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 & A verified reproducibility package for this paper is Macroeconomics, Trade and Investment available at http://reproducibility.worldbank.org, Global Practice click here for direct access. November 2024 Policy Research Working Paper 10971 Abstract When leaders face political economy constraints, is it best to delaying decarbonization efforts in a subset of sectors or delay all decarbonization initiatives until a sectorally coor- economy-wide. The paper shows that the cost difference dinated strategy can be implemented, or is it preferable between an economy-wide, coordinated decarbonization to implement an approach where sectors’ decarbonization strategy and an uncoordinated approach with heteroge- strategies are uncoordinated? This question underscores a neous carbon prices is smaller than the cost of delaying crucial trade-off—here coined the “timing versus allocation” action and implementing a coordinated policy in the future. trade-off—for politically constrained climate policymakers: This implies that it is preferable to implement some policy (i) to sacrifice the optimal timing of climate policies to pre- in each sector, insofar as this is politically feasible, with less serve the optimal allocation of emissions across economic politically challenged sectors compensating with a marginal sectors, or (ii) to preserve the optimal timing of abatement increase in policy ambition. The paper further elucidates investment to the detriment of the allocation of emissions how sectors with high annual emission rates, such as energy, across sectors. This paper systematically explores this trade- are more costly to delay in comparison to their mid- to off by presenting a modeling framework that elucidates low-emission counterparts, such as industry, despite these the economic implications of various sub-optimal policy sectors being nominally more costly sectors to decarbonize. approaches to decarbonization that involve relaxing or 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. 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Produced by the Research Support Team The Timing versus Allocation Trade-off in Politically Constrained Climate Policies∗ Adam Michael Bauer† ephane Hallegatte St´ 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).1 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 1 In 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 regu- 31 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.2 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.3 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 decar- 50 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 op- 52 timal.4 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. 2 Another possibility is that leaders may be waiting to implement decarbonization policies until the cost of abate- ment technologies decline owing to global knowledge spillovers from countries with climate policies, i.e., they are “free riders” (Nordhaus, 2021). 3 The 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). 4 This 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.5 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’ decarboniza- 74 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 anti- 77 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 5 This 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 Nord- 121 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) com- 126 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 command- 138 and-control regulation to address an environmental externality.6 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. 6 141 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 ıo Gonz´ many nations pursuing subsidy-based or other second-best policies; Del R´ 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.7 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 ¯i > 0; (iii) a a convex cost of investment in abatement technologies, ci (xi (t)); (ii) an emissions rate a 184 capital depreciation rate, 0 < δi < 1; and (iv) a start time, before which no investment occurs, t0,i ≥ 0. 185 The convexity of the cost function endows the system with adjustment costs (Lucas, 1967; Gould, 1968; 186 Mussa, 1977). 7 The remaining carbon budget is a geophysical quantity that relates the amount of carbon dioxide emissions the atmo- sphere 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 policy8 is the strategy where the planner immediately begins investing in 189 each economic sector, implying that for all i ∈ I , t0,i = 0. The planner then solves, ∞ min e−rζ ci (xi (ζ ))dζ, (II.1) {xi (t)}i∈I 0 i∈ I ˙ i (t) = xi (t) − δi ai (t), Subject to : a ˙ (t) = ψ ai − ai (t)) , (¯ i∈I 0 ≤ ai (t) ≤ a ¯i , 0 ≤ ψ (t) ≤ B. 190 This model was studied extensively in Vogt-Schilb et al. (2018). Using the optimal path of abatement, 191 a∗ i (t), we can determine the ex post optimal allocation of emissions to a sector i ∈ I as ∞ ∗ Bi := ai − a∗ (¯ i (ζ )) dζ, (II.2) 0 ∗ 192 which will be important for our discussion later. It is trivial to verify that Bi = B. i∈ I 193 In solving (II.1), sectors can be further characterized by an optimal decarbonization date, Ti∗ > 0. 194 The decarbonization date is determined endogenously as the time at which ai (t = Ti∗ ) = a ¯i , binding 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.9 8 Throughout, 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. 9 This 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),10 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 con- 217 strained policymaker as, ∞ min e−rζ ci (xi (ζ ))dζ , (II.3) {xi (t)}i∈I t0,i i∈I ˙ i (t) = xi (t) − δi ai (t), Subject to : a i ∈ I, ˙ j (t) = ψ ak − ak (t)) , (¯ j ∈ {C, N }, k∈j 0 ≤ ai (t) ≤ a ¯i , i ∈ I, ∗ 0 ≤ ψj (t) ≤ σj Bp + (Bk − t0,k a ¯k ) , j ∈ {C, N }, k ∈j   1  if j ∈ C σj = .  −1  if j ∈ N 218 It is clear from our formulation that if t0,i = 0 and Bp > 0 that we essentially have (II.1) with sub- 10 We 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′ (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 ∞ c′ i (xi (t)) = µj (t) e−δi (s−t) ds t =:Ei (t) ∞ −δi (s−t) −µj (t) e ds Ti =:Oi (t) + c′ ¯i )e−(r+δi )(Ti −t) , i (δi a (II.4) =:Li (t) 224 where c′ ¯i ) is the marginal investment cost in sector i in the steady-state, Ei (t) is the value of i (δi a 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 price11 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) homo- 238 geneous or (ii) heterogeneous policies across sectors. A homogeneous policy faces each economic sector 11 In our framework, the optimal carbon price is determined endogenously as the shadow value of emissions reductions, see Appendix A. 11 TABLE I. Political economy-constrained policy suites and how they influence model parameterization. Number Policy suite Policy type Policy description Problem Parameters Emissions pre- solved impacted mium, Bp 1 First-best Homogeneous Optimal, first-best policy (II.1) N/A N/A 2 Relaxed sectoral ac- Heterogeneous Decarbonization date delayed (II.3) Ti for i ∈ C Varies, see Ap- 12 tion pendix B 3 Delayed sectoral ac- Heterogeneous All action in politically chal- (II.3) t0,i for i ∈ C a ¯i t0,i tion lenged sectors delayed i∈ C 4 Delayed economy- Homogeneous All action in all sectors delayed (II.1)a t0,i for i ∈ I a ¯i t0,i wide action i∈I a For this case, (II.1) is solved with B → B − Bp , as all sectors are impacted by the political economy constraint, and the final costs are discounted by e−rt0 . 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.12 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 op- 244 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 con- 251 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 accom- 259 modates political headwinds by allocating a premium amount of emissions to the set of challenged sec- 260 tors, such that the decarbonization dates for the sectors i ∈ C are shifted by δT , making Ti = Ti∗ + δT . 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 poli- 263 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, t0,i = 0 for all 267 i ∈ I . This policy suite characterizes an approach that is focused on preserving the optimal timing of 12 We 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). 13 268 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, t0,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 t0,i = δT for i ∈ C . This sets the 273 emissions premium at Bp = ¯i δT and shifts all the decarbonization dates for i ∈ C by δT . This a 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 decar- 279 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 policy- 281 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 − ¯i δT , where the total costs are discounted by a 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 emis- 295 sions allocations are higher-than-optimal, leading to deflated carbon prices and longer decarbonization 14 296 dates.13 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, ∂Ti < 0, for i ∈ N, (IV.1) ∂Tk 299 implying non-challenged sectors’ decarbonization dates are sooner when decarbonization is delayed in 300 the sectors k ∈ C ; ∂µN > 0, (IV.2) ∂Tk 301 which means the carbon price in the non-challenged sectors increases; and ∂µC ≤ 0, (IV.3) ∂Tk 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 where Tk is the optimal decarbonization 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 where Tk is the optimal decarbonization date. Then using (II.4), we 13 The 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: ∂Ei (t) ∂µN r(t−t0,i ) ∞ −δi (s−t) = e e ds > 0; (IV.4) ∂Tk ∂Tk t ∂Oi (t) ∂µN r(t−t0,i ) ∞ −δi (s−t) ∂Ti = e e ds − µN er(t−t0,i ) e−δi (Ti −t) > 0; (IV.5) ∂Tk ∂Tk Ti ∂Tk ∂Li (t) ∂Ti = −(δi + r)e−(δi +r)(Ti −t) > 0, (IV.6) ∂Tk ∂Tk 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 emis- 321 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 bench- 324 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. 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 non- 346 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 characteris- tics. Global parameters: r = 2 % yr−1 B = 625 GtCO2 $ tCO− 1 Sector ¯ GtCO2 yr−1 a δ [% yr−1 ] c ¯ 2 GtCO2 yr−3 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 Theo- 372 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 con- 381 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) (Intergovern- 383 mental Panel on Climate Change, 2022b). The economic sectors we consider are: waste management, 384 buildings, forestry, industry, energy, agriculture, and transportation.14 In all of the results we show 385 below, the planner aims to decarbonize the world economy such that global average temperatures do 14 For 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. 18 386 not breach 1.7 °C in the long-run, in compliance with the Paris Agreement targets.15 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 chal- 392 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 FIGURE I. Change in decarbonization date for each sector in each policy suite. 410 V.2 Results 411 V.2.1 Impact on decarbonization dates and carbon prices 412 We first consider how the decarbonization date for each sector is changed when either energy or industry 413 is delayed by some number of years in Figure I (note, blue lines represent when energy is politically 414 challenged, and orange lines represent when industry is politically challenged). As previously discussed, 415 non-challenged sectors are decarbonized earlier when a challenged sector is decarbonized later, but the 416 magnitude of this change is highly dependent on both the non-challenged sector we are considering 417 and which sector is politically challenged. In particular, the observed change in the decarbonization 418 date are most pronounced for hard-to-abate sectors (industry, agriculture, and transport),16 where 419 decarbonization must be accelerated by as much as 40 years in policy suite 4. ‘Easy-to-abate’ sectors 420 (waste, forestry, energy, and buildings) experience more modest changes in their decarbonization dates. 421 Relative to the optimal policy, the decarbonization dates of ‘easy-to-abate’ sectors decrease by at most 422 7 years. 15 We do not consider 1.5 °C as our base case for the global temperature target, as state-of-the-art geophysical estimates of the remaining carbon budget distribution for the 1.5 °C target cannot rule out negative values, implying we have already passed the target based on present-day levels of emissions (Intergovernmental Panel on Climate Change, 2021; Dvorak et al., 2022). Therefore, it would be slightly unrealistic to present the economic costs of delaying enacting a policy to comply with this temperature threshold, as it may be too late already. 16 These sectors are often denoted as ‘hard-to-abate’ by virtue of their high marginal investment costs and their relatively large emissions intensities, see Table II. 20 FIGURE II. Carbon price indices for politically challenged and non-politically challenged groups in each policy suite. 423 An interesting finding shown in Figure I is that policy suite 4, where all economy-wide effort is 424 delayed, induces the most pronounced distortion in decarbonization dates, followed by policy suites 3 425 and 2, respectively. This owes to the fact that if decarbonization policy is implemented later, a larger 426 share of the carbon budget must be used early, which makes it necessary to achieve full decarbonization 427 earlier to respect a decreased carbon budget. This explains why the economy-wide delay strategy 428 actually, on aggregate, requires sooner decarbonization than the other policy suites we consider. 429 The influence of delay on the carbon price facing each set of sectors is shown in Figure II. We 430 find that carbon prices facing non-challenged sectors increase relative to the optimal level, while the 431 opposite occurs for the challenged sectors. The degree to which the carbon price facing non-challenged 432 sectors increases is controlled by the emissions intensity of the challenged sector (see the dashed lines 433 in Panels IIa) and b)). This can be explained by sectors with larger annual emissions needing a higher 434 emissions premium to accommodate the delay (see Table III), which increases the relative distortion 435 in carbon prices. For example, relaxing investment effort in energy to delay its decarbonization by a 436 a-vis policy suite 2 requires an emissions premium 1.5× larger than an equivalent delay in decade vis-` 437 industry; when all investment effort is delayed by a decade in energy as in policy suite 3, this relative 438 difference climbs to 2.19×, implying a much stronger carbon price distortion for delaying energy as 439 opposed to industry.17 17 These results also highlight the equivalency of our approach, where we change the emissions allocations between two ETSs to accommodate PECs, to a targeted change of the carbon price facing each set of sectors. 21 TABLE III. Emissions premiums (in GtCO2 ) for each policy suite and challenged sector. Policy suite Challenged sector(s) 5 years of delay 10 years of delay Policy suite 2 Energy 23.6 48.0 Industry 15.5 31.8 Policy suite 3 Energy 59.95 119.9 Industry 27.35 54.7 Policy suite 4 All 187.7 375.4 FIGURE III. Breakdown of investment paths: value of emissions reductions, forgone opportunity effect, and long-term value of abatement capital. 22 440 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 FIGURE IV. Optimal path of investment for every sector in each policy suite. 471 smooth investment over time (see dash-dot line in Panel IIIg)). We further find the long-term value of 472 abatement capital to be suppressed early on (because of an inflated decarbonization date), and grows at 473 a similar rate to the optimal case. In contrast, the stream of marginal investment costs in policy suite 474 3 is simply the optimal case shifted by a decade; there are no quantitative or qualitative differences in 475 the temporal distribution of investment between the optimal case and policy suite 3 beyond this shift. 476 Each of the changes discussed above in the stream of marginal investment costs for non-challenged 477 and challenged sectors materializes in changes in the stream of total investment effort for each sector, 478 shown in Figure IV. We find that non-challenged sectors (all but energy for policy suite 2 and 3 479 in Figure IV) have higher levels of investment initially relative to the optimal case owing to higher 480 near-term value in emissions reductions. For hard-to-abate sectors with “bell-shaped” investment 481 paths,18 investment effort grows quicker, peaks sooner, and declines sooner relative to the optimal case 482 when PECs are present (see Panels IVb), d), and e)) because of the enhanced influence of forgone 483 opportunities. For easy-to-abate sectors with strictly declining paths (see Panels IVa), c), f) and g)), 484 investment simply declines quicker in the case of policy suites 2-4, as these sectors experience a rush 485 of investment early on to decarbonize sooner before declining to the steady-state investment level.19 486 Note that examining the investment path of energy in policy suites 2-3 (dash-dot and solid blue 18 “Bell-shaped” investment paths have been reported in individual case studies, such as the building up the nuclear power fleet in France and the establishment of interstate highways in the United States (Lecocq and Shalizi, 2014). 19 Each sector having either a “bell-shaped” or a declining investment path is consistent with earlier work on optimal timing of abatement investment (Vogt-Schilb et al., 2018). 24 FIGURE V. Sectoral cost indices in each political economy policy suite. 487 lines in Panel IVf)) highlights the influence of PECs on the challenged sector(s): for policy suite 2, 488 investment is smoothed out over time to accommodate the political constraint. On the other hand, in 489 policy suite 3, the optimal investment strategy is shifted by a decade, and no changes are made to the 490 investment schedule once the policy is enacted a decade hence. 491 V.2.3 Impact on aggregate sectoral and total policy cost 492 To explore the relative costs of the different strategies to cope with PECs, and compare the cost 493 of delayed action to the cost of a sub-optimal allocation of effort across sectors, we compute the 494 sectoral policy cost index for every policy suite we consider (see Figure V). Across each sector, we 495 find the delayed economy-wide action policy suite (policy suite 4) leads the most drastic increase in 496 costs, followed by delayed sectoral action (policy suite 3) and relaxed sectoral action (policy suite 2), 497 respectively.20 This is in-line with the established hierarchy in the induced distortion by these policy 498 suites on decarbonization dates and carbon prices shown in Figures I and II, respectively. The degree 499 to which each policy increases sectoral costs, however, varies substantially across sectors. In hard-to- 500 abate sectors such as agriculture and industry, the cost of decarbonization increases by as much as 501 80% and 60% in the delayed economy-wide case, respectively (see Panels Vb) and Vd)). Relatively 20 The lone exception is in energy in policy suites 2 and 3, as this sector is politically challenged and therefore has less costs, see Corollary IV.4. 25 FIGURE VI. Aggregate cost implications of delay in each policy suite. 502 easier-to-abate sectors such as waste and forestry have smaller increases in overall cost. 503 The discrepancy of increased costs between sectors can be explained by considering the optimal 504 path of investment in each sector in each policy suite (for policy suites 2 and 3, energy is the chal- 505 lenged sector), shown in Figure IV. Prior to decarbonization, the investment path of hard-to-abate 506 sectors (such as transport, industry and agriculture) is characterized by building up investment over 507 time before peaking and declining to the steady-state level; investment is “bell-shaped” (Lecocq and 508 Shalizi, 2014; Vogt-Schilb et al., 2018). Maintaining the steady state also requires substantive financial 509 resources, as these sectors are nominally more expensive than other sectors while having high abate- 510 ment potentials (see Table II). By increasing the carbon price facing these sectors, thereby decreasing 511 their decarbonization date, the investment path starts higher in policy suites 2-4 and rises quicker than 512 in the optimal case. Moreover, the steady-state is reached sooner, requiring marginally more expenses 513 to maintain it. 514 On the other hand, the remaining sectors (waste, buildings, energy, and forestry) have high levels of 515 initial investment that decline over time. Relative to hard-to-abate sectors, this causes less integrated 516 investment effort over time, as (i) the distortion in investment is shorter-lived, and (ii) the steady-state 517 is much less expensive to maintain relative to hard-to-abate sectors. Therefore, the relative increase in 518 the cost of decarbonizing these sectors is less than hard-to-abate sectors. 519 Finally, we show how each increase in the sectoral price of decarbonization impacts the total aggre- 520 gate policy cost in Figure VI. We find that the aggregate economic cost of policy increases nonlinearly as 521 decarbonization in the challenged sector(s) is delayed. This can be shown most succinctly in Panel VIc), 26 522 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 a la policy suite With a smaller delay, or a sub-optimal set of policies in politically challenged sectors (´ 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 FIGURE VII. Relative policy cost after delaying each sector by 5 or 10 years, organized by their sectoral characteristics. 550 VI Discussion 551 VI.1 Implications for policy 552 Our quantitative results in Section V underscore the relative economic costs for different deployments 553 of delay as a policy instrument. On the one hand, we find that delaying decarbonization across sectors, 554 as in the case of policy suite 4, can significantly increase the cost of the green transition by as much 555 as 50%. On the other hand, relaxing action in politically challenging sectors (as in policy suite 2) can 556 have modest fiscal impacts, increasing the cost of policy by 1.7% for a decade of delay at the maximum. 557 In this way, our results highlight that the tactical use of delay, demonstrated in policy suite 2, can 558 have a low additional cost while complying with PECs, whereas the “brute force” application of delay 559 in policy suites 3 and 4 can have large additional costs. 560 Moreover, it is not clear that the “brute force” application of delay is, in the long run, politically 561 expedient. This is because, for policy suites 3 and 4, the policy implemented after the delay period 562 must be at least as stringent as the optimal policy in order for the policymaker to remain compliant 563 with the emissions cap; in the case of policy suite 4, the required policy could be significantly more 28 564 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.21 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.22 580 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.23 These results would suggest that policymakers using delay to assuage PECs should 21 One 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. 22 This point would suggest that the approach taken by the EU in their establishment of ETS2 is prudent from an economic and political perspective. 23 We 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. r2 values for regressions between parameter values and relative policy cost after delay, shown in Figure VII. Policy suite Delay amount (years) δ ¯ a ci ) log10 (¯ 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).24 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 endoge- 605 nous technology change. The first clear implication is that, since the rate of endogenous technolog- 606 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. 24 Although, 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 pri- 611 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 ¯i represents the quota is represented by B. The remaining parameters can be generalized as follows: a 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 FIGURE VIII. Paths of emissions and temperature rise for different policy suites. 637 2013)), it has the disadvantage of not teasing apart climate damages that arrive sooner or later as a 638 result of delaying decarbonization in politically challenged sectors.25 639 To gain insight into the timing of temperature rise and its implied climate damages, we compute 640 the stream of emissions and global average temperature in Figure VIII.26 We find that the rate of 641 global warming is highest in the economy-wide delay policy suite, followed by the delayed sectoral 642 decarbonization and relaxed sectoral decarbonization policy suites, with the optimal policy having the 643 lowest warming rate, see Table V. In particular, we find that while 1.6 °C is reached at roughly the 644 same time across baselines, 1.7 °C is reached as much as 40 years sooner under Baseline 4 compared 645 to Baseline 1. 646 The implied economic damages from the different warming rates observed in Figure VIII and 647 Table V will follow the same ranking as long as increases in temperature result in decreases in global 648 GDP. This implies that our approach is likely to underestimate the economic costs associated with 649 policy suite 4 (as we miss some of the damages associated with reaching higher temperatures sooner 650 than in the optimal case), and to a lesser extent, policy suite 3 and policy suite 2. This is particularly 651 the case as the rate of warming is higher in each of these policies compared to the optimal case, which 652 would strain the adaptation capacities of both ecosystems and human systems. In this way, each 653 political economy policy suite we consider is marginally more sub-optimal than presented in Section V, 25 This can be important for policy analysis; for example, Groom and Venmans (2023) show that the value of carbon offsets is zero in a cost-effectiveness setting. 26 To compute the global average temperature above preindustrial, we use the central estimate of the transient climate response to emissions from Dvorak et al. (2022). 32 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 arises27 ; 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 dis- 673 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 27 Bauer 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 con- 679 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,28 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. 28 Note 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.29 706 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 ephane Hallegatte St´ 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-bauer- 721 34/BHM-pol-econ-reprod. 29 Throughout, 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. 35 722 A Analytical solution to the general theoretical model 723 The present value Hamiltonian of (II.3) can be written as       ∗ H= ci (xi (t)) + µj (t)  ai − ai (t)) + (¯ ϕj (t)  Bi + σj Bp − ψj (t) i∈I j ∈{C,N } i∈ j j ∈{C,N } i∈ j + νi (t) (xi (t) − δi ai (t)) + ai − ai (t)) , λi (t) (¯ (A.1) i∈I i∈I 724 where µj (t) is the carbon price facing the sectors j ∈ {C, N }, and νi (t), λi (t) and ϕi (t) are the remaining 725 Lagrange duals. 726 The first order condition for the sector i ∈ I reads ∂H = c′ i (xi (t)) + νi (t) = 0 (A.2) ∂xi (t) 727 where ′ := ∂/∂xi (t). This implies that c′ i (xi (t)) = −νi (t) = |νi (t)|, (A.3) 728 where it follows from the fact that c′ (xi (t)) > 0 (convexity of the cost function) that νi (t) < 0 for all 729 i ∈ I . We can then alter (A.1) such that       ∗ H= ci (xi (t)) + µj (t)  ai − ai (t)) + (¯ ϕj (t)  Bi + σj Bp − ψj (t) i∈I j ∈{C,N } i∈ j j ∈{C,N } i∈ j + νi (t) (δi ai (t) − xi (t)) + ai − ai (t)) , λi (t) (¯ (A.4) i∈I i∈I 730 The remaining first order conditions for the costate variables are given by ∂H = −ϕj (t) = −µ ˙ j (t) + rµj (t), for j ∈ {C, N } (A.5) ∂ψj (t) ∂H dc′i (xi (t)) = −µj (t) + δi c′ ( i i x ( t)) − λ i (t ) = − rc′ (xi (t)), ∂ai (t) dt for j ∈ {C, N } and i ∈ j, (A.6) 731 where we have used the fact that |νi (t)| = c′ i (xi (t)) throughout and defined ˙ := d/dt. 732 We can use complementary slackness to simplify (A.5)–(A.6) before and after the decarbonization 36 733 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+ : ai (t) = a ¯i }. It 736 ¯i − ai (t) > 0 by definition, implying that immediately follows that, prior to the decarbonization date, a 737 λi (t) = 0 for all t < Ti by complementary slackness. 738 Lemma A.2. Consider (II.3). For all t < Tj∗ where j ∈ {C, N } and Tj∗ := max{Tk : k ∈ j }, ϕj (t) = 0. 739 Proof. Consider a set of sectors j ∈ {C, N } and the maximum decarbonization time for that set of 740 sectors Tj∗ := max{Tk : k ∈ j }. Given that ψj (t = 0) = 0 and a ¯i − ai (t) > 0 for all i ∈ j and t < Ti , 741 ˙ j (t) > 0 for all t < T ∗ . The constraint ψj (t) ≤ Bj , where Bj is given by (II.2) for j ∈ {C, N } binds at ψ j 742 the time Tj∗ = max{Tj : j ∈ {C, N }}, i.e., when the last unit of emissions has been emitted before total 743 decarbonization. Hence, for all t < Tj∗ , Bj − ψj (t) > 0, and by complementary slackness, ϕj (t) = 0 for 744 t < Tj∗ . 745 Using Lemmas A.1 and A.2, the first order conditions for the costate variables become, µ ˙ j (t) = rµj (t), for j ∈ {C, N } (A.7) dc′ i (xi (t)) (r + δi )c′ i (xi (t)) = + µj (t), for j ∈ {C, N } and i ∈ j (A.8) dt 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 µj (t) = µj (0)ert = µj ert , (A.9) 748 where we have defined µj (t = 0) = µj as the initial carbon price, which follows an exponentially in- 749 creasing path, in-line with the Hotelling rule (Hotelling, 1931).30 Using (A.9) in (A.8) and rearranging, 750 we have dc′ i (t) − (r + δi )c′ rt rt i (xi (t)) = −µj e = µj e , (A.10) dt 751 for each j ∈ {C, N } and i ∈ j , where we have used the fact that µj is interpreted as a tax to change 30 In an abuse of notation, we have relabeled t − t0 → 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, Ti c′ i (xi (t)) = e (r+δi )t Cx + (µj erζ )(e−(r+δi )ζ )dζ , t µj − δi t = e(r+δi )t Cx + e − e−δi Ti , (A.11) δi 753 where Cx ∈ R is a to-be-determined constant. Note we integrate from t → Ti as our boundary condition 754 ¯i ) – occurs at the decarbonization time Ti . for investment – that in the steady state, ci (xi (Ti )) = ci (δi a 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 µj rt c′ ′ ¯i )e(r+δi )(t−Ti ) + i (xi (t)) = c (δi a e 1 − eδi (t−Ti ) . (A.12) δi 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 ∞ ∞ c′ i (xi (t)) = µj e rt e−δi (s−t) ds −µj ert e−δi (s−t) ds + e−(r+δi )(Ti −t) c′ ¯i ) i (δi a (A.13) t Ti =:Li (t) =:Ei (t) =:Oi (t) 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, 1 2 ¯i x (t). ci (xi (t)) = c (A.14) 2 i 766 Using Assumption 1, we can solve (A.12) for the optimal investment path, x∗ i (t) as µj rt x∗ ¯i e(r+δi )(t−Ti ) + i (t) = δi a e 1 − eδi (t−Ti ) . (A.15) ¯i δi c 38 767 We can now use the optimal investment path (A.15) to solve for the optimal abatement path. Along 768 the optimal path we have ˙∗ a ∗ ∗ i (t) + δi ai (t) = xi (t), (A.16) 769 which can be solved using variation of parameters, t a∗ i (t) = e − δi t Ca + e δi ζ x ∗ i (ζ )dζ 0 µj et(δi +r) − 1 e−δi Ti et(2δi +r) − 1 ¯i δi et(2δi +r) − 1 e−Ti (δi +r) a = e−δi t − + Ca + , ¯i δi c δi + r 2δi + r 2δi + r (A.17) 770 with Ca ∈ R an undetermined constant. Using the boundary condition that ai (t = 0) = 0, we can solve 771 for Ca and write the optimal abatement path as µj et(δi +r) − 1 e−δi Ti et(2δi +r) − 1 ¯i δi et(2δi +r) − 1 e−Ti (δi +r) a a∗ i (t) = e − δi t − + . (A.18) ¯i δi c δi + r 2δi + r 2δi + r 772 Finally, we can derive the optimal path of cumulative emissions. Noting that t ∗ ψj (t) − ψj (t = 0) = ai − a∗ (¯ i (ζ )) dζ, (A.19) 0 i∈ j 773 we can write ∗ ψj (t) = (Ri (t) − µj Bi (t)) (A.20) i∈j 774 for each j ∈ {C, N }, with ¯i etδi − 1 e−Ti (δi +r)−tδi ra ¯i δi et(2δi +r) − 2etδi + 1 e−Ti (δi +r)−tδi a Ri (t) = ta ¯i + − , (A.21) (δi + r) (2δi + r) (δi + r) (2δi + r) 2 ert − 1 r etδi − 1 eδi Ti − 1 e−δi (Ti +t) Bi (t) = − 2 (δ + r ) (2δ + r ) ¯i (δi + r) (2δi + r) rc c¯i δi i i e−δi (t+Ti ) −et(2δi +r) + eδi (Ti +t)+rt + 2etδi − 3eδi (Ti +t) + 2eδi Ti − 1 + . (A.22) ¯i δi (δi + r) (2δi + r) c 775 Evaluating sector’s contribution to (A.20) at the decarbonization date of that sector yields an expression 776 for the carbon price in that set of sectors in terms of the decarbonization dates, ∗) + σ B (Ri (Ti ) − Bi i∈ j j p µj = (A.23) i∈j Bi (Ti ) 39 777 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 , then Bp > 0, and µC < µ∗ 779 where µ∗ 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∗ and Ti < Ti∗ , 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, ¯ 1 = a∗ a 1 (t = T1 , µj ) (A.24) . . . ¯ |j | = a ∗ a |j | (t = T|j | , µj ) (A.25) σj Bp + ∗ − Ri (Ti )) i∈j (Bi µj = (A.26) i∈j Bi (Ti ) 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 Ci (xi (t)) = e−Φi (t) ci (xi (t)) (A.27) 795 where Φi (t) is the technology growth rate in the sector. If technology growth is exogenous, constant, 796 and economy-wide, then Φi (t) = φ, and can be assimilated into the discounting term in the objective 797 function of (II.1), with the new discount rate ˜ := r + φ r (A.28) 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) ¯i δi a Abatement (A.18) ¯i a ∗ Cumulative emissions (A.20) σj Bp + Bi i∈j 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 min [T (Bp ) − T ∗ − δT ] , (B.1) Bp Subject to : T (Bp ) − T ∗ − δT > 0, (B.2) ⃗(T1 , µC ; Bp ) = ⃗ T (Bp ) = arg min f 0 , (B.3) T1 804 where (B.2) ensures we always get a positive T (Bp ), T ∗ is the decarbonization date in the optimal case 805 (i.e., without political constraints), δT is the delay amount, and    ¯1 − a∗ a 1 (T1 , µC )  ⃗(T1 , µC ; Bp ) =    f . (B.4) ∗ − R (T )     Bp + B1 1 1 µC − B1 (T1 ) 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 ∗ + 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: ∂ci > 0, (C.1) ∂xi ∂xi > 0, (C.2) ∂µj ∂Ti < 0, (C.3) ∂µj ∂xi < 0, (C.4) ∂Ti 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   ∞ −rt′ C= e  ci xi (t′ , Ti (µj ), µj )  dt′ . (C.5) 0 j ∈{C,N } i∈ j 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 ∂µN ≤0 (C.6) ∂µC 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 42 828 aggregate policy cost associated with policy suite 4, we can write ∞ ′ Cecon−wide = e−rt ci xi (t′ , Ti (µ + δµ), µ + δµ) dt′ . (C.7) 0 i∈ I 829 Taylor expanding (C.7) around δµ = 0 gives us ∞ ′ ∂ci ∂xi ∂Ti ∂xi Cecon−wide = Copt + δµ e−ρt + dt′ + O (δµ)2 . (C.8) 0 ∂xi ∂µ ∂µ ∂Ti i∈ I 830 Relaxed or delayed sectoral decarbonization (policy suites 2 and 3). For a relaxed de- 831 carbonization strategy or a sectoral delay strategy, we are, in effect, infinitesimally decreasing the 832 challenged sectors’ carbon price relative to the optimal price, µC = µ − δµC where 0 < δµC ≪ 1, while 833 also infinitesimally increasing the non-challenged sectors carbon price relative to the optimal by ∂µN µN = µ − δµC . (C.9) ∂µC 834 Note that (C.6) makes (C.9) positive, as expected. Using (C.5), we can write the total cost for the 835 policy suites 2 and 3 as Crelax and Csec−delay , respectively, such that ∞ ′ Crelax/sec−delay = e−rt ci xi (t′ , Ti (µ − δµC ), µ − δµC ) 0 i∈ C ∂µN ∂µN + ci xi (t′ , Ti µ − δµC ,µ − δµC ) dt′ . (C.10) ∂µC ∂µC i∈N 836 We again Taylor expand (C.10) around δµC = 0 to write ∞ ′ ∂µN ∂ci ∂xi ∂Ti ∂xi Crelax/sec−delay = Copt − δµC e−ρt + dt′ 0 ∂µC ∂xi ∂µ ∂µ ∂Ti i∈ N ∞ ′ ∂ci ∂xi ∂Ti ∂xi + e−ρt + dt′ + O (δµC )2 (C.11) 0 ∂xi ∂µ ∂µ ∂Ti i∈C 43 837 Synthesis. Consider policy suites 3 and 4. Neglecting higher order terms and taking the difference 838 between (C.8) and (C.11) and dividing by the change in carbon price,31 we find ∞ Cecon−wide − Csec−delay ∂µN ′ ∂ci ∂xi ∂Ti ∂xi = 1+ e−ρt + dt′ δµ ∂µC sec−delay 0 ∂xi ∂µ ∂µ ∂Ti i∈ N ∞ ∂µN ′ ∂ci ∂xi ∂Ti ∂xi = 1− e−ρt + dt′ (C.12) ∂µC sec−delay 0 ∂xi ∂µ ∂µ ∂Ti i∈N =:A 839 where we have cancelled out the contribution to the change in costs owing to the challenged sectors, C 840 and used (C.6). All that remains to prove the Proposition is to verify that A is positive. Using (C.1)- 841 (C.4), we see that every individual term in A is positive, ensuring that A is positive. Therefore, we 842 have Cecon−wide > Csec−delay (C.13) 843 proving that policy suite 4 is always more expensive than policy suite 3 for equivalent amounts of delay. 844 Carrying out the same procedure for policy suites 2 and 3, we can write ∞ Csec−delay − Crelax ∂µN ∂µN ′ ∂ci ∂xi ∂Ti ∂xi = − + e−ρt + dt′ δµ ∂µC sec−delay ∂µC relax 0 ∂xi ∂µ ∂µ ∂Ti i∈N ∞ ∂µN ∂µN ′ ∂ci ∂xi ∂Ti ∂xi = − e−ρt + dt′ . ∂µC sec−delay ∂µC relax 0 ∂xi ∂µ ∂µ ∂Ti i∈N (C.14) 845 Therefore, the sign of (C.14) depends on the relative magnitudes of the change in the non-challenged 846 carbon price owing to a distortion in the challenged sector carbon price between each policy suite. 847 Considering (A.23), in order for the change in the carbon price in policy suite 2 to be less than 848 that of policy suite 3 for an equivalent amount of delay, Bp,relax < Bp,sec−delay . It can readily be 849 seen that this condition is always satisfied; the emissions premium for policy suite 3, Bp,sec−delay can 850 be seen as an upper bound on emissions premiums for policy suite 2. For example, if the emissions 851 premiums were equal (i.e., Bp,relax = Bp,sec−delay ), then in the case of policy suite 2 some amount 852 of investment would happen immediately, and relative to policy suite 3, the challenged sectors would 853 be decarbonized marginally later as investment is smoothed out over time. Therefore, for equivalent 854 amounts of delay (as we are considering here), it must be that Bp,relax < Bp,sec−delay , otherwise the 31 In an abuse of notation, we relabel δµC ≡ δµ, as δµ and δµC are equivalent between the settings we consider (they represent the departure from the optimal carbon price for sectors facing higher carbon prices relative to the optimal). 44 855 assumption of equivalent amounts of delay for each policy suite would be violated. 856 Hence, we have ∂µN ∂µN − >0 (C.15) ∂µC sec−delay ∂µC relax 857 and by (C.14) we have Csec−delay > Crelax . (C.16) 858 Therefore, it must be the case that Cecon−wide > Csec−delay > Crelax , (C.17) 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. 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