Policy Research Working Paper 10743 How Delayed Learning about Climate Uncertainty Impacts Decarbonization Investment Strategies Adam Michael Bauer Florent McIsaac Stéphane Hallegatte Macroeconomics, Trade and Investment Global Practice & Climate Change Group March 2024 Policy Research Working Paper 10743 Abstract The Paris Agreement established that global warming remaining carbon budget impacts investment in three ways: should be limited to “well below” 2oC and encouraged (i) the cost of policy increases, especially when adjustment efforts to limit warming to 1.5oC. Achieving this goal pres- costs are present; (ii) abatement investment is front-loaded ents a significant challenge, especially given the presence of relative to the certainty policy; and (iii) the sectoral alloca- (i) economic inertia and adjustment costs, which penalize tion of investment changes to favor declining investment a swift transition away from fossil fuels, and (ii) climate pathways rather than bell-shaped paths. The latter effect uncertainty that, for example, hinders the ability to pre- is especially pronounced in hard-to-abate sectors, such as dict the amount of emissions that can be emitted before a heavy industry. Each of the effects can be traced back to the given temperature target is passed, which is often referred carbon price distribution inheriting a “heavy tail” when the to as the remaining carbon budget. This paper presents a remaining carbon budget is learned later in the century. The modeling framework that explores optimal decarbonization paper highlights how climate uncertainty and adjustment investment strategy when both delayed learning about the costs combined result in a more aggressive least-cost strategy remaining carbon budget and adjustment costs are pres- for decarbonization investment. ent. The findings show that delaying learning about the This paper is a product of the Macroeconomics, Trade and Investment Global Practice and the Climate Change Group. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/ prwp. The authors may be contacted at adammb4@illinois.edu, fmcisaac@worldbank.org, and shallegatte@worldbank.org. The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent. Produced by the Research Support Team How Delayed Learning about Climate Uncertainty Impacts Decarbonization Investment Strategies∗ Adam Michael Bauer† 1,2 , Florent McIsaac2 , and St´ ephane Hallegatte2 1 Department of Physics, University of Illinois Urbana-Champaign, 1110 W Green St Loomis Laboratory, Urbana, IL 61801 2 World Bank Group, Washington DC JEL: E10, Q50, Q54, Q58 Keywords: green investment, adjustment costs, stochastic modeling, climate risk, carbon price ∗ The authors thank Kevin Carey, Roy Dong, Sam Okullo, Cristian Proistosescu, Adrien Vogt-Schilb, and Gernot Wagner for helpful discussions related to this work. 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. † Corresponding author: adammb4@illinois.edu Contents 1 Introduction 3 2 Model description 4 2.1 Base models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Abatement investment model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.2 “Strawman” model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Adding delayed learning to the INV and MAC models . . . . . . . . . . . . . . . . . . . 6 2.2.1 Abatement investment model with delayed learning . . . . . . . . . . . . . . . . 6 2.2.2 “Strawman” model with delayed learning . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3.1 Global parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3.2 Marginal abatement and investment costs . . . . . . . . . . . . . . . . . . . . . . 10 3 Results 11 3.1 Impact of delayed learning on aggregate policy cost . . . . . . . . . . . . . . . . . . . . . 11 3.2 Impact of delayed learning on the temporal distribution of policy cost . . . . . . . . . . 14 3.3 Impact of delayed learning on the sectoral allocation of abatement investment . . . . . . 15 4 Discussion 17 A Including direct air capture technologies 19 A.1 Augmented model framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 A.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 B Increasing emissions baselines 21 B.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 C Alternative calibrations 24 C.1 High cost, linear calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 C.1.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 C.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 C.2 Nonlinear calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 C.2.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 C.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 D Computational details 26 2 1 Introduction The signing of the Paris Agreement marked a breakthrough commitment and consensus in the interna- tional community about the serious threat of human-caused climate change (United Nations Framework Convention on Climate Change, 2015). Coincident with this watershed moment in global climate policy arose an equally daunting challenge: confining planetary warming to the targets enshrined in the Paris Agreement, namely, “well-below” 2 ◦ C with effort towards limit warming to 1.5 ◦ C. These goals are ambitious, and present a major challenge to the international community. For example, estimates of the investment effort necessary to reduce United States emissions by 80% of its 2005 levels by 2050 range between $991 billion –$3.6 trillion1 , and are heavily dependent on the price of important technologies such as batteries (Heal, 2017). Clearly, transitioning from an economy based on “dirty” capital, such as coal-fired power plants, to “clean” capital, such as renewable energy and electricity-powered vehicles, will require significant in- vestment in abatement. A number of factors exist that complicate this process. First, the presence of adjustment costs (Lucas, 1967; Mussa, 1977) considerably raises the price tag of a swift transition to a clean economy, perhaps prohibitively so. Secondly, there are a number of options one can choose to abate fossil fuel emissions; how does one choose between abating emissions in the energy sector, say, versus heavy industry? Finally, economic inertia has been shown to favor climate policies with large near-term investments in abatement (Vogt-Schilb et al., 2018; Campiglio et al., 2022), which conflicts with a number of models in the literature which do not consider these effects (e.g., Nordhaus, 1992, 2017) and recommend an initially low, rising investment pathway. The challenge to financing decarbonization posed by each of these economic factors is compounded by the presence of uncertainty in the physical climate system. While the targets in the Paris Agreement are deterministic, in the sense that the targets themselves are not uncertain, the geophysical timing of their actualization is unclear owing to uncertain climate feedbacks (Sherwood et al., 2020; Intergovernmental Panel on Climate Change, 2021; Dvorak et al., 2022). For example, the true value of the remaining carbon budget (RCB), a geophysical quantity that corresponds to the amount of emissions one has left to emit before a given long-term global temperature target is nearly certain to be reached, is ambiguous owing to uncertainty in the zero-emissions commitment, future aerosol emissions, and the transient climate response to emissions (Matthews et al., 2009, 2018, 2021; Jenkins et al., 2022). The presence of climate uncertainty has been generally shown to increase the stringency of climate policy (see Ha-Duong et al., 1997; Lemoine and Rudik, 2017; Lemoine, 2021; Bauer et al., 2023, for a few examples); this is especially the case if so-called “climate tipping points” are considered (Lenton et al., 2008; Lemoine and Traeger, 2016; Cai and Lontzek, 2019; Dietz et al., 2021). In this paper, we amend an economic model of abatement investment that includes convex adjustment costs, economic inertia, and heterogenous sectors (Vogt-Schilb et al., 2018) with a representation of climate uncertainty to explore the joint influence of adjustment costs and climate uncertainty on optimal decarbonization investment strategies. In particular, we task the social planner with decarbonizing the economy prior to breaching some temperature target for the least cost when the true value of the RCB 1 Throughout, all monetary values are given in 2020 US dollars unless stated otherwise. These figures are the net-present discounted cost of investment, using a 2% discount rate. 3 is hidden until some point in time; once the RCB is known, the planner’s policy can be adjusted (either to be more stringent or lax) in accordance with the true value. The result is that the social planner has to formulate a policy which is robust under a number of potential future risk states that will be revealed later on (similar to the approach of, e.g., Ackerman et al., 2013; Crost and Traeger, 2014; Okullo, 2020). Our approach allows us to vary the time when the RCB is learned, and analyze how learning the true value of the RCB in 2030, for example, impacts the resulting policy in comparison to learning about the RCB in 2080 or later. We compare the results of this experiment with a “strawman” model, where the effects of economic inertia and adjustment costs are neglected, to isolate how climate uncertainty interacts with economic inertia and adjustment costs. We find that the effect of delayed learning about the RCB has three main impacts on optimal policy: (i) the presence of adjustment costs magnifies the impact of climate uncertainty, especially for stringent temperature targets, (ii) abatement investment is front-loaded relative to an equivalent policy without uncertainty, and (iii) sectoral allocation of abatement investment is affected to prioritize hard-to-abate sectors relative to certainty. These findings, taken together, imply that uncertainty and adjustment costs interact to exacerbate the influence of uncertainty, and overall advocate for a more aggressive least-cost investment strategy for decarbonization. We find these results are robust to a number of alternative model specifications, such as allowing for the deployment of direct air capture technologies (Appendix A), different emissions baselines (Appendix B), and alternative cost of investment calibra- tions (Appendix C). The remainder of this paper is structured as follows. We provide an overview of our modeling framework in § 2. We present our main findings in § 3, and close in § 4. 2 Model description In this section, we outline each of our model frameworks. In § 2.1, we define the base models; one that uses abatement investment and captures economic inertia (coined the INV model), and one that does not (coined the MAC model). In § 2.2, we add delayed learning about the RCB to the base models defined previously. Note the solutions to the models highlighted in § 2.1 are the “certainty” solutions of the delayed learning models in § 2.2. In § 2.3, we describe our calibration routine. 2.1 Base models 2.1.1 Abatement investment model The abatement investment model (referred to as the INV model herein) tasks a policymaker with decarbonizing a set of economic sectors, I , for the least cost over some time horizon, T := [0, T ] such that the carbon budget, B ∈ R+ , is not exceeded. Each sector has an abatement potential (or annual ¯i ∈ R+ , and a capital depreciation rate, 0 < δi ≤ 1. No sector has the capacity for emissions rate), a “negative emissions”. (We explore the influence of direct air capture on our results in Appendix A, and find that none of the results are significantly altered.) The policymaker controls the amount of abatement investment, xi,t , in each sector i ∈ I at each point 4 in time t ∈ T ; abatement investment leads to the abatement of emissions, ai,t , via the accumulation of abatement capital stocks. Global cumulative emissions are given by ψt . The cost of abatement investment in each sector i ∈ I , ci : R+ → R+ , is assumed to be an increasing and convex function of investment (i.e., its first and second derivatives with respect to investment are positive). The convexity requirement on our cost function endows the system with adjustment costs (Lucas, 1967; Mussa, 1977). The problem facing the social planner can therefore be formulated as, min βt ci (xi,t ) , (2.1) {xi,t }i∈I t∈T i∈I Subject to : ai,t+1 = ai,t + ∆t (xi,t − δi ai,t ) , ψt+1 = ψt + ∆t ai − ai,t ) , (¯ i∈I 0 ≤ ai,t ≤ a ¯i , 0 ≤ ψt ≤ B, ai,0 , ψ0 given, where ∆t > 0 is the timestep (assumed to be annual throughout), β := (1 + r)−1 is the discount factor, and 0 < r ≤ 1 is the discount rate. This model was extensively studied in Vogt-Schilb et al. (2018), and we refer readers to this paper for a full analytical treatment of the model. We quote one key theoretical result involving this model that is relevant for our discussion, which we reproduce as the following Lemma. Lemma 2.1 (Proposition 1 from Vogt-Schilb et al. (2018)). Along the optimal path of Eqn. (2.1), investment is either bell-shaped or decreasing over time. The formal proof of Lemma 2.1 can be found in Vogt-Schilb et al. (2018), but we quote the main implication for our purposes here. The shape of the investment pathway in a given sector relies on the inequality, µ ≷ (r + δi )c′ ¯ i ), i (δi a (2.2) that weighs the carbon price (given by µ) against the abatement investment costs. Note the carbon price is determined endogenously in the model as the shadow value of emissions reductions.2 When µ > (r + δi )c′ ¯i ), paying the carbon price is more expensive than a marginal unit of abatement i (δi a investment, and it follows from the proof of Lemma 2.1 that the optimal investment path is declining. If µ < (r + δi )c′ ¯i ), then initially paying the carbon price is more cost-effective than a marginal i (δi a unit of abatement investment, forcing the investment pathway to start low. The path then rises over time, commensurate with a rising carbon price, before declining again to the steady-state on long time horizons. This dynamic gives rise to the bell-shaped investment pathway mentioned in Lemma 2.1. These considerations will be important when we discuss the sectoral allocation of abatement when learning about the carbon budget is delayed in § 3.3. 2 Mathematically speaking, the shadow value of emissions reductions is the dual of the cumulative emissions time ˙ t := derivative, ψ ai − ai,t ). (¯ i∈I 5 2.1.2 “Strawman” model We formulate our “strawman” model (referred to as the MAC model throughout) as an analogous problem to that of (2.1) without adjustment costs and abatement capital stock accumulation. In this formulation, the social planner controls the abatement rate, rather than abatement investment; this approach is analogous to conventional approaches in climate-economic integrated assessment mod- els (Nordhaus, 2017; Barrage and Nordhaus, 2023) and has been used as a counterfactual case to isolate the effects of adjustment costs on decarbonization policy (see, e.g., Campiglio et al., 2022). Rather than using marginal investment costs, the social planner uses a marginal abatement cost function in each sector i ∈ I , γi : R+ → R+ , that maps abatement to monetary cost. The marginal abatement cost functions γi are increasing and convex functions of abatement. The remaining constraints are identical to those of the INV model. We again can formulate this problem as a cost-minimizing optimization problem, such that, min βt γi (ai,t ) , (2.3) {ai,t }i∈I t∈T i∈I Subject to : ψt+1 = ψt + ∆t ai − ai,t ) , (¯ i∈I 0 ≤ ai,t ≤ a ¯i , 0 ≤ ψt ≤ B. ψ0 given, This model is also treated in Vogt-Schilb et al. (2018) and we refer interested readers to their paper for a full analytical treatment. 2.2 Adding delayed learning to the INV and MAC models 2.2.1 Abatement investment model with delayed learning We now add delayed learning about the RCB to the model structure. Consider the INV model outlined by (2.1), and allow the remaining carbon budget B is uncertain; let pBT ∗ ∼ N B ¯T ∗ , σ 2 BT ∗ be the 3 ∗ ¯ 2 Gaussian distribution of the RCB for some temperature target T > 0 where BT ∗ > 0 and σB T∗ >0 ˜ are the average and variance of pBT ∗ , respectively. Let B := {B1 , B2 , ..., BN } be a discrete set of samples drawn from pBT ∗ , N := |B ˜ | be the number of possible future states, and J := {1, 2, ..., N } index the set of future states. Assume the social planner learns the true value of the remaining carbon budget at a time t∗ such that 0 < t∗ < T . Then we can write the social planner decarbonization 3 Note the distribution of the RCB is approximately Gaussian owing to the underlying distribution of the transient cli- mate response to emissions being approximately Gaussian (Intergovernmental Panel on Climate Change, 2021; Matthews et al., 2021; Dvorak et al., 2022). 6 problem as a 2-stage stochastic optimization problem, such that, t∗ −1 min βt ˜ Q xi,t ; B ci (xi,t ) + EB (j ) , (2.4) {xi,t }i∈I t=0 i∈I Subject to : ψt+1 = ψt + ∆t ai − ai,t ) , (¯ (2.5) i∈I ai,t+1 = ai,t + ∆t (xi,t − δi ai,t ) , 0 ≤ ai,t ≤ a ¯i , ai,0 , ψ0 , t∗ given, with T −1 j (j ) Q xi,t ; B := min βt ci xi,t , (2.6) (j ) xi,t t=t∗ i∈I i∈I ,j ∈J (j ) (j ) (j ) Subject to : ψt+1 = ψt + ∆t ¯i − ai,t a , i∈I (j ) (j ) (j ) (j ) ai,t+1 = ai,t + ∆t xi,t − δi ai,t , (j ) 0 ≤ ai,t ≤ a ¯i , (j ) 0 ≤ ψt ≤ B (j ) , (j ) ai,t∗ = ai,t∗ , (j ) ψt∗ = ψt∗ . (j ) Note ai,t represents the abatement in sector i ∈ I at time t ∈ T in future state j ∈ J (the same notation is applied for cumulative emissions and investment, where the sectoral index is dropped for cumulative emissions). The final two constraints in (2.6) require continuity in abatement and cumulative emissions (the state variables) across the learning time. In this framework, we can alter the value of t∗ ∈ T to change when the policymaker learns about the carbon budget. Note that if t∗ = 0, the problem reduces to no delayed learning, and the policymaker minimizes the expected policy cost in the usual way. 7 2.2.2 “Strawman” model with delayed learning The MAC model can be formulated with delayed learning in an analogous way to the INV model; using (2.3), we can write t∗ −1 min βt ˜ Q ai,t ; B γi (ai,t ) + EB (j ) , (2.7) {ai,t }i∈I t=0 i∈I Subject to : ψt+1 = ψt + ∆t ai − ai,t ) , (¯ i∈I 0 ≤ ai,t ≤ a ¯i , ψ0 , t∗ given, with T −1 (j ) Q ai,t ; B (j ) := min βt γi ai,t , (2.8) (j ) ai,t t=t∗ i∈I i∈I ,j ∈J (j ) (j ) (j ) Subject to : ψt+1 = ψt + ∆t ¯i − ai,t a , i∈I (j ) 0 ≤ ai,t ≤ a ¯i , (j ) 0 ≤ ψt ≤ B (j ) , (j ) ai,t∗ = ai,t∗ , (j ) ψt∗ = ψt∗ . 2.3 Calibration 2.3.1 Global parameters We calibrate the non-sector specific parameters in the following way. We lift the details of the RCB distribution from a recent publication which incorporates uncertainty into the RCB estimates for three temperature targets: 1.5 ◦ C, 1.7 ◦ C, and 2 ◦ C (Dvorak et al., 2022). Note for the 1.5 ◦ C target, we truncate the RCB distribution such that we always allow the planner a minimum of five years to decarbonize; any stricter budget for the 1.5 ◦ C temperature target causes unrealistically high investment levels, given that the planner is bound to always abide by the budget in our approach. Given our yearly emissions rate of ∼ 40 GtCO2 yr−1 , this equates to a minimum budget of ∼200 GtCO2 .4 We set the social discount rate to 2% yr−1 , in line with a recent international expert elicitation and the US EPA’s prevailing rate for their social cost of carbon estimates (Drupp et al., 2018; National Center for Energy Economics, 2022). Abatement potentials in each sector are taken from AR6 WGIII (Intergovernmental Panel on Climate Change, 2022). 4 In practice, the carbon budget for 1.5 ◦ C could be lower than 200 GtCO2 , and a negative carbon budget cannot be ruled out (Matthews et al., 2021) which would imply the world has already passed the 1.5 ◦ C target. In reality, it is much more likely that the planner would simply choose a higher temperature target if the carbon budget for 1.5 ◦ C is too low, and would of course be forced to do so if the RCB is negative when the policy is first implemented. 8 Figure 1: Calibrating marginal abatement costs. Marginal abatement costs broken down by sector using IPCC AR6 data (Intergovernmental Panel on Climate Change, 2022). Black lines represent a nonlinear marginal abatement cost (in particular, a quadratic fit for all sectors other than agriculture); gold lines represent a linear fit to lowest cost bracket marginal cost values; blue lines represent a linear fit to highest cost bracket marginal abatement cost data. 9 It remains to define a “business-as-usual” emissions scenario, a notoriously difficult task. Remarkably, the total abatement potential described by the IPCC is ∼40 GtCO2 yr−1 , which is approximately equal to the peak emissions of SSP2–4.5 (the “middle of the road” emissions scenario used by the IPCC (Intergovernmental Panel on Climate Change, 2021)) and the RFF–SPs used in the United States’ Environmental Protection Agency’s estimates of the social cost of carbon (Riahi et al., 2017; National Center for Energy Economics, 2022). In each of these baselines, emissions are expected to rise to just above 40 GtCO2 yr−1 before declining. It has been argued that higher emissions scenarios are unlikely (Hausfather and Peters, 2020). We therefore assume that, without abatement, emissions will maintain their peak levels in each of these emissions scenarios, i.e., at ∼40 GtCO2 yr−1 . This describes our “business-as-usual” scenario5 ; we probe the impact of assuming an increasing emissions baseline in Appendix B. Thus, in this calibration, once all emissions are abated, society is entirely decarbonized. We set the time horizon for decarbonization to 2100; this translates to solving the model for 80 years. 2.3.2 Marginal abatement and investment costs Throughout, we assume that our cost functions ci (xi,t ) and γi (ai,t ) are quadratics, such that 1 2 ¯i x , ci (xi,t ) = c (2.9) 2 i,t 1 2 ¯i a . γi (ai,t ) = γ (2.10) 2 i,t We calibrate these cost functions using IPCC marginal cost data (Intergovernmental Panel on Climate Change, 2022). Marginal abatement costs are given by, γ ′ (ai,t ) = γ ¯i ai,t , (2.11) which implies that marginal abatement costs are linear in abatement potential. Throughout the main text, we use the low cost, linear marginal cost calibration shown in Figure 1; we explore the impact of the other two alternative calibrations given in Figure 1 in Appendix C. We translate the marginal abatement cost into the marginal investment cost by requiring that between the two approaches (MAC and INV) the relative cost between each sector are equal, such that ¯i γ ¯i c = , (2.12) ¯j γ ¯j c for each unique pair (i, j ). This defines marginal costs up to one remaining parameter. The final cost coefficient is calibrated by demanding that the total discounted costs between both approaches are equal for a given target (Grubb, 1997) under the assumption of certainty about the carbon budget. Therefore, any differences in total policy costs once we add in the effects of uncertainty must come from the uncertainty itself, as the models are calibrated to be equally costly in the certainty case. See 5 Note that some green technologies will be cost-competitive with their dirty equivalents, and therefore not necessarily require a carbon price to be assimilated into the economy; indeed, these “costless” abatement technologies are a key source of uncertainty in projecting the costs of the green transition (Kotchen et al., 2023). Therefore, our assumption that emissions will remain at peak levels without a carbon price is conservative, as all abatement in our approach is spurred by the presence of a carbon price. 10 Table 1 for a full summary of our calibration results. 3 Results 3.1 Impact of delayed learning on aggregate policy cost We first explore the impact of delayed learning about the carbon budget on the aggregate cost of optimal policy for three temperature targets (1.5 ◦ C, 1.7 ◦ C, and 2 ◦ C) in Figure 2. In panel 2a, we show the total discounted cost of the optimal policy for each temperature target using the INV model (solid lines) and MAC model (dashed lines); we show the policy cost net of the equivalent certain policy cost in panel 2b. We find that across models and temperature targets, the later the policymaker learns the true value of the carbon budget, the higher the total policy cost relative to certainty. The most drastic increase in cost occurs for the 1.5 ◦ C temperature target; for example, if one learns about the carbon budget in 2050, the additional cost of the policy is ∼ $17 trillion, significantly higher than the ∼ $2 trillion (∼ $1 trillion, resp.) additional cost paid in the 1.7 ◦ C (2 ◦ C, resp.) policy. The strongly increasing relationship between total policy cost and the delay in learning about the RCB can be explained by how long the social planner has to formulate a policy that is robust to the worst-case scenarios for the RCB. If the policymaker learns about the RCB 10 years after the start date, then their policy must be robust to the worst-case carbon budgets for a relatively short amount of time. Therefore, there is some increased spending to preemptively plan for the worst-case, but the planner finds out relatively quickly if the worst-case ever materializes, and can relax the policy in the likely event it does not. On the other hand, if the policymaker learns about the RCB 50 years after the start date, then the policymaker must plan for the worst-case for much longer; by then, a significantly higher amount of precautionary spending has occurred, which considerably drives up the total policy cost, especially for ambitious temperature targets. A similar logic explains the “S”-shaped dynamics of the additional cost of uncertainty. Using the 1.5 ◦ C target as an example in panel 2b, the difference between learning in 2020 and 2030 is small owing to only a small amount of precautionary spending required for a robust policy. Yet this difference gets large after 2030 when the worst-case carbon budgets must be planned for over an increasingly long period of time. Finally, the difference between 2060 and 2080 is negligible, as there is only so much additional precautionary spending that can make the policy marginally more robust to learning of a catastrophe later on. This dynamic is present across temperature targets and model types, though to a lesser extent for more lenient temperature targets (as catastrophes are not so extreme). The “S”-shaped behavior of total policy cost can be clearly illustrated by computing the marginal value of learning about the carbon budget (see panel 2c); we find that the value of learning the carbon budget a year earlier can be as high as $1t yr−1 . This implies a potentially substantial cost-savings from learning the true value of the carbon budget, and furthermore, indicates that the value of waiting to invest in abatement in order to learn more about the carbon budget (Dixit and Pindyck, 1994) must be smaller than the increase in the expected net-present value of the social planner’s costs if they delay. We further find that, for each temperature target, the marginal cost of learning rises and falls, congruent with the “S”-shaped dynamics of the additional cost of uncertainty in panel 2b. 11 Table 1: Calibration parameters for the MAC and INV models for each temperature target using the low cost, linear calibration. T ∗ = 1.5 ◦ C T = 80 yr r = 2 % yr−1 ¯ = 440 GtCO2 B σB = 72 GtCO2 $ tCO− 1 $ tCO− 1 Sector ¯ γ 2 ¯ [GtCO2 yr−1 ] a δ [% yr−1 ] c ¯ 2 GtCO2 yr−1 GtCO2 yr−3 Waste 40.82 0.82 3.3 11557 Industry 17.54 5.47 4 4966 Forestry 7.12 8.25 0.8 2016 Agriculture 23.53 4.07 5 6662 Transport 6.12 3.74 6.7 1732 Energy 2.82 11.99 2.5 798 Buildings 12.99 3.2 1.7 3678 T ∗ = 1.7 ◦ C T = 80 yr r = 2 % yr−1 ¯ = 770 GtCO2 B σB = 75 GtCO2 $ tCO− 1 $ tCO− 1 Sector ¯ γ 2 ¯ [GtCO2 yr−1 ] a δ [% yr−1 ] c ¯ 2 GtCO2 yr−1 GtCO2 yr−3 Waste 40.82 0.82 3.3 14031 Industry 17.54 5.47 4 6029 Forestry 7.12 8.25 0.8 2447 Agriculture 23.53 4.07 5 8088 Transport 6.12 3.74 6.7 2104 Energy 2.82 11.99 2.5 969 Buildings 12.99 3.2 1.7 4465 T ∗ = 2 ◦C T = 80 yr r = 2 % yr−1 B¯ = 1340 GtCO2 σB = 110 GtCO2 $ tCO− 1 $ tCO− 1 Sector ¯ γ 2 ¯ [GtCO2 yr−1 ] a δ [% yr−1 ] c ¯ 2 GtCO2 yr−1 GtCO2 yr−3 Waste 40.82 0.82 3.3 17475 Industry 17.54 5.47 4 7509 Forestry 7.12 8.25 0.8 3048 Agriculture 23.53 4.07 5 10073 Transport 6.12 3.74 6.7 2620 Energy 2.82 11.99 2.5 1207 Buildings 12.99 3.2 1.7 5561 12 Figure 2: Effect of delayed learning on aggregate policy cost. In panel a, we show the total discounted policy cost for each temperature target (see the legend) for the INV model that includes the impact of adjustment costs and economic inertia (solid lines) and MAC model, with does not include these factors (dashed lines). Panel b is as a, net of the cost of the corresponding policy without uncertainty. Panel c shows the marginal cost of delaying learning by one additional year. Panel d shows the ratio of the total policy cost of uncertainty in the INV model divided by the total policy cost of uncertainty in the MAC model; we call this ratio the uncertainty-inertia interaction index. 13 Figure 3: Effect of delayed learning on the temporal distribution of spending. In panel a, we show the fraction of total policy spending before and after 2030, relative to the certainty policy for the 1.5 ◦ C target. A value of unity implies that the spending pre- or post-2030 in the certain model and the model with delayed learning are identical. Panels b–c are as a, but for the 1.7 ◦ C and 2 ◦ C targets, respectively. We further find that economic inertia accentuates the impact of delayed learning about the RCB. In panel 2d, we show the ratio of the total policy cost in the INV model to the MAC model, deemed the uncertainty-inertia interaction index (UIII) (i.e., the ratio of the solid and dashed lines in panel 2a). We find that for all temperature targets, the UIII is greater than unity, showcasing the additional price of delayed learning when adjustment costs and inertia are present. This is easily explained: adjustment costs significantly penalize rapid decarbonization, which is exactly what is necessary if the worst-case arises. Put differently, adjustment costs and economic inertia make catastrophes far worse than when they are absent. Note the UIII decreases towards the end-of-century because the total cost of policy in the MAC model rises, while the cost of the INV policy flatlines, collectively acting to shrink the UIII (see panel 2a). 3.2 Impact of delayed learning on the temporal distribution of policy cost If delayed learning about the RCB increases the total cost of policy, how does the social planner distribute the additional spending in time? We explore the temporal shift in spending in Figure 3, where we show the fraction of spending before and after 2030 relative to the certain policy for each temperature target (called the additional cost of uncertainty index in Figure 3). We find that the inclusion of uncertainty leads to a front-loading of abatement investment effort relative to the certainty policy across temperature targets. In addition, we find that the degree to which spending is shifted depends strongly on the desired temperature target; for a 1.5 ◦ C target, one may spend as much as ∼3× more prior to 2030 relative to certainty, but for more lenient temperature targets, the maximum increase in spending pre-2030 is much smaller. The dependency on the temporal shift in abatement investment effort across temperature targets can be explained by the relative urgency of the worst-case carbon budgets in each case. In the 1.5 ◦ C 14 Figure 4: Effect of delayed learning on sectoral allocation of abatement investment. In each panel we show the average investment effort in each sector (see the titles). The color of each curve corresponds to the year the information about the carbon budget is revealed. Notice the regime change from bell-shaped to declining in agriculture and industry, while already declining investment path sectors such as energy get steeper relative to their no uncertainty case. Note these investment pathways are for the 1.5 ◦ C temperature target. target case, the worst-case carbon budget could be as low as ∼ 200 GtCO2 , which given our baseline levels of emissions (∼ 40 GtCO2 yr−1 ) would imply the temperature target could be reached in ∼ 5 yrs in the absence of abatement. Hence, much of the precautionary spending is front-loaded, given the overall urgency of the temperature target. On the other hand, for the 1.7 ◦ C and 2 ◦ C targets, in the worst-case the policymaker would have ∼ 14 and ∼ 25 years to decarbonize in the absence of abatement (respectively). Hence the overall urgency of the worst-case is lower for higher temperature targets, which lessens the front-loading of precautionary investments in abatement. The front-loading of abatement investment effort shown in Figure 3 provides further insight into the finding that delaying learning about the RCB causes total policy costs to increase, as shown in Figure 2. As an increasing amount of abatement investment is front-loaded to account for uncertainty, the less impact that discounting has on the total policy cost, as spending occurs much sooner than in the certainty case. 3.3 Impact of delayed learning on the sectoral allocation of abatement investment We explain the front-loading of abatement investment effort by sectorally disaggregating investment ef- fort in Figure 4. We find that across sectors, as learning about the RCB happens later, more investment effort occurs earlier; this is a form of precautionary spending, as the social planner prepares for the worst-case scenario, as discussed above. For example, easy-to-abate sectors such as energy and forestry 15 Figure 5: Effect of delayed learning on the carbon price. In panel a, we plot the remaining carbon budget distribution when the information is revealed (yellow bars) against the carbon price distribution (blue bars) for the case where information is revealed in 2020. The black dots are the simulated values of the carbon price for each carbon budget; the pink star is the average carbon price. Note reading the gold bars with the top x-axis dictates the initial distribution of the RCB, while reading the gold bars with the bottom x-axis dictates how much of the carbon budget remains when the true value is revealed. Panel b is as a, but for when the carbon budget is revealed in 2030. Note the data used in this figure is for a 1.5 ◦ C temperature target. see a ∼ 4 − 5× increase in investment effort relative to certainty in 2020 when the carbon budget is revealed in mid-century. In hard-to-abate sectors – such as industry and agriculture (panels 4b,d) – we observe a regime change in the optimal investment pathway, where the shape of the path shifts from bell-shaped to declining as learning occurs later, further buttressing the front-loading of investment effort observed in Figure 3. The change in sectoral allocation of abatement investment can be explained by the influence of delayed learning about the RCB on the carbon price; see Figure 5. We find that delayed revelation about the carbon budget has important implications for the distribution of possible carbon prices after one learns. If the policymaker learns about the carbon budget in 2020, we find an approximately power-law relationship between the RCB and the carbon price (i.e., an approximately linear relationship on log- scale in Figure 5a); it follows that since the underlying RCB distribution is Gaussian that the carbon price is approximately log-normally distributed. If the policymaker learns in 2030, we find the log-normal distribution of carbon prices breaks down; the carbon price distribution inherits a “heavy tail” and becomes positively skewed, as shown in Figure 5b. This owes to the amount of the carbon budget left to be used can be as low as ∼ 30 GtCO2 in the worst case, since some amount of emissions has occurred between 2020 and 2030. If this worst-case scenario 16 materializes – akin to a so-called “green swan” scenario (Bolton et al., 2020) – the social planner has no choice but to impose an incredibly stringent carbon price to meet the 1.5 ◦ C temperature target. (Recall the social planner is constrained to always meet the target in our framework; in actuality, the planner may simply choose to target a higher temperature threshold if the required carbon price is prohibitively high.) This heavy tail makes the average carbon price higher than in the case of learning in 2020, and explains why the average carbon price (pink star in Figure 5b) is outside the main body of the carbon price distribution in 2030. Furthermore, Figure 5 demonstrates why the “value of learning” – that is, the value in delaying invest- ment to learn about the details of the carbon budget distribution (Dixit and Pindyck, 1994; Lemoine and Rudik, 2017; Barnett et al., 2021) – in our framework is negative: the carbon price distribution is only heavy tailed on the high end, leading to an asymmetry between the worst-case and best-case responses. In the worst-case, the social planner must impose an incredibly stringent carbon price to accommodate the lower-than-expected carbon budget, whereas in the best-case, the social planner only relaxes the carbon price slightly. This asymmetry gives rise to the precautionary investment strategy undertaken by the social planner. The carbon price influences the shape of the optimal investment pathway via the inequality shown in Lemma 2.1. When the carbon price increases, the relative cost of either paying the carbon price or a marginal unit of abatement investment is shifted in favor of investment. This leads to the social planner investing more aggressively in easy-to-abate sectors, which causes their already declining investment pathways to have an even steeper profile. In hard-to-abate sectors, the bell-shaped profile shifts to a declining path as information is learned later and the average carbon price increases. This explains the regime change observed in the sectoral allocation of abatement investment shown in Figure 4. This behavior sheds further light on our previous findings that later learning about the carbon budget leads to the front-loading of abatement investment, shown in Figure 3. The front-loading of investment relative to certainty is caused by a regime change in the optimal investment pathway for hard-to- abate sectors from a bell-shaped path to a declining path, along with the already-declining investment pathways of easy-to-abate sectors declining more steeply for higher carbon prices. The high carbon prices, in turn, are caused by the remaining carbon budget when the information is revealed to be smaller than the initial distribution; this leads to a risk of extremely high carbon prices when the carbon budget is revealed to be low, which drives up the average carbon price. In other words: the later bad news arrives, the more extreme the policy response must be to accommodate the bad news, here in the form of an extremely high carbon price, which drives up the cost of policy. 4 Discussion In this work, we explored the impact of delayed learning about the remaining carbon budget (RCB) on optimal abatement investment pathways when adjustment costs are present. We posit three main takeaway points: (i) that delayed learning about the RCB increases the overall costliness of policy, especially when adjustment costs are included, (ii) delayed learning causes abatement investment to be front-loaded relative to an equivalent policy without uncertainty, and (iii) the sectoral allocation of abatement is significantly impacted by delayed learning, especially in hard-to-abate sectors, where we 17 find a regime change from a bell-shaped investment pathway to a declining pathway. The increasing relationship between total policy cost and delayed learning about the RCB can be explained by how long the social planner has to formulate a policy that is robust to extremely low values of the RCB. The later the policymaker learns, the more precautionary spending they must commit to; this increases the aggregate amount of spending, while also driving down the influence of discounting on the total policy cost. The precautionary spending undertaken by the policymaker to account for worst-case scenarios in the RCB explains the front-loading of abatement investment shown in Figure 3. The temporal redistribution of spending as a result of delayed learning about the RCB can be explained by considering the sectoral response (Figure 4). We find that the optimal investment pathway for easy- to-abate sectors, which have a declining investment pathway in the certainty case, becomes more steeply declining when delayed learning is present. For hard-to-abate sectors, that have a bell-shaped investment pathway in the certainty case, we observe a regime change where the optimal abatement pathway becomes declining. Both of these changes in sectoral investment dynamics can be explained by the carbon price distribution being significantly impacted by delayed learning (Figure 5). Learning later about the RCB causes the distribution of potential carbon prices to inherit a heavy tail and become positively skewed, which drives up the average carbon price; this increase in the carbon price interacts with Lemma 2.1 to alter the optimal investment pathway shape in both easy-to-abate and hard-to-abate sectors. Moreover, Figure 5 highlights how the asymmetric responses to the worst-case and best-case scenarios for the carbon budget gives rise to the precautionary motive for investment and a negative “value of learning” (Dixit and Pindyck, 1994). A key takeaway from this work for policymakers is that, if one takes meeting a given temperature target seriously, then considering the worst-case scenario for carbon budgets can have a sizable influence on optimal policy. This is especially the case when considering stringent temperature targets, when learning the precise value of the carbon budget (or its uncertainty is lessened to the point of near- certainty) is significantly delayed, or when both of these effects are present.6 Moreover, the presence of adjustment costs – which make a swift transition to a clean economy especially costly – makes the worst-case scenario even more important to consider, as if the worst-case scenario materializes, adjustment costs will make an aggressive decarbonization strategy especially expensive. There are two further takeaways from this work (demonstrated in Appendices A and B, respectively). The first is that the inclusion of direct air capture only leads to significant cost-savings as a function of learning time in the case of the 1.5 ◦ C target. Importantly, the role of DAC in this case is as a cost-offset for more expensive sectors such as agriculture and industry. This gives context to the role of DAC in the decarbonization discussion: DAC can offset the costs of decarbonizing hard-to-abate sectors, but should not be considered a substitute for decarbonizing other relatively easy-to-abate sectors such as energy. The second point is that solving the model with an increasing baseline level of emissions reveals that the investment pathway for achieving the 2 ◦ C target is approximately the same in terms of cost as the 1.5 ◦ C investment path without the increasing baseline. This highlights the significant cost of 6 It is known that the uncertainty in the carbon budget for a given temperature target will decline as the target is approached (owing to arguments akin to those laid out in Myhre et al., 2015), but a precise quantification of how quickly the decline occurs is, as of writing, unknown. 18 building more “dirty” capital stocks that have to be decarbonized down the line. A final takeaway from our work is that narrowing the uncertainty in the RCB could lead to non-trivial monetary savings for policymakers. Indeed, the logic of the marginal cost of uncertainty shown in Figure 2c cuts both ways; learning a year earlier about the RCB could also lead to high levels of savings, especially if we are able to learn before 2030. This could be a fruitful direction for research for climate scientists that would complement many policy discussions. We note that our framework is idealized, and lends itself to a number of future studies. For example, we consider here the effects of moving from a deterministic framework with static and dynamic abatement costs (our MAC and INV models, respectively) to one with uncertainty about the carbon budget, conditional on the deterministic costs being identical. One could carry out a future study where the expected costs between the two frameworks are identical, and analyze the dynamical differences when expected costs are set to be the same (i.e., when the dashed and solid lines in Figure 2 are required to be equal). Moreover, a number of additional considerations beyond those mentioned in this paper are necessary in formulating a robust climate policy suite. As an example, we compute a globally optimal investment strategy, and future work would consider the cost implications of sub-optimal policies, or heterogenous policy mixes (as explored in Clausing and Wolfram, 2023). Data availability statement All code and data used in this study is publicly available on the corresponding author’s Github page, found at the following URL: https://github.com/adam-bauer-34/BMH-delayed-learning-reprod. A Including direct air capture technologies In this Appendix, we include direct air capture technologies (DAC) in our modeling framework. We ¯DAC = 0) and some treat DAC like an additional economic sector with zero annual emissions (i.e., a −1 abatement potential, that we call Z > 0 (in GtCO2 yr ). DAC is unique in that its emissions rate is not the same as its abatement potential, as was assumed in every other sector. 19 A.1 Augmented model framework A social planner tasked with decarbonization with DAC technologies at their disposal solves the fol- lowing cost-minimization problem, min βt ci (xi,t ) , (A.1) {xi,t }t∈T ,i∈I t∈T i∈I Subject to : ai,t+1 = ai,t + ∆t (xi,t − δi ai,t ) , ψt+1 = ψt + ∆t ai − ai,t ) , (¯ i∈I 0 ≤ ai,t ≤ a ¯i , 0 ≤ ψt ≤ B, 0 ≤ aDAC,t ≤ Z, ai,0 , ψ0 given. Extending this formulation to a model with uncertainty about the RCB is straightforward. We take the marginal abatement cost of DAC from International Energy Agency (2022) as $100–$300 tCO− 1 2 . In each simulation we show below, we assume a marginal abatement cost of $125 tCO− 1 2 . Any lower marginal cost we would judge to be unrealistically cheap, and any higher cost causes model behavior to be completely unchanged in all of our simulations. We translate this marginal abatement cost to a marginal investment cost using (2.12). We assume the capital depreciation rate of DAC is 5% yr−1 and that, at maximum, the abatement potential from DAC technologies is 5 GtCO2 yr−1 (Terlouw et al., 2021). A.2 Results We find that the impact of including DAC in our model for abatement investment is heterogenous across temperature targets, see Figure 6. For lenient temperature targets, we find that there is practically no change in the value of learning; in other words, the investment strategies with and without DAC are essentially identical. This owes to the carbon price in the lenient temperature target cases being too low to induce the social planner to invest heavily in DAC; it is cheaper to simply abate emissions at their source, given the relatively cheap cost of abatement investment in other sectors compared to DAC. For the 1.5 ◦ C temperature target, however, a different pattern emerges. We find significant cost- savings relative to the equivalent set of policies without DAC. This is because, as shown in Figure 5, the average carbon price can be significantly higher in the 1.5 ◦ C target case, which provides the social planner with sufficient incentive to invest in DAC technologies. The additional investment in DAC allows the social planner to invest slightly less resources in hard-to- abate sectors, see Figure 7. We find that most sectors – waste, forestry, transport, energy, and buildings – have minimal changes to their investment pathway when DAC technologies are introduced. However, hard-to-abate sectors, such as industry and agriculture, have non-trivial reductions in investment early 20 Figure 6: Effect of delayed learning on aggregate policy cost including direct air capture technologies. In panel a, we show the total discounted policy cost for each temperature target (see the legend) for the INV model without DAC (solid lines) and INV model with direct air capture technologies (dashed lines). Panel b is as a, net of the cost of the corresponding policy without uncertainty. on; these resources have been redirected to DAC. Therefore, the social planner uses DAC as a way to offset the costs in hard-to-abate sectors only, while relatively easy-to-abate sectoral investment is unchanged with and without DAC technologies. B Increasing emissions baselines In this Appendix, we allow for the baseline emissions to increase. We assume that each sector grows at some annual rate, 0 < ζ ≤ 1, until 2050, after which emissions are kept at level. We can therefore write the emissions baseline in each sector as,  a¯i,2020 (1 + ζ )t , t < 30, ¯i,t = a (B.1) a¯ i,2020 (1 + ζ )30 , t ≥ 30, where a¯i,2020 is the 2020 emissions levels quoted in Table 1. We choose ζ = 1.5% yr−1 in the results that follow; this puts 2050 emissions on par with SSP3–7.0 (Riahi et al., 2017). The model equations with a time dependent emissions baseline are straightforward to formulate based off our discussion in § 2 by simply replacing a ¯i,t . ¯i with a B.1 Results After incorporating an increasing emissions baseline, we find that the cost of achieving a given tem- perature target unilaterally increases; this owes to the assumption of more ‘dirty’ capital stocks being 21 Figure 7: Impact of delayed learning on sectoral allocation of abatement investment when direct air capture technologies are present. In each panel we show the average investment effort in each sector (see the titles). The black line is the certainty policy, the gold line is the average investment when the carbon budget is learned in 2060 without direct air capture technologies (DAC), and the blue line is the average investment under the same experimental setup as the gold line but with DAC included. Note these investment pathways are for the 1.5 ◦ C temperature target. 22 Figure 8: Effect of delayed learning on aggregate policy cost, growing emissions baseline. In panel a, we show the total discounted policy cost for each temperature target (see the legend) for the INV model that includes the impact of adjustment costs and economic inertia (solid lines) and MAC model, with does not include these factors (dashed lines). Panel b is as a, net of the cost of the corresponding policy without uncertainty. Panel c shows the marginal cost of delaying learning by one additional year. Panel d shows the ratio of the additional cost of uncertainty in the INV model divided by the additional cost of uncertainty in the MAC model; we call this ratio the uncertainty-inertia interaction index. 23 Figure 9: Effect of delayed learning on the temporal distribution of spending, growing emissions baseline. In panel a, we show the fraction of total policy spending before and after 2030, relative to the certainty policy for the 1.5 ◦ C target. A value of unity implies that the spending pre- or post-2030 in the certain model and the model with delayed learning are identical. Panels b–c are as a, but for the 1.7 ◦ C and 2 ◦ C targets, respectively. turned over to ‘clean’ capital stocks, which of course come at a price. The key conclusions of our paper, however, still hold: learning later about the carbon budget causes the overall cost of policy to increase, especially for the 1.5 ◦ C target (see Figure 8), and spending is relatively front-loaded when compared to the certainty policy (see Figure 9). One important note is that, when an increasing emissions baseline in considered (i.e., we consider a world trying to decarbonize while some sectors are building new ‘dirty’ capital simultaneously), the investment cost of decarbonizing and meeting the 2 ◦ C policy is similar to the cost of decarbonizing a world with a constant emissions baseline that meets the 1.5 ◦ C policy. This emphasizes the significant cost-savings of decarbonization that simply arise by not building any new ‘dirty’ capital stocks, such as coal-fired power plants or inefficient buildings. C Alternative calibrations C.1 High cost, linear calibration In this subsection, we use the high cost, linear calibration of the marginal abatement cost curves (blue lines in Figure 1) in our optimization scheme. We find that while the quantitative implications of our analysis are altered somewhat, the qualitative results are robust to this alternative calibration. C.1.1 Calibration We use the same scheme as outlined in § 2 to calibrate the INV and MAC model. We use (2.12) to ensure the relative costs between each sector are equal in both models, and that the total cost of each policy without uncertainty is equal. The numerical values for the marginal abatement cost and 24 Table 2: Calibration parameters for the MAC and INV models for each temperature target using the high cost, linear calibration. T ∗ = 1.5 ◦ C T = 80 yr r = 2 % yr−1 ¯ = 440 GtCO2 B σB = 72 GtCO2 $ tCO− 1 $ tCO− 1 Sector ¯ γ 2 ¯ [GtCO2 yr−1 ] a δ [% yr−1 ] c ¯ 2 GtCO2 yr−1 GtCO2 yr−3 Waste 243.9 0.82 3.3 78497 Industry 36.6 5.47 4 11767 Forestry 24.24 8.25 0.8 7801 Agriculture 24.57 4.07 5 7907 Transport 26.7 3.74 6.7 8605 Energy 16.68 11.99 2.5 5368 Buildings 62.5 3.2 1.7 20114 T ∗ = 1.7 ◦ C T = 80 yr r = 2 % yr−1 ¯ = 770 GtCO2 B σB = 75 GtCO2 $ tCO− 1 $ tCO− 1 Sector ¯ γ 2 ¯ [GtCO2 yr−1 ] a δ [% yr−1 ] c ¯ 2 GtCO2 yr−1 GtCO2 yr−3 Waste 243.9 0.82 3.3 88683 Industry 36.6 5.47 4 13924 Forestry 24.24 8.25 0.8 8815 Agriculture 24.57 4.07 5 8934 Transport 26.7 3.74 6.7 9722 Energy 16.68 11.99 2.5 6065 Buildings 62.5 3.2 1.7 22725 T ∗ = 2 ◦C T = 80 yr r = 2 % yr−1 B¯ = 1340 GtCO2 σB = 110 GtCO2 $ tCO− 1 $ tCO− 1 Sector ¯ γ 2 ¯ [GtCO2 yr−1 ] a δ [% yr−1 ] c ¯ 2 GtCO2 yr−1 GtCO2 yr−3 Waste 243.9 0.82 3.3 95912 Industry 36.6 5.47 4 14378 Forestry 24.24 8.25 0.8 9533 Agriculture 24.57 4.07 5 9661 Transport 26.7 3.74 6.7 10514 Energy 16.68 11.99 2.5 6559 Buildings 62.5 3.2 1.7 24577 25 marginal investment cost coefficients are shown in Table 2. C.1.2 Results In the high cost, linear calibration, we find that the key implications of our paper hold: learning later about the carbon budget causes the overall cost of policy to increase, especially for the 1.5 ◦ C target (see Figure 10), and spending is relatively front-loaded when compared to the equivalent certainty policy (see Figure 11). The only differences with the results of the low cost, linear calibration are quantitative, which is obviously the case, given that costs have increased in each sector significantly. C.2 Nonlinear calibration In this subsection, we use the nonlinear calibration of the marginal abatement cost curves (black lines in Figure 1) in our optimization scheme. Similar to the high cost, linear calibration shown above, using the nonlinear calibration causes our quantitative results to change, while leaving our qualitative conclusions about the effects of delayed uncertainty on policy unchanged. C.2.1 Calibration In our nonlinear calibration we fit a quadratic curve to the marginal abatement costs in Figure 1 to all sectors except agriculture, which is obviously linear. Therefore, we use ′ γi ¯i aξ (ai,t ) = γ i i,t , (C.1) in conjunction with (2.12) to calibrate our MAC and INV models. See Table 3 for the numerical values for this calibration. C.2.2 Results In the nonlinear calibration, we find that the cost of achieving a given temperature target increases across each target relative to the low cost, linear calibration; this owes to the nonlinear case being significantly more expensive than the low cost, linear calibration. The key conclusions of our paper, however, still hold: learning later about the carbon budget causes the overall cost of policy to increase, especially for the 1.5 ◦ C target (see Figure 12), and spending is relatively front-loaded when compared to the equivalent certainty policy (see Figure 13). D Computational details We solve our model using CVXPY, a domain-specific language written in Python and Julia for convex optimization problems (Diamond and Boyd, 2016). CVXPY uses disciplined geometric programming to interpret model equations and constraints and cast them into generic forms that can be used by open-source solvers (Agrawal et al., 2019). We use the GUROBI solver (Gurobi Optimization, LLC, 26 Figure 10: Effect of delayed learning on aggregate policy cost, high-bound calibration. In panel a, we show the total discounted policy cost for each temperature target (see the legend) for the INV model that includes the impact of adjustment costs and economic inertia (solid lines) and MAC model, with does not include these factors (dashed lines). Panel b is as a, net of the cost of the corresponding policy without uncertainty. Panel c shows the marginal cost of delaying learning by one additional year. Panel d shows the ratio of the total policy cost of uncertainty in the INV model divided by the total policy cost of uncertainty in the MAC model; we call this ratio the uncertainty-inertia interaction index. 27 Table 3: Calibration parameters for the MAC and INV models for each temperature target using the nonlinear calibration. T ∗ = 1.5 ◦ C T = 80 yr r = 2 % yr−1 B¯ = 440 GtCO2 σB = 72 GtCO2 $ tCO− 1 $ tCO− 1 Sector ¯ γ 2 ¯ [GtCO2 yr−1 ] a δ [% yr−1 ] c ¯ 2 ξ [–] GtCOξ 2 yr −ξ GtCOξ 2 yr −3ξ Waste 242.1 0.82 3.3 552744 2 Industry 5.03 5.47 4 11487 2 Forestry 2.54 8.25 0.8 5802 2 Agriculture 25.4 4.07 5 58037 1 Transport 4.83 3.74 6.7 11017 2 Energy 1.04 11.99 2.5 2369 2 Buildings 19.4 3.2 1.7 44173 2 T ∗ = 1.7 ◦ C T = 80 yr r = 2 % yr−1 B¯ = 770 GtCO2 σB = 75 GtCO2 $ tCO− 1 $ tCO− 1 Sector ¯ γ 2 ¯ [GtCO2 yr−1 ] a δ [% yr−1 ] c ¯ 2 ξ [–] GtCOξ 2 yr −ξ GtCOξ 2 yr −3ξ Waste 242.1 0.82 3.3 1282032 2 Industry 5.03 5.47 4 26644 2 Forestry 2.54 8.25 0.8 13458 2 Agriculture 25.4 4.07 5 134610 1 Transport 4.83 3.74 6.7 25552 2 Energy 1.04 11.99 2.5 5495 2 Buildings 19.4 3.2 1.7 102712 2 T ∗ = 2 ◦C T = 80 yr r = 2 % yr−1 B¯ = 1340 GtCO2 σB = 110 GtCO2 $ tCO− 1 $ tCO− 1 Sector ¯ γ 2 ¯ [GtCO2 yr−1 ] a δ [% yr−1 ] c ¯ 2 ξ [–] GtCOξ 2 yr −ξ GtCOξ 2 yr −3ξ Waste 242.1 0.82 3.3 2019340 2 Industry 5.03 5.47 4 41967 2 Forestry 2.54 8.25 0.8 21198 2 Agriculture 25.4 4.07 5 212025 1 Transport 4.83 3.74 6.7 40247 2 Energy 1.04 11.99 2.5 8656 2 Buildings 19.4 3.2 1.7 161377 2 28 Figure 11: Effect of delayed learning on the temporal distribution of spending, high- bound calibration. In panel a, we show the fraction of total policy spending before and after 2030, relative to the certainty policy for the 1.5 ◦ C target. A value of unity implies that the spending pre- or post-2030 in the certain model and the model with delayed learning are identical. Panels b–c are as a, but for the 1.7 ◦ C and 2 ◦ C targets, respectively. 2023), which can be accessed for free by academics and researchers, that solves the base model in under 10−2 seconds and models with uncertainty in the range of 10−1 seconds, depending on how many samples we draw from the RCB distribution. Using this software allows us to circumvent solving the Bellman equation (Bellman, 1957) directly using techniques such as value function iteration (Judd, 1998), which can be computationally challenging (Cai, 2019), especially with high-dimensional state spaces and constraints on each state variable as is present in our model setup. 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