i_ , _Li_ ( _t_ ) is the 760 long-term value of the abatement capital in sector _i_ , and _Oi_ ( _t_ ) is the forgone opportunity of future 761 investments after decarbonization in sector _i_ . The same expression was reached by Vogt-Schilb et al. 762 (2018), though in a setting without political economy constraints.

763 In order to continue with an analytically tractable model for the remaining state variables, we must 764 make the following assumption about investment costs.

765 **Assumption 1.** _The cost of investment takes a quadratic form,_ 766 Using Assumption 1, we can solve (A.12) for the optimal investment path, _x_<sup>_∗_</sup> _i_<sup>(</sup><sup>_t_)as</sup> 38
( J) ~~-(~~ — ~~+)~~ ~~(+)~~ J> > ~~i~~ <u>( ) ( )(</u> ~~ee~~ <u>)</u>
777 With (A.23) in hand, we can prove Proposition IV.1.

778 _Proof of Proposition IV.1._ Consider a challenged sector _k ∈ C_ . If _Tk > Tk_<sup>_∗_,then</sup><sup>_Bp>_0,and</sup><sup>_μC< μ∗_</sup> 779 where _μ_<sup>_∗_</sup> is the optimal carbon price (given by setting _Bp_ = 0) immediately follows from (A.23), 780 proving (IV.3). The same logic implies that for _j ∈ N_ and _Bp >_ 0, that is, _μN > μ_<sup>_∗_</sup> and _Ti < Ti_<sup>_∗_,</sup> 781 proving the proposition.

782 Combining (A.15), (A.18), and (A.20) results in the optimal solution to our model; the equivalent 783 expressions for the non-challenged sectors are straightforward analogs. What remains is to determine 784 numerical values for the decarbonization dates and the carbon price. We cannot hope to find solutions 785 analytically, given the immense complexity of the equations involved. We therefore solve the following 786 system of nonlinear, implicit equations for the carbon price and decarbonization dates, 787 using a numerical root-finding algorithm. This completes the analytical solution to the model equations, 788 see Table VI.

789 A final result is the following Lemma, showing the influence of exogenous technology change on the 790 model.

791 **Lemma A.3.** _Consider_ (II.1) _with an exogenous, economy-wide, constant technology growth rate given_ 792 _by φ >_ 0 _. This has the equivalent impact on policy as a shift in the social discount rate by φ._ 793 _Proof._ Consider (II.1), and allow the technology-adjusted cost of investment in a sector _i ∈ I_ to be 794 given by 795 where Φ _i_ ( _t_ ) is the technology growth rate in the sector. If technology growth is exogenous, constant, 796 and economy-wide, then Φ _i_ ( _t_ ) = _φ_ , and can be assimilated into the discounting term in the objective 797 function of (II.1), with the new discount rate 40
**TABLE VI. Full analytic solutions of** (II.3) **subject to Assumption 1 for each sector set** _j ∈{C, N }_ **and** _i ∈ j_ **.**

|Variable|_t < Ti_|_t > Ti_|
|---|---|---|
|Investment|(A.15)|_δi_ ̄_ai_|
|Abatement|(A.18)| ̄_ai_|
|Cumulative emissions|(A.20)|_σjBp_+<br>�<br>_i∈j_<br>_B_<sup>_∗_</sup><br>_i_|

798 as desired.

# 799 **B Determining emissions premiums for policy suite 2**

800 We now provide an algorithm for determining how much emissions premium, _Bp_ , is required to delay a 801 sector’s decarbonization date, _Ti_ , by some number of years, _δTi_ . For simplicity, consider one challenged 802 sector, such that _|C|_ = 1. The system (A.24)–(A.26) can be formulated as a constraint to an additional 803 root-finding algorithm, such that

804 where (B.2) ensures we always get a positive _T_ ( _Bp_ ), _T_<sup>_∗_</sup> is the decarbonization date in the optimal case 805 (i.e., without political constraints), _δT_ is the delay amount, and 806 In words, the approach is to specify some _Bp_ , and solve (A.24)–(A.26) using a root-finder. This yields 807 the decarbonization date of the challenged sector, _T_ ( _Bp_ ); comparison with this decarbonization date 808 and the target date _T_<sup>_∗_</sup> + _δT_ informs the next _Bp_ choice. Carrying out this process iteratively results 809 in a _Bp_ such that the new decarbonization date in the challenged sector, _T_ ( _Bp_ ) is exactly equal to 810 _T_ + _δT_ , as desired.

41
811

# **C Proof of Theorem IV.5**

812 **Forerunners.** Consider (II.1) and (II.3). We first note the signs of each of the following partial 813 derivatives: 814 where (C.1) follows from the definition of convexity, (C.2) follows from (A.15), (C.3) immediately from 815 Lemma (II.1), and (C.4) follows from (A.15). The main equation we need to consider the following, 816 which represents the aggregate policy costs, given by 817 Note if _C_ = _∅_ and _μj_ = _μ_ where _μ_ is the optimal carbon price, (C.5) is the optimal cost, which we 818 denote as _Copt_ . Now, to the proposition. With these forerunners, we can prove the theorem.

819 _Proof of Theorem IV.5._ Consider (II.1) and (II.3). Assume that there are _C_ challenged sectors facing 820 a carbon price _μC_ and a set of non-challenged sectors, _N_ , facing a carbon price _μN_ , such that the 821 decarbonization of the challenged sectors is delayed by some amount _δT >_ 0. One key inequality to 822 note is that 823 which follows from Lemma II.1. We will prove the theorem via a perturbative approach. Throughout, 824 let _μ_ be the optimal carbon price.

825 **Economy-wide delay (policy suite 4).** In an economy-wide delay strategy, we are effectively 826 perturbing the economy-wide carbon price _μ → μ_ + _δμ_ where 0 _< δμ ≪_ 1 by infinitesimally shrinking 827 our pollution quota, driving up the shadow value of abatement infinitesimally. If _Cecon−wide_ is the 42
, Xt | f ( ~~EGC —))~~ ) ©) [ (et | r{ ( ~~—~~ ) ~~—~~ }) — ~~(z~~ - ~~(- —))~~ ) fi (e ~~-G~~ —))) o>
~~———~~ ( ~~-|~~ jf ( ~~Ee —))~~ ) ( ~~KH~~ yf <u>(</u> ~~<u>eG —)</u>~~ <u>)</u>

~~—— (+ +~~ )/ ( ~~ee —))~~ ) ~~(I~~ yf ( ~~e-C —)~~ )
855 assumption of equivalent amounts of delay for each policy suite would be violated.

856 Hence, we have 857 and by (C.14) we have 858 Therefore, it must be the case that 859 as desired.

860

# **D Calibration of numerical experiments**

861 We follow the prescriptions of Vogt-Schilb et al. (2018) and Bauer et al. (2024a) to calibrate our 862 numerical experiments. We lift marginal abatement cost and abatement potential rates from Intergov863 ernmental Panel on Climate Change (2022b), and utilize the algorithm presented in Vogt-Schilb et al. 864 (2018) to compute the marginal investment costs for each sector. We use estimates for the capital 865 depreciation rates of capital in each sector from Philibert (2007) and World Bank (2012). We lift the 866 central estimate of the RCB for the 1.7 °C temperature target from Friedlingstein et al. (2023). This 867 approach gives us the parameter values found in Table II.

868

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