metadata: authors: 0: author: Bauer,Adam Michael
1: author: Mcisaac,Florent John
2: author: Hallegatte,Stephane
count: World,Multi-Regional
docna: 0: docna: How Delayed Learning about Climate
            Uncertainty Impacts Decarbonization Investment Strategies
docty: Policy Research Working Paper
abstracts: cdata!: The Paris Agreement established that
            global warming should be limited to &#x201c;well
            below&#x201d; 2&#xb0;C and encouraged efforts to
            limit warming to 1.5&#xb0;C. Achieving this goal
            presents a challenge, especially given (i) economic inertia
            and adjustment costs, which penalize a swift transition away
            from fossil fuels, and (ii) climate uncertainty that, for
            example, hinders the ability to predict the amount of
            emissions that can be released before a given temperature
            target is exceeded. This paper presents a modeling framework
            that explores optimal decarbonization investment strategy
            when both adjustment costs and climate uncertainty are
            present. The findings show that climate uncertainty impacts
            investment in three ways: (i) the cost of policy increases,
            especially when adjustment costs are present; (ii) abatement
            investment is front-loaded relative to the certainty policy;
            and (iii) the sectoral allocation of investment changes to
            favor declining investment pathways rather than bell-shaped
            paths. The latter effect is especially pronounced in
            hard-to-abate sectors, such as heavy industry. Each of these
            effects can be traced back to the carbon price distribution
            inheriting a &#x201c;heavy tail&#x201d; when climate
            uncertainty is present. The paper highlights how climate
            uncertainty and adjustment costs combined result in a more
            aggressive least-cost strategy for decarbonization investment.
display_title: How Delayed Learning about Climate
            Uncertainty Impacts Decarbonization Investment Strategies
listing_relative_url: /research/2024/03/34290394/delayed-learning-climate-uncertainty-impacts-decarbonization-investment-strategies
topicv3: Climate Change m1326829 567,Investment Climate(IC) m1327235 401,Carbon Pricing m1327308 1846,Macroeconomic and Structural Policies
            m1327380 439
publishtoextweb_dt: 2024-03-28T00:00:00Z
dois: 10.1596/1813-9450-10743
origu: Off of Sr VP Dev Econ/Chief Econ (DECVP)
no_of_pages: 47
author_keys: 0: author_key: 2311156
1: author_key: 2040293
2: author_key: 1103816
model_extractions: 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 April 2024 A verified reproducibility package for this paper is available at http://reproducibility.worldbank.org, click **here** for direct access.
Policy Research Working Paper 10743

# **Abstract**

The Paris Agreement established that global warming should be limited to “well below” 2°C and encouraged efforts to limit warming to 1.5°C. Achieving this goal presents a challenge, especially given (i) economic inertia and adjustment costs, which penalize a swift transition away from fossil fuels, and (ii) climate uncertainty that, for example, hinders the ability to predict the amount of emissions that can be released before a given temperature target is exceeded. This paper presents a modeling framework that explores optimal decarbonization investment strategy when both adjustment costs and climate uncertainty are present. The findings show that climate uncertainty impacts investment in three ways: (i) the cost of policy increases, especially when adjustment costs are present; (ii) abatement investment is front-loaded relative to the certainty policy; and (iii) the sectoral allocation of investment changes to favor declining investment pathways rather than bell-shaped paths. The latter effect is especially pronounced in hard-to-abate sectors, such as heavy industry. Each of these effects can be traced back to the carbon price distribution inheriting a “heavy tail” when climate uncertainty is present. The paper highlights how climate uncertainty and adjustment costs combined result in a more aggressive least-cost strategy for decarbonization investment.

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. A verified reproducibility package for this paper is available at http://reproducibility.worldbank.org, click **here** for direct access.

<!-- Start of picture text --> TRANSPARENT<br>ANALYSIS<br>G<br>R E<br>N<br>S<br>I P<br>Y E<br>K A<br>C A<br>R P<br>LI R<br>O E<br>O C<br>W R<br>H<br>S<br>P<br><!-- End of picture text --> _The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# How Delayed Learning about Climate Uncertainty Impacts Decarbonization Investment Strategies<sup>∗</sup>

Adam Michael Bauer<sup>†1,2</sup> , Florent McIsaac<sup>2</sup> , and St ́ephane Hallegatte<sup>2</sup> 1 _Department of Physics, University of Illinois Urbana-Champaign, 1110 W Green St Loomis Laboratory, Urbana, IL 61801_

> 2 _World Bank Group, Washington DC_

Originally published in the Policy Research Working Paper Series on _April 2024._ This version is updated on _June 2024._ To obtain the originally published version, please email prwp@worldbank.org.

**JEL:** E10, Q50, Q54, Q58

**Keywords:** green investment, adjustment costs, stochastic modeling, climate risk, carbon price

> ∗The authors thank Yongyang Cai, Kevin Carey, Simon Dietz, Roy Dong, Sam Okullo, Cristian Proistosescu, Adrien Vogt-Schilb, and Gernot Wagner for helpful discussions related to this work. The authors also thank Mar ́ıa Reyes Retana Torre for verifying the reproducibility of our results. 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.

> †Corresponding author: adammb4@illinois.edu
# **Contents**

|**1**<br>**Intr**|**oduction**|**3**|
|---|---|---|
|**2**<br>**Mo**|**del description**|**4**|
|2.1|Base models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .|. . .<br>4|
||2.1.1<br>Abatement investment model . . . . . . . . . . . . . . . . . . . . . . . . . .|. . .<br>4|
||2.1.2<br>“Strawman” model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .|. . .<br>6|
|2.2|Adding delayed learning to the INV and MAC models . . . . . . . . . . . . . . . .|. . .<br>6|
||2.2.1<br>Abatement investment model with delayed learning<br>. . . . . . . . . . . . .|. . .<br>6|
||2.2.2<br>“Strawman” model with delayed learning<br>. . . . . . . . . . . . . . . . . . .|. . .<br>8|
|2.3|Calibration<br>. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .|. . .<br>8|
||2.3.1<br>Global parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .|. . .<br>8|
||2.3.2<br>Marginal abatement and investment costs . . . . . . . . . . . . . . . . . . .|. . .<br>10|
|**3**<br>**Res**|**ults**|**12**|
|3.1|Impact of delayed learning on aggregate policy cost . . . . . . . . . . . . . . . . . .|. . .<br>12|
|3.2|Impact of delayed learning on the temporal distribution of policy cost<br>. . . . . . .|. . .<br>14|
|3.3|Impact of delayed learning on the sectoral allocation of abatement investment . . .|. . .<br>15|
|**4**<br>**Dis**|**cussion**|**17**|

2
# **1 Introduction**

The signing of the Paris Agreement marked a breakthrough commitment and consensus in the international 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<sup>_◦_</sup> C with effort towards limit warming to 1.5<sup>_◦_</sup> 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 trillion<sup>1</sup> , 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 investment 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 that 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; Cai, 2021; Bauer et al., 2024, for a few examples) and imply more climate damages (Calel et al., 2020); 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 decar-

> 1Throughout, 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
bonizing the economy prior to breaching some temperature target for the least cost when the true value of the RCB 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 leastcost 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 (Section 1 of the _Supplementary Materials_ ), different emissions baselines (Section 2 of the _Supplementary Materials_ ), and alternative cost of investment calibrations (Section 3 of the _Supplementary Materials_ ).

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. 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<sup>+</sup> , is not exceeded. Each sector has an abatement potential (or annual emissions rate), _a_ ̄ _i ∈_ R<sup>+</sup> , and a capital depreciation rate, 0 _< δi ≤_ 1. No sector has the capacity for “negative emissions”.<sup>2</sup>

> 2We explore the influence of direct air capture on our results in Section 1 of the _Supplementary Materials_ , and find that none of the results are significantly altered.

4
The policymaker controls the amount of abatement investment, _xi,t_ , in each sector _i ∈I_ at each point 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<sup>+</sup> _→_ R<sup>+</sup> , 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, Subject to : _ai,t_ +1 = _ai,t_ + ∆ _t_ ( _xi,t − δiai,t_ ) _,_ where ∆ _t >_ 0 is the timestep (assumed to be annual throughout), _β_ := (1+ _r_ )<sup>_−_1</sup> is the discount factor, and 0 _< r ≤_ 1 is the discount rate.<sup>3</sup> 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, that weighs the carbon price (given by _μ_ ) against the marginal abatement investment costs. Note the carbon price is determined endogenously in the model as the shadow value of emissions reductions.<sup>4</sup> When _μ >_ ( _r_ + _δi_ ) _c_<sup>_′_</sup> _i_<sup>(</sup><sup>_δia_ ̄</sup><sup>_i_), paying the carbon price is more expensive than a marginal unit of abatement</sup> investment, and it follows from the proof of Lemma 2.1 that the optimal investment path is declining. If _μ <_ ( _r_ + _δi_ ) _c_<sup>_′_</sup> _i_<sup>(</sup><sup>_δia_ ̄</sup><sup>_i_),theninitiallypayingthecarbonpriceismorecost-effectivethanamarginal</sup> unit of abatement investment, forcing the investment pathway to start low. The path then rises over

> 3Our framework does not explicitly capture technology change over time; however, including exogenous technology change, as done in a number of other integrated assessment models (e.g., Nordhaus, 2017) would simply amount to a transformation in the discount factor, such that _β_ = (1 + _r_ )<sup>_−_1</sup> _→_ (1 _− φ_ ) _/_ (1 + _r_ ), where 0 _≤ φ <_ 1 is the exogenous rate of technology change.

> 4Mathematically speaking, the shadow value of emissions reductions is the dual of the cumulative emissions time derivative, _ψ_<sup> ̇</sup> _t_ := � ( ̄ _ai − ai,t_ ). _i∈I_

5
time, commensurate with a rising carbon price (Hotelling, 1931), before declining again to the steadystate 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 investment in § 3.3.

## **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 models (Nordhaus, 2017; Barrage and Nordhaus, 2024) 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 investment costs, the social planner uses an abatement cost function in each sector _i ∈I_ , _γi_ : R<sup>+</sup> _→_ R<sup>+</sup> , that maps abatement to monetary cost. The abatement cost functions _γi_ are increasing and convex functions of abatement. The remaining constraints are identical to those of the INV model.

We can formulate this problem as a cost-minimizing optimization problem such that, 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_ to be uncertain. We use a recent study (Dvorak et al., 2022) that estimates the RCB probability distribution using a state-of-the-art, simplified climate emulator, FaIR (Smith et al., 2018; Leach et al., 2021), which was also used by the IPCC for the same purpose (Intergovernmental Panel on Climate Change, 2021). Their RCB distribution is approximately Gaussian<sup>5</sup> , and we sample the 1<sup>st</sup> _−_ 99<sup>th</sup> percentile range of their distribution using a truncated normal

> 5Note the distribution of the RCB is approximately Gaussian owing to the underlying distribution of the transient climate response to emissions being approximately normal (Intergovernmental Panel on Climate Change, 2021; Matthews et al., 2021; Dvorak et al., 2022).

6
( ) (SE [tj Y gy EO) ~() ( )
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<!-- Start of picture text --> Energy Industry Buildings Agriculture<br>200 . .<br>ae 175<br>8~ 150<br>Sam<br>$<br>3 125<br>N<br>o<br>= 100 . °. . 6<br>i rae San<br>fo} - 7<br>o 75 /-- fo<br>oO 7<br>£& g on<br>2 50 . f°? ° a<br>25 eee ae rae “<br>0.0 2.5 5.0 7.5 10.0 12.5 0 2 4 0 1 2 3 0 1 2 3 4<br>Abatement pot. (GtCO2 yr-?) Abatement pot. (GtCO2 yr-?) Abatement pot. (GtCO2 yr-?) Abatement pot. (GtCO2 yr-?)<br>Transport Forestry Waste<br>200 . g<br>=iss 175 * IPCC AR6 Data<br>= 150 — Nonlinear<br>eyA --- Low cost, linear<br>8 125 High. cost, linear.<br>& 100 2 . fe<br>ny<br>fo}<br>o 75<br>oO<br>£-<br>2<br>oO 50 . . -" ae e<br>= uect a<br>25 _-e-- -— uecc7<br>0 1 2 3 0 2 4 6 8 0.0 0.2 0.4 0.6 0.8<br>Abatement pot. (GtCO2 yr-?) Abatement pot. (GtCO2 yr-?) Abatement pot. (GtCO2 yr-?)<br><!-- End of picture text -->
National Center for Energy Economics, 2022). In each of these baselines, emissions are expected to rise to just above 40 GtCO2 yr<sup>_−_1</sup> 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<sup>_−_1</sup> . This describes our “business-as-usual” scenario<sup>9</sup> ; we probe the impact of assuming an increasing emissions baseline in Section 2 of the _Supplementary Materials_ . 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 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 Section 3 of the _Supplementary Materials_ . In this calibration, we assume that our cost functions _ci_ ( _xi,t_ ) and _γi_ ( _ai,t_ ) are quadratics, such that We calibrate these cost functions using IPCC marginal cost data (Intergovernmental Panel on Climate Change, 2022). Marginal abatement costs are given by, which implies that marginal abatement costs are linear in abatement potential. 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 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 Table 1 for a full summary of our calibration results.

> 9Note 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: Calibration parameters for the MAC and INV models for each temperature target using the low cost, linear calibration.**

|_T_ = 80 yr|_T _<sup>_∗_</sup>= 1_._7 <sup>_◦_</sup>C<br>_r_ = 2 % yr<sup>_−_1</sup><br> ̄_B_ = 770 GtCO2|_σB_ = 220 GtCO2|
|---|---|---|
|Sector| ̄_γ_<br>�<br>$ tCO<sup>_−_1</sup><br>2<br><br>�<br> ̄_a_[GtCO2yr<sup>_−_1</sup>]|_δ_[%yr<sup>_−_1</sup>]<br> ̄_c_<br>�<br>$ tCO<sup>_−_1</sup><br>2<br><br>�|
||GtCO2 yr<sup>_−_1</sup><br><br>|<br><br><br>GtCO2 yr<sup>_−_3</sup><br>|
|Waste|40.82<br>0.82|3.3<br>14031|
|Industry|17.54<br>5.47|4<br>6029|
|Forestry|7.12<br>8.25|0.8<br>2447|
|Agriculture|23.53<br>4.07|5<br>8088|
|Transport|6.12<br>3.74|6.7<br>2104|
|Energy|2.82<br>11.99|2.5<br>969|
|Buildings|12.99<br>3.2|1.7<br>4465|
||_T _<sup>_∗_</sup>= 2 <sup>_◦_</sup>C||
|_T_ = 80 yr|_r_ = 2 % yr<sup>_−_1</sup><br> ̄_B_ = 1340 GtCO2|_σB_ = 380 GtCO2|
|Sector| ̄_γ_<br>�<br>$ tCO<sup>_−_1</sup><br>2<br><br>�<br> ̄_a_[GtCO2yr<sup>_−_1</sup>]|_δ_[%yr<sup>_−_1</sup>]<br> ̄_c_<br>�<br>$ tCO<sup>_−_1</sup><br>2<br><br>�|
||GtCO2 yr<sup>_−_1</sup><br><br>|<br><br><br>GtCO2 yr<sup>_−_3</sup><br>|
|Waste|40.82<br>0.82|3.3<br>17475|
|Industry|17.54<br>5.47|4<br>7509|
|Forestry|7.12<br>8.25|0.8<br>3048|
|Agriculture|23.53<br>4.07|5<br>10073|
|Transport|6.12<br>3.74|6.7<br>2620|
|Energy|2.82<br>11.99|2.5<br>1207|
|Buildings|12.99<br>3.2|1.7<br>5561|

11
# **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 each temperature target (1.7<sup>_◦_</sup> C and 2<sup>_◦_</sup> C) in Figure 2. In Panel 2 **a** , we show the total discounted cost of the optimal policy 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 2 **b** . 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 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 1.7<sup>_◦_</sup> C target as an example in Panel 2 **b** , 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 2080 and 2090 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 the curves are less steep 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 2 **c** ); we find that the value of learning the carbon budget a year earlier can be as high as $900b yr<sup>_−_1</sup> . 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 2 **b** .

We further find that economic inertia accentuates the impact of delayed learning about the RCB. In Panel 2 **d** , 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 2 **a** ). We find that for both 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 12
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1.7<sup>_◦_</sup> 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 5 **b** ) 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 investment 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 that 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 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 17
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, become more steeply declining when delayed learning is present. For hard-to-abate sectors that have a bellshaped 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 inheriting a heavy tail and become positively skewed, which drives up the average carbon price (see Figure 3); 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 nearcertainty) is significantly delayed, or when both of these effects are present.<sup>10</sup> 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 Section 1 and 2 of the _Supplementary Materials_ , respectively). The first is that the inclusion of direct air capture only leads to marginal cost-savings as a function of learning time for both the 1.7<sup>_◦_</sup> C target and the 2<sup>_◦_</sup> 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<sup>_◦_</sup> C target exceeds that of the 1.7<sup>_◦_</sup> C investment path without the increasing baseline. This highlights the significant cost of building more “dirty” capital stocks that have to be decarbonized or stranded down the line.

A final takeaway from our work is that narrowing the uncertainty in the RCB could lead to nontrivial monetary savings for policymakers. Indeed, the logic of the marginal cost of uncertainty shown

> 10It 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
in Figure 2 **c** 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 are publicly available on the corresponding author’s Github page, found at the following URL: `https://github.com/adam-bauer-34/BMH-delayed-learning-reprod` , see Release 1.1, or via the World Bank Group’s Reproducibility Package repository.

# **Author ORCIDs**

Adam Michael Bauer: 0000-0002-7471-8934 Florent McIsaac: 0000-0002-5169-8256 St ́ephane Hallegatte: 0000-0002-1781-4268

# **Author Contributions**

AMB, FM, and SH conceived the study. AMB wrote the code, gathered and analyzed data, made the figures, and wrote the first draft of the paper. FM and SH provided guidance and advising. All authors assisted in editing this draft and approve the submitted version.

# **Competing interests**

AMB: I have no conflicts of interest to disclose.

FM: I have no conflicts of interest to disclose.

SH: I have no conflicts of interest to disclose.

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24
Supplementary Materials for: “How Delayed Learning about Climate Uncertainty Impacts Decarbonization Investment Strategies” Adam Michael Bauer<sup>∗1,2</sup> , Florent McIsaac<sup>2</sup> , and St ́ephane Hallegatte<sup>2</sup> 1 _Department of Physics, University of Illinois Urbana-Champaign, 1110 W Green St Loomis Laboratory, Urbana, IL 61801_

> 2 _World Bank Group, Washington DC_

May 22, 2024

# **Contents**

|**1**<br>**Incl**|**uding direct air capture technologies**|**3**|
|---|---|---|
|1.1|Augmented model framework . . . . . . . . . . . . . . . . . . . .|. . . . . . . . . . . . .<br>3|
|1.2|Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .|. . . . . . . . . . . . .<br>3|
|**2**<br>**Inc**|**reasing emissions baselines**|**4**|
|2.1|Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .|. . . . . . . . . . . . .<br>7|
|**3**<br>**Alt**|**ernative calibrations**|**8**|
|3.1|High cost, linear calibration . . . . . . . . . . . . . . . . . . . . .|. . . . . . . . . . . . .<br>8|
||3.1.1<br>Calibration . . . . . . . . . . . . . . . . . . . . . . . . . .|. . . . . . . . . . . . .<br>8|
||3.1.2<br>Results<br>. . . . . . . . . . . . . . . . . . . . . . . . . . . .|. . . . . . . . . . . . .<br>9|
|3.2|Nonlinear calibration . . . . . . . . . . . . . . . . . . . . . . . . .|. . . . . . . . . . . . .<br>9|
||3.2.1<br>Calibration . . . . . . . . . . . . . . . . . . . . . . . . . .|. . . . . . . . . . . . .<br>9|
||3.2.2<br>Results<br>. . . . . . . . . . . . . . . . . . . . . . . . . . . .|. . . . . . . . . . . . .<br>9|
|3.3|1.5 <sup>_◦_</sup>C temperature target . . . . . . . . . . . . . . . . . . . . . .|. . . . . . . . . . . . .<br>9|
||3.3.1<br>Calibration . . . . . . . . . . . . . . . . . . . . . . . . . .|. . . . . . . . . . . . .<br>15|
||3.3.2<br>Results<br>. . . . . . . . . . . . . . . . . . . . . . . . . . . .|. . . . . . . . . . . . .<br>15|

- ∗Corresponding author: adammb4@illinois.edu

1
|**4**<br>**Co**|**mputational details**|**18**|
|---|---|---|
|4.1|Truncated Gauss-Hermite approximation. . . . . . . . . . . . . .|. . . . . . . . . . . . .<br>18|
|4.2|Solving the model<br>. . . . . . . . . . . . . . . . . . . . . . . . . .|. . . . . . . . . . . . .<br>19|

2
# **1 Including direct air capture technologies**

In this Section, we include direct air capture technologies (DAC) in our modeling framework. We treat DAC like an additional economic sector with zero annual emissions (i.e., _a_ ̄ _DAC_ = 0) and some abatement potential, that we call _Z >_ 0 (in GtCO2 yr<sup>_−_1</sup> ). DAC is unique in that its emissions rate is not the same as its abatement potential, as was assumed in every other sector.

## **1.1 Augmented model framework**

A social planner tasked with decarbonization with DAC technologies at their disposal solves the following cost-minimization problem, Subject to : _ai,t_ +1 = _ai,t_ + ∆ _t_ ( _xi,t − δiai,t_ ) _,_ 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<sup>_−_</sup> 2<sup>1.</sup> In each simulation we show below, we assume a marginal abatement cost of $200 tCO<sup>_−_</sup> 2<sup>1.Anylower</sup> 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<sup>_−_1</sup> and that, at maximum, the abatement potential from DAC technologies is 5 GtCO2 yr<sup>_−_1</sup> (Terlouw et al., 2021).

## **1.2 Results**

We find that the impact of including DAC in our model for abatement investment is consistent across temperature targets, see Figure 1. For both temperature targets we explored, 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 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.

The marginal investments in DAC allows the social planner to invest slightly less resources in hardto-abate sectors, see Figure 2. We find that most sectors – waste, forestry, transport, energy, and 3
<!-- Start of picture text --> 40 a b<br>c<br>onl or<br>hd > e<br>© 30 —— £<br>7) == o<br>Ssc > a oor 5 15<br>E25 5°<br>3fo} BSo= 10<br>5 20 vE<br>6 oe<br>% 5<br>8=<br>sre} 15 3 5<br>10 =<br>0 S<br>5 e_—=<=<—«=——=>RrevQeEeeEeEeEeEeEEEElQV—0QQVV——_—_—_et tn —eECoCoC70C(C—u_Q<br>2020 2030 2040 2050 2060 2070 2080 2090 2020 2030 2040 2050 2060 2070 2080 2090<br>Year information is revealed Year information is revealed<br>T*=1.5°C —_ T*=17°C<br><!-- End of picture text -->
<!-- Start of picture text --> Waste Industry Forestry Agriculture<br>\ 350 \ 200<br>804 | |<br>| 300 8007 \<br>5<br>oT 250 150<br>2560i—<ee 200 600<br>Ba | 100 j<br>2 5 40 150 | 400 |<br>30 \ |<br>sée 100 200 50 A \<br>0 —|<br>TN 500 _— 0 Ne 0 a<br>| a | re | a|<br>400 Transport 1400 Energy Buildings DAC<br>| 3507 |<br>350 12004 | |<br>\ 300 20<br>~ 300 1000<br>e4 250<br>oh 250 15<br>Ca 800<br>£ Peat‘5 200 200<br>22150aaoe 600 150 10<br>ga3 ~ 100 400 100<br>ai 5<br>504 200 aN 50 Ni<br>0 0 0 0<br>2020eer2040 2060 2080oO2100 2020eT2040 2060 2080oO2100 2020eT2040 2060 2080oO2100 2020eT2040 2060 2080oO2100<br>Year Year Year<br>— Certainty policy —— Average investment without DAC Average investment with DAC<br><!-- End of picture text -->
<!-- Start of picture text --> 70 a b<br>30<br>ae ><br>‘S 60 € 25<br>2G<br>85<br>Ee= Siese 20 voc --<br>350 Sa ae<br>3 - 85 a<br>a oer 8 = 15 “<br>5 ae wo Ee “<br>a “ i 0 v<br>S 40 a Es 1 /<br>rc)5 ¢“ cS}<x 1<br>“ 5 4<br>“ of<br>30 Pae uo"<br>nn 0 Sse eee<br>rr | ee|<br>c 1.40 d<br>0.8 % 1.35<br>Tt<br>on<br>> 5B 1.30<br>aa oN go:<br>os 0.6 - s oo<br>ow \ \ 5 = 1.25<br>gs /‘ N‘ os#2<br>ow 1 \ $6 1.20<br>3604 ! ‘\ oa”<br>£= i \ £Y<br>Qe ! \ >= 115<br>=~ F \ £1<br>0.2 ; \ 551.10<br>/ re)<br>/ NS c<br>; =)<br>4 1.05<br>Se eee<br>0.0 o” 1.00<br>Le | Le|<br>2020 2030 2040 2050 2060 2070 2080 2090 2020 2030 2040 2050 2060 2070 2080 2090<br>Year information is revealed Year information is revealed<br>T*=1.7°C — T*=2°C<br><!-- End of picture text -->
T*=1.7°C T*=2°C <!-- Start of picture text --> a b<br>3.5<br>x<<br>oO<br>e 3.0<br>ray<br>&<br>£25<br>oO<br>UV<br>Cc<br>a<br>S 2.0<br>ar)<br>fe}<br>U<br>215<br>2<br>Ww<br>ne}<br>me}<br>0.5<br>2020 2025 2030 2035 2040 2045 £2050 2020 2025 2030 2035 2040 2045 # £2050<br>Year information is revealed Year information is revealed<br>— Pre-2030 Spending —— Post-2030 Spending<br><!-- End of picture text -->
**Table 1: Calibration parameters for the MAC and INV models for each temperature target using the high cost, linear calibration.**

|_T_ = 80 yr|_T _<sup>_∗_</sup>= 1_._7 <sup>_◦_</sup>C<br>_r_ = 2 % yr<sup>_−_1</sup><br> ̄_B_ = 770 GtCO2|_σB_ = 220 GtCO2|
|---|---|---|
|Sector| ̄_γ_<br>�<br>$ tCO<sup>_−_1</sup><br>2<br><br>�<br> ̄_a_[GtCO2yr<sup>_−_1</sup>]|_δ_[%yr<sup>_−_1</sup>]<br> ̄_c_<br>�<br>$ tCO<sup>_−_1</sup><br>2<br><br>�|
||GtCO2 yr<sup>_−_1</sup><br><br>|<br><br><br>GtCO2 yr<sup>_−_3</sup><br>|
|Waste|243.9<br>0.82|3.3<br>88683|
|Industry|36.6<br>5.47|4<br>13924|
|Forestry|24.24<br>8.25|0.8<br>8815|
|Agriculture|24.57<br>4.07|5<br>8934|
|Transport|26.7<br>3.74|6.7<br>9722|
|Energy|16.68<br>11.99|2.5<br>6065|
|Buildings|62.5<br>3.2|1.7<br>22725|
||_T _<sup>_∗_</sup>= 2 <sup>_◦_</sup>C||
|_T_ = 80 yr|_r_ = 2 % yr<sup>_−_1</sup><br> ̄_B_ = 1340 GtCO2|_σB_ = 380 GtCO2|
|Sector| ̄_γ_<br>�<br>$ tCO<sup>_−_1</sup><br>2<br><br>�<br> ̄_a_[GtCO2yr<sup>_−_1</sup>]|_δ_[%yr<sup>_−_1</sup>]<br> ̄_c_<br>�<br>$ tCO<sup>_−_1</sup><br>2<br><br>�|
||GtCO2 yr<sup>_−_1</sup><br><br>|<br><br><br>GtCO2 yr<sup>_−_3</sup><br>|
|Waste|243.9<br>0.82|3.3<br>95912|
|Industry|36.6<br>5.47|4<br>14378|
|Forestry|24.24<br>8.25|0.8<br>9533|
|Agriculture|24.57<br>4.07|5<br>9661|
|Transport|26.7<br>3.74|6.7<br>10514|
|Energy|16.68<br>11.99|2.5<br>6559|
|Buildings|62.5<br>3.2|1.7<br>24577|

# **3 Alternative calibrations**

## **3.1 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, the qualitative results are robust to this alternative calibration.

### **3.1.1 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 marginal investment cost coefficients are shown in Table 1.

8
## **3.1.2 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 and spending is relatively frontloaded when compared to the equivalent certainty policy (see Figures 5 and 6). 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.

# **3.2 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.

## **3.2.1 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 in conjunction with (2.12) to calibrate our MAC and INV models. See Table 2 for the numerical values for this calibration.

## **3.2.2 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 and spending is relatively front-loaded when compared to the equivalent certainty policy (see Figures 7 and 8).

# **3.3 1.5**<sup>_◦_</sup> **C temperature target**

In this subsection, we explore our model implications when we use a remaining carbon budget that implies warming stays at or below 1.5<sup>_◦_</sup> C. We emphasize that our approach differs from the notion of “keeping 1.5<sup>_◦_</sup> C possible”, that is, formulating a global investment strategy that maintains a nonzero probability of 1.5<sup>_◦_</sup> C warming by 2100. Our model goes much further, and enforces that the planner must formulate a strategy that is guaranteed to keep warming below 1.5<sup>_◦_</sup> C. Hence, in the results below we find that the cost of achieving the 1.5<sup>_◦_</sup> C target is incredibly high even for our lowest cost calibration, because the policy is robust to even very tiny carbon budgets, which causes the cost of 9
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<!-- Start of picture text --> T*=1.7 °C T*=2°C<br>a b<br>6<br>x<<br>wo<br>xe}<br>£<br>>5<br><<br>8<br>u4<br>Cc<br>a]<br>oS<br>2 3<br>U<br>ry<br>Cc<br>22<br>2<br>xe}<br>xt<br> VVCVoOV030303 0 VVCVoOCV0505u0u030C 0<br>2020 2025 2030 2035 2040 2045 #2050 2020 2025 2030 2035 2040 2045 £2050<br>Year information is revealed Year information is revealed<br>— Pre-2030 Spending — Post-2030 Spending<br><!-- End of picture text -->
~~[-—~~ ~~[-—~~ ~~[-— [-—]~~
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<!-- Start of picture text --> T*=1.7 °C T*=2°C<br>10 a b<br>x<<br>()<br>28<br>><br><<br>8<br>v6<br>Cc<br>a<br>So<br>i)<br>[o}<br>o4<br>©<br>Cc<br>2<br>#<br>To<br>~ ? ee<br>—<br>re ee ee ee ee es | re ee ee ee ee es |<br>2020 2025 2030 2035 2040 2045 £2050 2020 2025 2030 2035 2040 2045 £2050<br>Year information is revealed Year information is revealed<br>— Pre-2030 Spending —— Post-2030 Spending<br><!-- End of picture text -->
**Table 3: Calibration parameters for the MAC and INV models for the 1.5**<sup>_◦_</sup> **C temperature target using the low cost, linear calibration.**

|_T_ = 80 yr|_T _<sup>_∗_</sup>=<br>_r_ = 2 % yr<sup>_−_1</sup>|1_._5 <sup>_◦_</sup>C<br> ̄_B_ = 440 GtCO2|_σB_ = 180 G|tCO2|
|---|---|---|---|---|
|Sector| ̄<br>�<br>$ tCO<sup>_−_1</sup><br>2<br>�| ̄_a_[GtCOr<sup>_−_1</sup>]|_δ_[%r<sup>_−_1</sup>]| ̄_c_<br>�<br>$ tCO<sup>_−_1</sup><br>2<br>�|
||_γ_<br><br>GtCO2 yr<sup>_−_1</sup><br>|2 y|y|GtCO2 yr<sup>_−_3</sup><br>|
|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|

policy to skyrocket. For context, the worst case carbon budget we sample is just higher than 100 GtCO2, which would give the social planner just north of two years to decarbonize in the absence of any abatement. Furthermore, we cannot rule out zero or negative RCBs for the 1.5<sup>_◦_</sup> C temperature target (Matthews et al., 2021; Dvorak et al., 2022), implying that even our sampling of the RCB is optimistic from a geophysical perspective. This framing provides context to our results: they should not be interpreted as “ruling out” limiting warming to 1.5<sup>_◦_</sup> C, but rather should demonstrate the urgency of this target in terms of global investment needs.

# **3.3.1 Calibration**

We calibrate our 1.5<sup>_◦_</sup> C target model using the low cost, linear calibration for the marginal abatement costs, i.e., the gold lines in Figure 1. We follow the same prescription as laid out in the main text, see Table 3.

# **3.3.2 Results**

We find that for the 1.5<sup>_◦_</sup> C temperature target, the main qualitative findings of our paper hold, as was the case in each other alternative calibration; see Figures 9 and 10).

However, we do find some differences with other temperature targets. For example, the overall cost of policy as a function of learning time does not follow an “S”-curve as much as an inverted “L”-curve, owing to even a marginal delay in learning about the RCB causing a large increase in cost. This is because the worst-case carbon budget is as low as 100 GtCO2, causing an immediate surge in precautionary investment. However, overall costs level out as the budget is learned later in the century, consistent with other temperature targets.

Furthermore, we find that the marginal cost of delay for the 1.5<sup>_◦_</sup> C target is more than twice that of the 1.7<sup>_◦_</sup> C or 2<sup>_◦_</sup> C targets, again owing to the urgency of the worst-case scenario.

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# **4 Computational details**

## **4.1 Truncated Gauss-Hermite approximation**

In our formulation of the RCB, we use a truncated normal distribution, as negative values of the RCB make our optimization problem infeasible. Geophysical modeling suggests that the RCB distribution is approximately normal, justifying our approach (Intergovernmental Panel on Climate Change, 2021; Dvorak et al., 2022).

To approximate this distribution, we use Gauss-Hermite quadrature nodes (Cai and Judd, 2010) transformed to a truncated normal distribution following the approach of Cai and Judd (2015). We do so using the following prescription. For a random variable _x ∼N_ ( ̄ _x, σx_ ), we can generate a random draw _x_<sup>_∗_</sup> using where _θ ∼N_ (0 _,_ 1). Therefore, formulating the Gauss-Hermite quadrature approximation for a truncated normal distribution, _x_ , equates to formulating the Gauss-Hermite quadrature approximation for _θ_ .

Therefore, consider the random variable _θ_ . We can transform _θ_ to a bounded random variable using the sigmoid function where _κ,_ Υ _∈_ R<sup>+</sup> . One can immediately see that Ψ has zero mean, is symmetric around the mean, and is bounded between ( _−_ Υ _,_ Υ). We can choose a number such that our distribution is bounded between _±_ Υ and select a _κ_ such that Ψ has unit variance.<sup>1</sup> An intuitive way to select Υ is for Υ to be the number of standard deviations away from _x_ ̄ one wishes to sample; for example, if _x_ 1 is the first percentile of the distribution of _x_ , then Assuming one has calculated _κ_ for their choice of Υ, a draw _x_<sup>_∗_</sup> _trunc_<sup>canbegeneratedusing</sup> It remains to apply the Gauss-Hermite approximation. We can approximate the distribution of _θ_ using _N_ Gauss-Hermite nodes with locations _{θ_ 1 _, ..., θN }_ , where _θi_ is the _i_<sup>th</sup> root of the physicist’s Hermite polynomials (Cai and Judd, 2010). Therefore, the locations of the Gauss-Hermite quadrature nodes for the truncated normal distribution are given by as originally desired.

We choose Υ to be the 1<sup>st</sup> percentile for the RCB distribution for the 1.7<sup>_◦_</sup> C and 2<sup>_◦_</sup> C targets,

> 1This is done using a root-finding algorithm, that can be found in the reproducibility package for this paper.

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**Table 4: Truncated Gauss-Hermite parameters for each temperature target.**

|_T _<sup>_∗_</sup>|Υ|_κ_|
|---|---|---|
|1.5|1.8809|1.04536|
|1.7|2.3254|1.04534|
|2|2.3262|1.04505|

and the 3<sup>rd</sup> percentile for the 1.5<sup>_◦_</sup> C target. We give the numerical values for Υ and _κ_ in each case in Table 4.

## **4.2 Solving the model**

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, 2023), which can be accessed for free by academics and researchers, that solves the base model in under 10<sup>_−_2</sup> seconds and models with uncertainty in the range of 10<sup>_−_1</sup> seconds, depending on how many quadrature nodes we use or 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. Note that we use Gauss-Hermite quadrature methods to evaluate the expectation operator in our objective functions, and use 8 approximation nodes throughout (Cai, 2019); only in generating Figure 5 do we take a large number of samples (500) from the RCB distribution to solve the model for illustrative purposes. The results of this study were checked to be the same using both methods.

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