The entanglement of climate and energy policies

Abstract

Increased CO₂ in the atmosphere increases tropospheric temperatures, thereby changing other climate attributes. However, cutting CO₂ emissions by reducing fossil fuel use raises concerns about economic development and energy security. Additional CO₂ is also directly beneficial for plants. Adaptation measures are an alternative, but those that are public goods are unlikely to be efficiently deployed without explicit government action. A model is developed to illustrate some of these complex policy trade-offs. The analysis is simplified by focusing on CO₂ accumulation in the atmosphere as a sufficient statistic for both the external effects of CO₂ and the global transition from fossil fuels to alternative energy sources.

JEL Classification: D62 Externalities; H23 Taxation and Subsidies: Externalities; Redistributive Effects; Environmental Taxes and Subsidies; O13 Economic Development: Agriculture; Natural Resources; Energy; Environment; Other Primary Products; Q32 Exhaustible Resources and Economic Development; Q42 Alternative Energy Sources; Q52 Pollution Control Adoption and Costs; Distributional Effects; Employment Effects; Q54 Climate; Natural Disasters and Their Management; Global Warming.

1. Introduction

Decades of attempts to prevent CO₂ accumulating in the atmosphere have failed. Regressing the logarithm of annual average concentrations of CO₂ at Mauna Loa against time, a decade at a time, reveals yearly growth rates of 0.46% from 1991 to 2000, 0.54% from 2001 to 2010, and 0.62% from 2011 to 2020. The failure can be traced to three problems.

The first is the global scope of the externality. Policies to limit CO₂ accumulation require international cooperation, but each nation wants others to bear the costs of control.

Second, many benefits from CO₂ emission control are delayed while the costs are immediate. As a practical matter, resource allocation decisions, whether made in the marketplace or via political mechanisms, tend to weigh short-term costs and benefits much more heavily than longer-term ones.

The third problem, which exacerbates the other two, is that fossil fuel combustion dominates both world energy supply and anthropogenic CO₂ emissions. According to the EI Statistical Review of World Energy, fossil fuels provided over 82% of world primary energy consumption in 2021, while IPCC (2001) says that fossil fuel combustion accounts for around 75% of anthropogenic CO₂ emissions. The link between energy use and CO₂ emissions makes issues apart from climate change relevant to evaluating policy. Since energy enables a force to do work, it is an essential input to production. Affordable, reliable, controllable, storable, transportable, and versatile energy sources are indispensable to a modern lifestyle. Hence, governments are reluctant to implement policies widely understood to raise energy prices or reduce energy use, supply reliability, or convenience. The dependence of economic output on energy input and the use of oil products by modern armed forces also ties energy supply to national security.

Section 4 develops a model that formalizes these trade-offs. Since our focus is the connections between climate and energy policies, we ignore anthropogenic emissions of greenhouse gases other than CO₂ and anthropogenic sources of CO₂ apart from using fossil fuels to provide energy. The data cited above suggest that the simplification may also be reasonable. To motivate the modeling assumptions, Section 2 summarizes relevant scientific issues, while Section 3 outlines some critical economic considerations. Section 5 compares the model to the class of aggregated Integrated Assessment Models (IAMs), called “benefit-cost IAMs” by Weyant (2017). The model developed herein more closely resembles these IAMs than the more disaggregated models dubbed “detailed process IAMs” by Weyant.

2. Overview of Relevant Scientific Issues

CO₂ is a colorless, odorless, non-toxic ¹ gas that is directly beneficial to plants as an input into photosynthesis and hence essential for almost all life on earth. CO₂ is also absorbed and released by oceans and the soil and, in the long term, sequestered in rocks and fossil fuels. The release of CO₂ via fossil fuel combustion raises its concentration in the atmosphere. The rates of removal of CO₂ by all sinks, including plants, then also increase, but not by enough to prevent the stock from accumulating at about one-half of one percent per year.

2.1. CO₂ as a Greenhouse Gas

The harm from CO₂ is indirect and depends on its stock in the atmosphere. CO₂ absorbs and re-radiates some frequencies of infrared radiation emitted from the earth’s surface and atmosphere. Increased CO₂ thereby impedes the transmission of long-wave radiation to space. Temperatures in the lower troposphere then must increase to balance incoming solar radiation not directly reflected to space by clouds, ice, and other highly reflective surfaces.

While interactions between individual gases and electromagnetic radiation are well understood, the altitudinal and latitudinal distributions of greenhouse gases alter the incremental effects of CO₂. Both absorption and emission are temperature- and pressure-dependent, and radiation bands affected by greenhouse gases overlap, with those affected by CO₂ already quite saturated. Despite the complexities, many researchers have concluded that the additional radiative forcing accompanying increased CO₂ in the atmosphere can be approximated by a logarithmic function $\mathrm{\Delta }F\approx K\ln x$ where $K$ is a constant and $x$ is the ratio of the current concentration of CO₂ in the atmosphere to the so-called “pre-industrial” level and $\mathrm{\Delta }F$ is measured in watts per square meter. ²

2.2. Feedback Effects

Climate models incorporate feedbacks that reinforce the initial warming from increased CO₂. The most important is that the initial warming increases evaporation and raises the amount of water vapor, the most significant greenhouse gas, in the Earth’s atmosphere. The modeled final effect from $\Delta F$ increased radiative forcing can be well-approximated by $\Delta F/(1-f)$ where $f$ is the feedback parameter. A fact that is critically important for policy is that $f$ is uncertain and differs substantially between models. ³

2.3. Climate, Weather, and Temperatures

Adverse effects from CO₂ accumulation arise partly from temperature increases in the atmosphere, on the land, and in the oceans. ⁴ Others are forecasted to arise from what is often called “climate change” as higher temperatures affect other meteorological phenomena. Climate is defined as the joint probability distribution of various meteorological measurements – such as temperature, rainfall, snowfall, humidity, frost-free days, hours of sunlight, cloudiness, and wind speed and direction – at a given location. Because of the substantial and predictable variation of these measurements within a year, the distributions are defined as functions of the ordinal day number within a year. A sample of multiple, traditionally thirty, years of observations is needed to measure climate. “Climate change” can then be defined as a change in the distributions of the meteorological measurements at one or more locations. There are several consequences of this definition.

First, climates constantly change from one 30-year period to the next since statistical measurements based on a sample size of thirty observations will have sampling variation. Imprecision in the measurements adds randomness. Equipment failures, poor siting, and other factors can also impart bias to measurements. An undetected change in the bias would also be measured as “climate change.”

Second, the phrase “climate change” often designates a change in “climate,” thought of as a singular object independent of location, occurring only as a result of human actions. It is often used even more narrowly to mean a change in “climate” (singular) only because of human emissions of greenhouse gases. In practice, humans affect climates (plural) in many other ways, including via urbanization, land clearance, introduction of invasive plants, and large-scale irrigation. ⁵ Climates also change independently of human actions. Volcanic eruptions have been shown to have short-run effects. Possible candidates for long-term natural forces that could affect temperatures and other climate attributes include long cycles in ocean currents and changes in the amount or distribution of incident solar radiation and other solar effects. ⁶ Some of these changes may affect climates by directly changing the type, geographic distribution, and amount of cloud cover.

Third, damaging weather events are often cited as the main threats from CO₂. However, the occurrence of many of these events is consistent with the distributions of weather variables from decades prior to when CO₂ accumulation could have had any significant effects. They are better characterized as extreme weather rather than climate change.

2.4. Measuring Harm from Changes in Climates

IAMs assume that harm from CO₂ monotonically increases in the departure of globally-averaged surface temperature anomalies ⁷ (GSTA) from a base value. Different IAMs use different base values to measure damages. The chosen base value is rarely explicitly justified as “ideal,” but the pre-industrial (Little Ice Age) average would surely be too low. ⁸

Another important fact for climate policy is that changes in distributions of meteorological phenomena following an enhanced greenhouse effect will differ by location. Furthermore, the net costs of those changes will depend on location. In some regions, extreme hot or cold temperatures impact welfare most, while extreme rainfall, high winds, or droughts are most impactful in others. It would seem most unlikely that the same GSTA would be ideal for every location. ⁹

2.5. Direct External Effects of CO₂

Increased CO₂ accumulation also has direct external effects. Thousands of laboratory and open-air experiments have shown that CO₂ enrichment increases plant growth and makes plants more tolerant of dryness by reducing transpiration (Pospisilova and Catsky 1999). Extra CO₂ has already been shown to increase agricultural output. Costs are incurred to pump CO₂ into commercial greenhouses to enhance productivity. More relevant for CO₂ emission control policy, Taylor and Schlenker (2021) examined the effect of increased ambient CO₂ on maize, soybean, and wheat yields in the main U.S. growing regions for these crops. They estimated a panel model of average yields at the county level with ambient CO₂ concentrations measured by the NASA Orbiting Carbon Observatory-2 (OCO-2) satellite as the main explanatory variable. They also include measures of temperature, precipitation, criteria pollutants (CO, NO₂, O₃, PM₁₀, SO₂), and county fixed effects and time trends. ¹⁰ They summarize the results as implying that, independent of any climate impacts, a 1 ppm increase in CO₂ increases yields for maize, soybean, and wheat by 0.5%, 0.6%, and 0.8%. Additional CO₂ in the oceans should also assist phytoplankton growth, thereby enhancing marine ecosystem productivity more generally. ¹¹

2.6. Natural Absorption of CO₂

CO₂ is absorbed not only by the growth of photosynthesizing organisms but also by other organisms exploiting that growth. In addition, oceans absorb more CO₂ as its concentration in the atmosphere increases, although the interactions are affected by seawater temperature and alkalinity. National Oceanic and Atmospheric Administration (NOAA) Global Monitoring Laboratory (2022) has developed a model of global CO₂ sources and sinks called CarbonTracker. They find that about half of anthropogenic CO₂ emitted into the atmosphere is absorbed by natural sinks, but there are uncertainties associated with the underlying mechanisms and measurements. While the modeled mean annual increases in CO₂ (in ppm/year) over 2000 to 2018 are close to the measured increases, the standard errors in the modeled annual increases are about the same magnitude as the measured increases.

3. Economic Issues Aaffecting CO₂ Emission Policy

3.1. Marginal Damages

The potentially harmful effects of CO₂ depend on the stock of CO₂ in the atmosphere, which accumulates gradually. This has two implications for the marginal damage curve from CO₂ emissions in any one year. First, the curve will have a positive intercept, as even trivial emissions that year will add to the existing harm and thus have a strictly positive effect. Second, the curve will be quite elastic as additional emissions in that year add only slightly to the damage from past and anticipated future emissions.

3.2. Marginal Costs of Emission Reduction

Emissions of conventional air pollutants (such as SO _x , NO _x , CO, mercury, or particulates) can be reduced in many ways. Changes in the production process or the type of fuel used can alter flue gas composition. Devices like precipitators, scrubbers, or catalytic converters can clean the pollutant from the flue gas before it is vented. Cutting output also will cut emissions at a cost of control equal to the forgone benefit from lost consumption minus the explicit (excluding the external costs) cost saving from reduced production. When many methods are available, emissions should be reduced using the lowest marginal cost method first. Once the marginal costs of different methods are equal, further emission reductions should use all those methods while keeping their marginal costs equal. This is known as the equimarginal principle. ¹² It will give a function relating the minimum marginal cost of control to emissions.

3.2.1. CO₂ as a Global Externality

The potentially harmful effects of CO₂ depend on global emissions. The equimarginal principle then requires the marginal cost of emission reduction to be equal in all locations. In a world of sovereign nations with different resource endowments and goals, such an outcome will be difficult, if not impossible, to achieve in practice.

Most of the growth in CO₂ emissions over coming decades is forecasted to come from large population developing countries. These forecasts are well-founded for several reasons. First, the large populations mean that a small increase in per capita energy use greatly increases total energy use. Second, the mechanization of agriculture, industrialization, and urbanization accompanying the early stages of rapid economic growth greatly increase the energy intensity of production. Third, in prioritizing economic growth, these countries have preferred fossil fuels. Not only are they more affordable than the alternatives. They are also more reliable, versatile, transportable, and storable.

Controlling CO₂ emissions in developed countries alone could raise CO₂ emissions growth in the short run. Many energy-intensive industries would move to exempt locations, where CO₂ emissions per unit of energy used are often higher. International transportation of bulky commodities also may increase.

3.2.2. Initial Marginal Costs of Reducing CO₂ Emissions

It is commonly assumed that the marginal costs of reducing emissions of air pollutants start at zero at the uncontrolled level. In the case of CO₂, and using the notation in Section 4, $C^{\prime }\left(0\right)$ would be zero if the marginal benefits of fossil fuel use equaled the marginal costs of supplying fossil fuel, ignoring any climate externalities. That would, in turn, follow if fossil fuels were supplied in competitive markets with no non-climate externalities, taxes, or subsidies. However, none of these conditions is valid, and it is possible that $C^{\prime }\left(0\right)\gtrless 0$ .

Focus first on the oil market, where transportation fuel is the largest demand component. Most OECD countries tax vehicle fuel more heavily than many other goods or services. Usually, a tax would leave the marginal benefits of consumption above the marginal costs of production, implying $C^{\prime }\left(0\right)>0$ . Parry and Small (2005) examined gasoline taxes in the U.K. and the U.S. in detail, noting that the tax in the U.K. is among the highest in industrial countries while that in the U.S. is among the lowest.

Parry and Small give three arguments supporting fuel taxes apart from any effect on CO₂ emissions. First, they are a “second best” ¹³ way of internalizing environmental externalities from local toxic pollutants. ¹⁴ Second, by raising driving costs, fuel taxes are a “second best” ¹⁵ way of reducing traffic congestion and accident externalities. Third, the relatively inelastic demand for fuel may reduce the efficiency losses from taxing it. Parry and Small find the congestion externality is the largest component of a “second-best optimal” tax, while the pollution component is the smallest (and would remain so even without the CO₂ externality). They also find that the U.S. tax rate is about half the “optimal” rate, while the U.K. rate is about double it. U.S. fuel efficiency and other mandates – including emission tests for vehicle registration – partially address the pollution externality, while increasing use of road and bridge tolls, parking taxes, and public transport subsidies partially address the congestion one. In summary, these arguments suggest $C^{\prime }\left(0\right)\approx 0$ for transportation fuel use in OECD economies.

Many oil-exporting countries subsidize oil products, which would normally cause the marginal cost of production to exceed the marginal benefits of consumption. However, many of the subsidizing countries are OPEC members restricting supply. Even the subsidized domestic price likely exceeds their relatively low marginal cost of production. Then $C^{\prime }\left(0\right)>0$ if emissions are reduced by cutting OPEC oil production. However, maintaining production while shifting consumption from OPEC to non-OPEC countries by eliminating subsidies would raise efficiency. It probably would also reduce total oil consumption and production because increased OPEC exports would back out some non-OPEC production. Reduced OPEC consumption also would exceed the rise in non-OPEC consumption from smaller decreased prices. If CO₂ emissions fell, the policy would have $C^{\prime }\left(0\right)<0$ .

In the case of natural gas and coal, their main uses are as inputs into electricity generation, industrial processes (fertilizers, petrochemicals, and metallurgy), and space and water heating. Subsidies and anti-competitive behavior are less significant in these markets. In developed countries, environmental externalities from local toxic pollutants are largely internalized by anti-pollution devices installed due to regulations or as responses to taxes or tradable permit schemes. Coal combustion in many developing countries produces significant environmental externalities from local toxic pollutants. This would make $C^{\prime }\left(0\right)<0$ for CO₂ emission reductions achieved by reducing coal use in those countries.

Summarizing, $C^{\prime }\left(0\right)$ could be positive or negative depending on the fossil fuel and the country where its use is curtailed. The main conclusion, however, is that $C^{\prime }\left(0\right)$ is likely to vary across economies. The equimarginal principle implies that so-called “no regrets” policies, where $C^{\prime }\left(0\right)<0$ , should be implemented first.

3.2.3. Elasticity of Marginal Ccost

The marginal cost of controlling CO₂ emissions will likely rise steeply as emissions decrease. Current technologies for removing CO₂ from flue gases are more costly than comparable technologies for removing conventional pollutants. This may be partly the result of CO₂ being a much larger constituent of the flue gases. ¹⁶ In addition, once the CO₂ has been captured, it currently is costly to sequester in bulk. ¹⁷

Given current technologies, the marginal cost of reducing fossil fuel use will likely rise steeply. The low elasticity of energy demand implies that the foregone marginal benefit of reduced energy use will rise steeply. Instead, fossil fuels could be displaced by alternative energy sources that release less CO₂ per unit of energy supplied. Pigouvian taxes or marketable CO₂ emission permits could achieve reductions at minimum cost, but governments have instead relied on mandates and subsidies for electrifying transportation, industry, and space and water heating and for generating electricity using wind and solar. These policies have substantially increased energy costs while doing little to slow the growth in CO₂ emissions. For example, Hartley (2018) examined the effect of taxes on natural gas in Texas. At moderate tax rates, the elasticity of the electricity cost with respect to the natural gas price was roughly 40 times the elasticity of emission reductions. ¹⁸

Wind and solar raise system-wide electricity costs for several reasons. The average load factor for wind farms is at best around one-third, and for utility-scale solar plants at best around one quarter, compared to over 90% for nuclear plants and up to 80% for coal plants operated to supply base load as intended. ¹⁹ Lower load factors and a shorter lifespan for wind and solar plants raise capital costs per MWh of electricity produced.

Long transmission lines are often needed to get the power from wind and solar plants to market. This is partly because the best resources are often distant from load centers. In addition, the low energy density of wind and solar energy means wind and solar plants require substantial land per MW of capacity and sites with lower land costs are also often distant from load centers. Transmission lines from renewable plants also tend to be used at low load factors, further increasing their costs per MWh supplied.

The wide range of realized wind and solar load factors, from 0% to above 90%, means that substantial backup capacity, often enough to meet the entire load, is needed to ensure system operation. A separate issue is that realized wind (and, to a lesser extent, solar) load factors dramatically fluctuate over brief intervals. As unanticipated blackouts are very costly, more quick-start generators are needed to keep voltage, frequency, and reactive power within fine tolerances. Grids where wind generators are even 20% of total capacity, but more significant locally because transmission capacity constrains electricity trades, have experienced reliability problems. Intermittent generation currently needs backup thermal capacity unless hydroelectricity based on stored water is available. ²⁰ The thermal technologies with the lowest marginal costs are base load plants, but they are unsuitable for quick-start backup. Traditional nuclear plants are designed to operate at full capacity except when refueling, and it can be costly and hazardous, unless specifically allowed for, to quickly cycle them up and down (Lokhov 2011). Cycling baseload coal plants can increase SO _x and NO _x pollution because control equipment is tuned to operate when the plant runs at full capacity (Bentek Energy LLC 2010). Cycling any plants using steam turbines can waste energy as a head of steam is maintained without producing electricity output. As a result, open-cycle gas turbines, which have relatively high operating costs, are most often used to back up intermittent renewables. As backup plants are offloaded when wind and solar plants operate, they are used at reduced load factors, which increases capital and other fixed costs per unit of output. Rapid ramping of plants can also increase stress and raise maintenance costs.

Qvist and Brook (2015) argue that only nuclear power and hydroelectricity have proved capable of deeply cutting CO₂ emissions by displacing fossil fuels. Hartley (2018) examined strategies for displacing fossil fuels in Texas and found that supplying the Texas 2016 load with wind and storage would be about 28% more expensive than using nuclear and storage. The explanation is that electricity storage is very costly, and because wind generation is intermittent and poorly correlated with load, it requires 96% more storage capacity. At a low CO₂ tax rate, wind backed up with natural gas generation was less costly than nuclear with natural gas backup. However, even moderate increases in tax rates made nuclear with natural gas backup, and then nuclear with storage, less costly.

There is also doubt that mineral production could support a substantial displacement of fossil fuels by wind, solar, and batteries. According to a World Bank Report (Arrobas et al. 2017), wind and solar require about 30 times more copper than nuclear per MW of generating capacity, and wind 375 times more rare earths, 3 times more molybdenum, 2.3 times more nickel, and 1.8 times more chromium. Solar PV requires 90 times more tin, 80 times more cadmium, 60 times more lead, and 25 times more indium per MW of capacity than nuclear. The much lower load factors and much shorter life spans for wind and solar generators imply mineral requirements per unit of energy supplied would be far greater.

3.2.4. The Energy Security Externality

As discussed in Hartley (2017), reductions in the energy security of democracies could be another external cost of reducing fossil fuel use. If nations such as the US, Canada, Australia, and Norway reduce fossil fuel output, imports of fossil fuels from less dependable suppliers, like Russia and the Middle East, will rise. Furthermore, China dominates the production of many critical components or inputs for wind and solar plants. According to the International Energy Agency Photovoltaic Power Systems Program, in 2019, China accounted for 68% of global polysilicon production, 96% of global photovoltaic (PV) wafers production, 76% of PV cells production, and 71% of PV module production. The GWEC Global Wind Blade Supply Chain Update for 2020 ranks China as the largest producing country for wind turbines. Chinese firms, either as independents or OEMs for large international firms, control more than 50% of global wind blade production capacity. In a 2021 report, the U.S. International Trade Commission reported that China is now the leading exporter of wind-powered generating sets, accounting for about 10% of the market outside China. Finally, USGS data shows that China dominates production, especially refining, of many minerals critical to manufacturing wind turbines, solar PV, and batteries.

3.3. Alternatives to Emission Controls

Since CO₂ is not directly harmful, reducing its accumulation in the atmosphere is not the only option for reducing damages. Actions classified as self-protection by Ehrlich and Becker (1972) can reduce the likelihood of harmful consequences from extreme weather. Other actions, classified as self-insurance by Ehrlich and Becker (1972), can lower the costs of adverse weather events after they have occurred.

Individuals and firms will take some insurance and/or protective actions in response to changing damages. For example, they may reduce heat stress by adopting air conditioning, improving insulation, or altering working conditions. Farmers may maintain agricultural output by increasing irrigation and changing crop types. Property owners may install sprinkler systems or change building materials to reduce damage from wildfires. These are termed “endogenous adaptation” in IAMs. Damages in those models assume that any efficient private self-protection and self-insurance measures are implemented.

Self-protection actions that are public goods require explicit policy to be efficiently deployed. Other actions may require changes in regulations or the elimination of subsidies to achieve an efficient outcome. Examples of such policies include: building dikes to protect vulnerable coasts; building levees, dredging rivers, and improving urban drainage; building dams, reservoirs and aqueducts to help protect against droughts, wildfires, and floods; undertaking controlled burns or building firebreaks on public land; mitigating urban heat island antecedents by changing zoning laws; removing subsidies to living on vulnerable flood plains or coasts or in areas subject to wildfires; planning evacuation procedures ahead of threatening weather events; training and equipping rescue services; improving weather forecasts to give better warnings to take precautions; changing building codes to increase structural integrity; and investing in research to develop crops more resilient to weather extremes. These measures can defend against extreme weather events within the natural range of variation in addition to shocks that result from a change in the distribution of weather due to either CO₂ accumulation or natural forces.

Similarly, some self-insurance measures are public goods. Examples include better disaster relief (including improved cooperation between different jurisdictions), better policing to reduce criminal activities after disasters, improved coordination between emergency medical facilities, and improved civil reconstruction capability. Such measures can also reduce the costs of other disasters that have nothing to do with adverse weather events, such as earthquakes, tsunamis, volcanic eruptions, terrorist attacks, major industrial accidents, and pandemics. The fact that public self-insurance measures are useful in these other contexts raises their benefit/cost ratios.

Self-protection and self-insurance policies have several advantages relative to reducing CO₂ emissions. They do not increase energy costs. Global agreement is not required to implement them, and each country can tailor measures to the risks they face. Any beneficial changes in weather resulting from increased CO₂ can be retained. Finally, self-protection and self-insurance measures do not impinge on energy security.

4. Model

To limit the scope of the analysis, we ignore greenhouse gas emissions other than CO₂ and anthropogenic sources of CO₂ other than fossil fuel combustion. In addition, although intertemporal trade-offs make climate policy an optimal control problem, the model avoids that issue ²¹ by assuming that CO₂ accumulation in the atmosphere is a sufficient statistic for both the external effects of CO₂ and the comprehensive transition from fossil fuels to alternative energy sources. Specifically, the expected path of fossil fuel extraction cost is assumed to track cumulative fossil fuel use. Absent CCS (which, for simplicity, ²² we ignore), cumulative fossil fuel use also will determine the stock of CO₂ that accumulates in the atmosphere. Use $X_t$ to denote the CO₂ level at $t\ge 0$ measured as ratios to 295 ppm (so 420 ppm corresponds to $X_0\approx 1.42$ ).

The solid line in Figure 1 illustrates the expected path of fossil fuel extraction cost in the absence of climate policy. Serendipitous discoveries and new production technologies can temporarily increase the supply and/or reduce the costs of producing fossil fuels. Nevertheless, expanding demand for fossil fuels from large population developing countries (Section 3.2.1) will deplete lower-cost deposits, and raise the real prices of fossil fuel energy. For a given cost of alternative (or “backstop”) energy (initially taken to be exogenous), define $\hat{X}$ as the stock of CO₂ corresponding to the cumulative fossil fuel extraction at which the relative costs of the different energy sources are equal. Let $\hat{T}$ be the expected transition time. Beyond $\hat{T}$ , $X_t$ should stop increasing. For simplicity, we assume niche uses of fossil fuel energy beyond $\hat{T}$ continue to produce sufficient emissions of CO₂ to offset natural sequestration, so $E{X}_t=\hat{X}$ for all $t\ge \hat{T}$ . Continued fossil fuel extraction will, however, continue to increase fossil fuel extraction costs beyond $\hat{T}$ , albeit at a much-diminished rate as illustrated by the solid line in Figure 1. To date, CO₂ accumulation has tracked IPCC scenario RCP4.5 most closely. Taking an average of scenarios RCP4.5 and 6 as determining the transition cost would imply a maximum concentration of 650 ppm of CO₂-equivalent greenhouse gases at $\hat{T}$ , or $\hat{X}\approx 2.32$ .

Figure 1.

Expected path of fossil fuel extraction cost.

Climate policy, such as taxing CO₂ emissions, reduces the CO₂-intensity of GDP by encouraging fuel substitution or increased energy efficiency. ²³ Policy implemented at $t=0$ thus reduces $X_0$ to $x$ (with a random component that reflects uncertainty in natural sequestration, as discussed in Section 2.6). Since population and economic growth determine the growth in energy use, however, the policy leaves the expected growth rate $g$ of $X_t$ unchanged. The shift from the solid to the dashed line in Figure 1 thus illustrates the effect of the policy on extraction costs. As the dashed line shows, although $g$ is unchanged, future levels of $X$ will decline unless a future government reverses current policy. Contrary to an optimal control model, this does not mean that current governments bind the policy decisions of their successors. Rather, alterations to the CO₂-intensity of GDP are permanent unless they are later reversed. As in the IAMs, we measure the expected cost of reducing $X_0$ to x as a fraction of GDP and denote it by $C(X_0-x)$ . From Section 3.2.2, we assume that $C^{\prime }\left(0\right)\gtrless 0,C^{\prime \prime }>0$ .

Based on a regression of the log of Mauna Loa annual average CO₂ from 1959 to 2022 against time, we assume that $\ln X_t$ for $0<t\le \hat{T}$ can be well approximated as a linear function of time (implying a constant ²⁴ mean growth rate $g\approx 0.0045$ )

$$\ln X_t=x+\mathrm{gt}+{\epsilon }_t$$

(1)

where $x\le X_0$ is the expected effect of current policy on CO₂ concentration at $t=0$ . Randomness in actual accumulation is part of ${\epsilon }_t$ , which also reflects randomness in energy demand growth. For simplicity, we assume that ${\epsilon }_t$ is an independently identically distributed random variable with mean zero and variance ${\omega }^2$ .

Using (1) and the observation that $\hat{T}$ will be endogenously determined by the time when $E{X}_t=\hat{X}$ , $\hat{T}$ can be solved in terms of $\hat{X},x$ and $g$ ²⁵ :

$$\hat{T}=\frac{1}{g}\ln \left(\frac{\hat{X}}{x}\right)$$

(2)

The model thus incorporates a version of the “green paradox” ²⁶ prevalent in optimal control models of energy transition whereby reductions in current fossil fuel use lower fossil fuel extraction costs and extend the time they dominate energy supply.

From sections 2.1 and 2.2, the impact of $X_t$ on the GSTA at $t$ is approximated by

$$y_t=\tau \ln X_t+n_t$$

(3)

where $\tau =R/(1-f)$ with $R\ln X$ representing the direct temperature impact of accumulated CO₂ and $f$ the net feedback effect. Assume $\tau$ is random, independently identically distributed with mean $\overline{\tau }$ and variance ${\nu }^2$ . If $X_t=1$ , (3) implies $y_t=n_t$ , so $n_t$ represents the natural temperature expected at $t$ . For simplicity, we assume that $n_t$ is random, independently identically distributed with mean $\overline{n}$ and variance ${\sigma }^2$ . For simplicity, we also assume that $\tau ,\epsilon$ , and $n$ are statistically independent. ²⁷

Following Section 2.4, we assume that the costs of adverse weather events at $t$ depend partly on deviations of GSTA $y_t$ from some ideal value $y_1$ . Departures of $y$ either side of $y_1$ should be considered harmful. For analytical convenience, we use a quadratic loss function, $A{(y_t-y_1)}^2$ , to characterize the ratio to GDP of the net damage arising from temperature change. ²⁸ Losses then depend on the means and variances of the underlying random variables in (3). Specifically, using (1) and (3), the independence assumption, and the fact that the expectation of the square of a random variable equals the mean squared plus the variance, expected losses at $t<\hat{T}$ can be written:

$$A\{K_0+2K_1\ln x+K_2{\left(x\right)}^2+2g[K_1+K_{2}x]t+K_2{g}^2{t}^2\}$$

(4)

where $K_0={\omega }^2({\overline{\tau }}^2+{\nu }^2)+{\sigma }^2+{(\overline{n}-y_1)}^2>0$ , $K_1=\overline{\tau }(\overline{n}-y_1)\gtrless 0$ , and $K_2={\overline{\tau }}^2+{\nu }^2>0$ are constants. Analogously, expected damages for $t\ge \hat{T}$ can be written

$$A[K_0+2K_1\ln \hat{X}+K_2{\left(\hat{X}\right)}^2]$$

(5)

Let $d$ denote public ²⁹ defensive measures aimed at reducing the costs of adverse weather events. As discussed in Section 3.3, many such public defensive measures are available. An increase in $d$ can represent increased deployment of such measures on either the extensive or intensive margins. The coefficient $A$ in (4) and (5) becomes a function $A\left(d\right)$ with $A^{\prime }<0$ and declining marginal effectiveness of $d$ implies that $A^{\prime \prime }>0$ . Denoting the cost (measured as a fraction of GDP) of $d$ by $I\left(d\right),I^{\prime }>0$ , increasing marginal cost of $d$ implies $I^{\prime \prime }>0$ .

Increases in $d$ also defend against adverse weather events and changes in their distributions that occur without departures of $y$ from $y_1$ (Section 2.3). Some measures $d$ could also defend against non-weather disasters. ³⁰ Let the sum of these expected damages (measured annually as a fraction of GDP) be $D\left(d\right)$ , with $D^{\prime }<0,D^{\prime \prime }>0$ .

To convert expected damages (4), (5) and $D$ to present values, define discount factors:

$${\int }_0^{\hat{T}}{e}^{-\mathrm{rt}}\mathrm{dt}=\frac{1}{r}[1-{\left(\frac{x}{\hat{X}}\right)}^{r/g}]\equiv P{V}_0\mathrm{and}{\int }_{\hat{T}}^{\infty }{e}^{-\mathrm{rt}}\mathrm{dt}=\frac{1}{r}-P{V}_0$$

(6)

where we have used (2) to write $P{V}_0$ as a function of $\hat{X}$ and $x$ . Since terms involving $t$ and $t^2$ appear in (4), we also need:

$${\int }_0^{\hat{T}}t{e}^{-\mathrm{rt}}\mathrm{dt}=\frac{1}{\mathrm{rg}}(x-\hat{X})(1-\mathrm{rP}{V}_0)+\frac{1}{r}P{V}_0$$

(7)

$${\int }_0^{\hat{T}}{t}^2{e}^{-\mathrm{rt}}\mathrm{dt}=\frac{x-\hat{X}}{r^2{g}^2}[2g-r(x-\hat{X})](1-\mathrm{rP}{V}_0)+\frac{2}{r^2}P{V}_0$$

(8)

Using (6) to (8) with (4) and (5) the present value of expected damages can be written:

$$\frac{A\left(d\right)}{r}\{K_0-\frac{2g{K}_2\ln \hat{X}}{r}{\left(\frac{x}{\hat{X}}\right)}^{r/g}+2(K_1+\frac{g}{r}{K}_2)(\ln x+\mathrm{gP}{V}_0)+K_2{\left(x\right)}^2\}+\frac{D\left(d\right)}{r}$$

(9)

From Sections 2.5 and 3.2.4, increased fossil fuel use and atmospheric CO₂ also provide direct positive external ³¹ net benefits. The marginal external benefits will likely decline as CO₂ concentration increases. For simplicity, assume the welfare value of these external benefits as a fraction of GDP is proportional to $\ln X_t$ with a constant of proportionality $B$ . Using (1), (2), (6), and (7), the expected present value of these direct external benefits is

$$\frac{B}{r}\{\frac{g}{r}[1-{\left(\frac{x}{\hat{X}}\right)}^{r/g}]+\ln x\}$$

(10)

4.1. Global Model

Optimal climate policy in a global model involves choosing $x$ (a reduction $X_0-x$ ) and $d$ to maximize expected public benefits minus expected costs. Letting $\lambda \ge 0$ be the multiplier on constraint $X_0\ge x$ , and using (9) and (10) we obtain the Lagrangian:

$$\begin{array}{c}\begin{array}{c}\mathcal{L}=-\frac{A\left(d\right)}{r}\{K_0-\frac{2g{K}_2\ln \hat{X}}{r}{\left(\frac{x}{\hat{X}}\right)}^{r/g}+2(K_1+\frac{g}{r}{K}_2)(\ln x+\mathrm{gP}{V}_0)+K_2{\left(x\right)}^2\}\hfill \\ \multicolumn{1}{c}{-\frac{D\left(d\right)}{r}+\frac{B}{r}\{\frac{g}{r}[1-{\left(\frac{x}{\hat{X}}\right)}^{r/g}]+\ln x\}-I\left(d\right)-C(X_0-x)+\lambda (X_0-x)}\end{array}\hfill \end{array}$$

(11)

The first order necessary conditions for a maximum of (11) can be simplified to

$$\begin{array}{c}\begin{array}{c}\frac{B}{\mathrm{rx}}[1-{\left(\frac{x}{\hat{X}}\right)}^{r/g}]-\frac{2A\left(d\right)}{\mathrm{rx}}\{K_2\ln \left(\frac{x}{\hat{X}}\right)+[K_1+K_2(\frac{g}{r}+\hat{X})][1-{\left(\frac{x}{\hat{X}}\right)}^{r/g}]\}\hfill \\ \multicolumn{1}{c}{+C^{\prime }(X_0-x)-\lambda =0,\mathrm{with}\lambda \ge 0,X_0\ge x,\lambda (X_0-x)=0}\end{array}\hfill \end{array}$$

(12)

$$\begin{array}{c}\begin{array}{c}-\frac{A^{\prime }\left(d\right)}{r}\{K_0-\frac{2g{K}_2\ln \hat{X}}{r}{\left(\frac{x}{\hat{X}}\right)}^{r/g}+2(K_1+\frac{g}{r}{K}_2)(\ln x+\mathrm{gP}{V}_0)+K_2{\left(x\right)}^2\}-\frac{D^{\prime }\left(d\right)}{r}=I^{\prime }\left(d\right)\hfill \end{array}\hfill \end{array}$$

(13)

When $x<X_0$ is optimal, the inequalities in (12) imply $\lambda =0$ , so $x$ solves

$$\begin{array}{c}\frac{2A\left(d\right)}{\mathrm{rx}}\{K_2\ln \left(\frac{x}{\hat{X}}\right)+[K_1+K_2(\frac{g}{r}+\ln \hat{X})][1-{\left(\frac{x}{\hat{X}}\right)}^{r/g}]\}-\frac{B}{\mathrm{rx}}[1-{\left(\frac{x}{\hat{X}}\right)}^{r/g}]=C^{\prime }(X_0-x)\hfill \end{array}$$

(14)

where $d$ simultaneously solves (13). When $x=X_0$ , (12) involves $C^{\prime }\left(0\right)$ , which, as was noted in Section 3.2.2, will vary by location. We assume $C^{\prime }\left(0\right)=0$ until Section 4.2 where costs can vary by location. Then from (12) it would be optimal to set $x=X_0$ when

$$B\ge 2A\left(d\left(X_0\right)\right)\{K_1+K_2(\frac{g}{r}+\ln \hat{X})+K_2\left(\frac{X_0}{\hat{X}}\right){[1-{\left(\frac{X_0}{\hat{X}}\right)}^{r/g}]}^{-1}\}$$

(15)

for $d\left(X_0\right)$ solving (13) with $x=X_0$ .

There is no need to allow for a $d\ge 0$ constraint if it is desirable to undertake public defensive measures against natural severe weather and other disasters, that is when $x=1=\hat{X}$ . Then $\ln \left(x\right)=\left(\hat{X}\right)=P{V}_0=0$ and $-A^{\prime }\left(d\left(1\right)\right)K_0-D^{\prime }\left(d\left(1\right)\right)=r{I}^{\prime }\left(d\left(1\right)\right)$ must have a solution for $d\left(1\right)>0$ . The assumptions $A^{\prime },D^{\prime }<0$ and $A^{\prime \prime },D^{\prime \prime },I^{\prime \prime }>0$ then imply (13) will yield $d\left(X_0\right)\ge d\left(x\right)>d\left(1\right)>0$ . From (15), large external direct benefits $B$ and high $d\left(X_0\right)$ , and thus low $A\left(d\left(X_0\right)\right)$ , could make $x=X_0$ optimal.

From (13), high $d\left(X_0\right)$ is encouraged by the ability to reduce not only $A$ but also $D$ . A large $K_0$ also incentivizes public defensive measures without directly affecting $x$ . Thus, higher $d$ is encouraged by more variable CO₂ accumulation rates ( ${\omega }^2$ ), more variable feedback from CO₂ accumulation to temperature ( ${\nu }^2$ ), and more variable natural temperatures ( ${\sigma }^2$ ). This reflects the fact that $d$ is a form of insurance. Equation (14) shows that reducing $x$ below $X_0$ also is a form of insurance, but only against uncertainty in temperature feedback (via $K_2$ ). Similarly, a higher deviation of expected natural temperature above or below $y_1$ incentivizes higher $d$ by increasing $K_0$ . By contrast, (14) implies that only high $\overline{n}$ relative to $y_1$ encourages lower $x$ (via $K_1$ ).

The bound on the right of (15) increases in $X_0$ . Hence, even if $B$ is large enough to make $x=X_0$ optimal, a future government solving the same optimization problem (11) but with $X_t>X_0$ could find it optimal to choose $x_t<X_t$ . However, since government views on the ideal temperature $y_1$ , the energy security benefits of fossil fuel use, and other aspects of the objective function are likely to change, a current government cannot reliably expect that a future government would solve the same optimization problem.

4.1.1. A Numerical Illustration of Solutions to (13) and (14)

Regressing the logarithm of the Manua Loa average annual CO₂ level against time we find $g\approx .0045$ , with a mean squared error of the regression of only .00007. The larger component of ${\epsilon }_t$ is the randomness in accumulation attendant upon a change in emissions. To calibrate that component of ${\omega }^2$ , changes in the annual average CO₂ levels from Mauna Loa were regressed on global CO₂ emissions from fossil fuel use obtained from the EI Statistical Review of World Energy (converted from millions of tons of CO₂ to ppm concentration). The estimated coefficient of 0.525 accords with the observation in Section 2.6 that the CarbonTracker model finds that about half of emissions are absorbed by natural sinks. The mean squared error of that regression (added to .00007) yields an approximate value for ${\omega }^2\approx .209$ .

To calibrate $\tau$ , we estimated equation (3) by regressing UAH MSU/AMSU global satellite lower troposphere temperature data on the natural logarithm of the Mauna Loa CO₂ monthly series (both seasonally adjusted). We took the resulting coefficient estimate of approximately ³² 2.7 as the mean $\overline{\tau }$ and the squared standard error of the coefficient as ${\nu }^2\approx 0.016$ . We also take the mean squared error of this regression as the estimate of the variability ${\sigma }^2\approx .32$ of temperatures apart from changes in CO₂.

Assume initially that $K_1=0$ , that is, the ideal temperature $y_1$ equals the mean natural temperature $\overline{n}$ . We later discuss the effect of changing $y_1$ . For the above parameter values, $K_0\approx 1.555$ and $K_2\approx 7.287$ .

Equations (13) and (14) also involve the discount rate $r$ . To obtain an efficient allocation of savings across investments, the real discount rate should match that for other investments of equivalent risk. ³³ Most proposals for transitioning off fossil fuels involve increasing the share of electricity in final energy consumption while avoiding fossil fuels in electricity production. A real weighted average cost of capital used in power market investments $r=.05$ , may thus be a suitable discount rate, but we examine the effect of reducing $r$ to 0.02 at the end of this section (Table 1).

Table 1.

Variables, Parameters, and Functions in the Numerical Example.

Variables	Description
$X_t$	Atmospheric CO2 level at $t$ relative to 295 ppm
$\hat{T}$	Time of bulk displacement of fossil fuels
$x$	Chosen current level of $X$ , carried forward at growth rate $g$
$y_t$	Expected temperature anomaly at $t$ resulting from the choice of $x$
$d$	Chosen public defensive measures against adverse weather events
Parameters	Description, base value in the numerical example
$g$	Growth rate of CO2, 0.45%
$X_0$	Initial value of $X$ , 1.42
$\hat{X}$	Transition value of $X$ , 2.32
$\overline{\tau }$	Mean of multiplier τ of $\ln X_t$ in affecting $y_t$ , 2.7
${\nu }^2$	Variance of multiplier τ of $\ln X_t$ in affecting $y_t$ , 0.016
${\sigma }^2$	Variance of temperatures independent of CO2, 0.032
${\omega }^2$	Variance in CO2 accumulation from emissions, 0.209
$\overline{n}$	Mean natural temperature, 0.0
$y_1$	Ideal temperature, 0.0
$r$	Discount rate, 5%
$P{V}_0$	Present value of a constant annuity from $t=0$ to $t=\hat{T}$ , 19.915
$K_0$	${\omega }^2({\overline{\tau }}^2+{\nu }^2)+{\sigma }^2+{(\overline{n}-y_1)}^2$ , 1.555
$K_1$	$\overline{\tau }(\overline{n}-y_1)$ , 0.0
$K_2$	${\overline{\tau }}^2+{\nu }^2$ , 7.287
B	Multiplier of $\ln \left(X_t\right)$ for direct external benefits (% GDP) of CO2, 0.0004
${\alpha }_A$	Constant in $A\left(d\right),D\left(d\right)$ and $I\left(d\right)$ functions, 0.0004
${\alpha }_C$	Constant in marginal cost of emission reduction function, 0.7
Functions	Description, form used in the numerical example
$A\left(d\right)$	Coefficient linking ${(y_t-y_1)}^2$ to damages as % of GDP, ${\alpha }_A{(1-d)}^{1.1}$
$D\left(d\right)$	Natural weather and other damages reduced by $d$ , $2{\alpha }_A{(1-d)}^2$
$I\left(d\right)$	Cost of public defensive measures $d$ , $10{\alpha }_A/{(1-d)}^2$
$C^{\prime }(X_0-x)$	Marginal cost of CO2 emission reductions, ${\alpha }_C(X_0-x)/(x-0.98X_0)$

The functions $A\left(d\right),D\left(d\right),I\left(d\right),\mathrm{and}C\left(\Delta x\right)$ need to have $A^{\prime },D^{\prime }<0$ and $I^{\prime },I^{\prime \prime },A^{\prime \prime },D^{\prime \prime },C^{\prime \prime }>0$ . The ADDICE model (de Bruin et al. 2009) assumed adaptation reduced the damage from higher temperatures linearly. To approximate this while fulfilling the restrictions on the signs of the derivatives of $A$ , set $A\left(d\right)={\alpha }_A{(1-d)}^{1.1}$ . For ${\alpha }_A=.0004$ , the present value of expected damages from CO₂ will be around 1% to 4% of output in the relevant ranges of x and $d$ (as in the graphs in de Bruin et al. 2009). To begin with, we set the coefficient $B$ equal to ${\alpha }_A$ . We discuss the effect of changing $B$ following Figure 2 below.

Figure 2.

Equations (14) for $x$ and (13) for $d$ in the numerical example.

Since $D$ includes damages from the far larger number of extreme weather events not associated with GSTA plus damages from non-weather disasters, we expect $D\left(0\right)>A\left(0\right)$ . Also, since public defensive measures can be tailored to the risks that dominate each locale, we might expect them to be more effective at reducing $D$ than $A$ . For this illustration we chose $D\left(d\right)=2{\alpha }_A{(1-d)}^2$ .

We assume that the cost $I\left(d\right)$ of $d$ becomes unbounded as $d\to 1$ , implying that public defensive measures cannot eliminate damages. Since we also want $I^{\prime },I^{\prime \prime }>0$ , we chose $I\left(d\right)={\alpha }_{I}d/(1-d)$ (with $I^{\prime }\left(d\right)={\alpha }_I/{(1-d)}^2$ ) and we will take ${\alpha }_I=10{\alpha }_A$ . When $d=0.5$ , for example, this would imply that $I\left(d\right)$ is about 20 times both $A\left(d\right)$ and $D\left(d\right)$ .

IAMs typically define the cost of reducing CO₂ as a function of emission reductions, whereas we need them as a function of reduced concentration of CO₂ in the atmosphere. Using the estimated coefficient 0.525 relating emissions to concentration, average emissions over the last Fifteen years in the EI data of 4.13 ppm would change expected CO₂ concentration measured as a ratio to 295 ppm by around 0.0074. We conclude, consistent with the arguments in Section 3.2.3 and the small effects of events such as the COVID lockdowns in 2020 and the 2008 financial crisis, that reducing $x$ from 1.42 to even $0.98\times 1.42\approx 1.39$ would be very costly. We assume, therefore, that $C^{\prime }(X_0-x)={\alpha }_C(X_0-x)/(x-0.98X_0)\to \infty$ as $x\to 0.98X_0$ . Setting ${\alpha }_C=.7$ , and integrating $C^{\prime }$ , we find the total cost of reducing $X_0$ to 1.40 would be close to 0.01 (or 1% of GDP). According to World Bank data, 1% of global GDP in 2022 would be about $US 9 $\times {10}^{11}$ . Using the estimated coefficient 0.525 relating emissions to concentration, a reduction $X_0-x=0.02$ would require reduced emissions of 11.24 ppm, which corresponds to about 8.8 $\times {10}^{10}$ tons of CO₂. The implied average cost of the reduction $X_0-x=0.02$ is thus about $10 per ton of CO₂, or $38 per ton of carbon embedded in the CO₂.

Figure 2 graphs equations (13) and (14) for the above functional forms and parameter values. The graph on the left illustrates the solution to (14) evaluated at the solution for $d\approx 0.660645$ from the graph on the right, while the graph on the right similarly embodies the solution for $x\approx 1.419786$ . The implied marginal cost of control at A is slightly above 0.53% of GDP. From the calculations above, that is equivalent to a tax ³⁴ of about $20 per ton of carbon embedded in the CO₂.

The additional elements in this model are the direct net benefits of CO₂ represented by (10), the ability to reduce expected damages by spending on $d$ , and the effect of $d$ on damages not associated with GSTA. Eliminating these by setting $B=0$ and constraining $d=0$ , (14) yields $x\approx 1.418618$ . The resulting $X_0-x$ is almost 6.5 times larger than in the full model. The marginal cost of reducing $x$ is almost 3.6% of GDP, which is 6.75 times the marginal cost in the full model. This is equivalent to a tax on CO₂ emissions that is about double the tax rate for 2025 calculated by IWG (2021) for a discount rate of 5%.

Point B in the left graph of Figure 2 illustrates the solution for $x$ for $B=0$ while retaining d = 0.660645. The difference between points A and B shows that recognizing direct external benefits from CO₂ accumulation while keeping public defensive measures fixed approximately halves the desired reduction in $x$ . From condition (15), increasing $B$ from 0.0004 to slightly below 0.00078, or increasing the ideal temperature $y_1$ by ³⁵ $0.{64}^{°}$ C while keeping $B=0.0004$ yields the corner solution $x=X_0$ .

Reducing $r$ increases the present values of public defensive measures or the benefits from lowering future temperatures $y$ by reducing CO₂. Ceteris paribus, this should make higher $d$ and lower $x$ more desirable. However, a reduction in $r$ also increases the present value of the direct external benefits of CO₂, given by (10), which makes higher $x$ more desirable. Table 2 gives the solutions for $d$ and $x$ , the resulting values for the two components of (9), and the value of (10) for a range of interest rates. The non-monotonic and small changes in $x$ reflect the two offsetting direct effects, the reduced marginal benefit of reducing $y$ by decreasing $x$ when $d$ is larger, and the steepness of the marginal cost of control curve. Since the optimal $x$ is insensitive to $r$ , so also is the implied emissions tax rate.

Table 2.

Solutions for a Range of Discount Rates.

$r$	$d$	$x$	Implied present values as % GDP
			$\mathrm{\Delta }y$ cost	$D\left(d\right)/r$	$\mathrm{\Delta }x$ benefit
0.05	0.660645	1.419786	0.7365	0.1843	0.3521
0.04	0.696053	1.419777	0.8605	0.1848	0.4616
0.03	0.738220	1.419780	1.0568	0.1827	0.6598
0.02	0.790318	1.419828	1.4144	0.1759	1.1003

4.2. Allowing for Geographic Variations

Section 2.4 noted that higher temperatures could be less harmful or even beneficial in some locations. Introduce an index $\ell =1,2,\dots ,L$ to represent location and use $y_{1\ell }$ to denote the GSTA people at $\ell$ deem as most desirable. The natural random temperature anomaly at $\ell$ in year $t$ , $n_{\ell t}$ with mean ${\overline{n}}_{\ell }$ and variance ${\sigma }_{\ell }^2$ , will also vary by location. Then, since $K_0$ depends on ${\sigma }_{\ell }^2$ and $K_0$ and $K_1$ depend on ${\overline{n}}_{\ell }-y_{1\ell }$ , these also will become functions, $K_{0\ell }$ and $K_{1\ell }$ , of $\ell$ . Let public defensive measures at $\ell$ be $d_{\ell }$ . The functions $A_{\ell }$ , $D_{\ell }$ , and costs $I_{\ell }$ will also depend on location. Heterogeneous fertilizer benefits of CO₂, and different energy security benefits of fossil fuels by location will make $B_{\ell }$ differ by location. From Section 3.2.2, costs of reducing fossil fuel use will also vary across locations, with $C_{\ell }^{\prime }\left(0\right)\ge 0$ for some locations and $C_{\ell }^{\prime }\left(0\right)\le 0$ for others.

4.2.1. Cooperative Solution

Under international cooperation, policies maximize a weighted sum of expected net benefits with $w_{\ell }$ , $\sum _{\ell }{w}_{\ell }=1$ , the weight of location $\ell$ . Maximizing efficiency corresponds to ³⁶ $w_{\ell }=1/L$ but weights could instead reflect equity concerns or the outcome of a bargaining process. If reduced emissions at $\ell$ reduce expected aggregate CO₂ at $t=0$ by $\mathrm{\Delta }{x}_{\ell }\ge 0$ then

$$x=X_0-\sum _{\ell =1}^L\mathrm{\Delta }{x}_{\ell }$$

(16)

For ${\lambda }_{\ell }$ the multiplier on constraint $\mathrm{\Delta }{x}_{\ell }\ge 0$ and $\mu$ the multiplier on (16), the objective is:

$$\begin{array}{c}\begin{array}{c}\underset{x,\{\Delta x_{\ell },d_{\ell }{\}}_{\ell =1}^L}{\max }\sum _{\ell =1}^L{w}_{\ell }\{\frac{B_{\ell }}{r}[\frac{g}{r}[1-{\left(\frac{x}{\hat{X}}\right)}^{r/g}]+x]-\frac{D_{\ell }\left(d_{\ell }\right)}{r}-C_{\ell }\left(\mathrm{\Delta }{x}_{\ell }\right)-I_{\ell }\left(d_{\ell }\right)\hfill \\ \multicolumn{1}{c}{-\frac{A_{\ell }\left(d_{\ell }\right)}{r}[K_{0\ell }-\frac{2g{K}_2\ln \hat{X}}{r}{\left(\frac{x}{\hat{X}}\right)}^{r/g}+2(K_{1\ell }+\frac{g}{r}{K}_2)(\ln x+\mathrm{gP}{V}_0)+K_2{\left(x\right)}^2]\}}\\ \multicolumn{1}{c}{+\sum _{\ell =1}^L{\lambda }_{\ell }\mathrm{\Delta }{x}_{\ell }+\mu [\sum _{\ell =1}^L\mathrm{\Delta }{x}_{\ell }+x-X_0]}\end{array}\hfill \end{array}$$

(17)

First order necessary conditions for the choice of public defensive measures $d_{\ell }$ are:

$$\begin{array}{c}\begin{array}{c}-\frac{A_{\ell }^{\prime }\left(d_{\ell }\right)}{r}\{K_{0\ell }-\frac{2g{K}_2\ln \hat{X}}{r}{\left(\frac{x}{\hat{X}}\right)}^{r/g}+2(K_{1\ell }+\frac{g}{r}{K}_2)(\ln x+\mathrm{gP}{V}_0)+K_2{\left(x\right)}^2\}\hfill \\ \multicolumn{1}{c}{-\frac{D_{\ell }^{\prime }\left(d_{\ell }\right)}{r}=I_{\ell }^{\prime }\left(d_{\ell }\right),\ell =1,2,\dots ,L}\end{array}\hfill \end{array}$$

(18)

In the interior case, the first order necessary conditions for $x$ and each $\Delta x_{\ell }>0={\lambda }_{\ell }$ are:

$$\begin{array}{c}\begin{array}{c}\frac{1}{\mathrm{rx}}[1-{\left(\frac{x}{\hat{X}}\right)}^{r/g}]\sum _{\ell =1}^L{w}_{\ell }{B}_{\ell }-\frac{2}{\mathrm{rx}}[1-{\left(\frac{x}{\hat{X}}\right)}^{r/g}]\sum _{\ell =1}^L{w}_{\ell }{A}_{\ell }\left(d_{\ell }\right)K_{1\ell }\hfill \\ \multicolumn{1}{c}{-\frac{2K_2}{\mathrm{rx}}\{\ln \left(\frac{x}{\hat{X}}\right)+[\frac{g}{r}+\hat{X}][1-{\left(\frac{x}{\hat{X}}\right)}^{r/g}]\}\sum _{\ell =1}^L{w}_{\ell }{A}_{\ell }\left(d_{\ell }\right)+\mu =0}\end{array}\hfill \end{array}$$

(19)

$$\mu =w_{\ell }{C}_{\ell }^{\prime }\left(\mathrm{\Delta }{x}_l\right),\ell =1,2,\dots ,L$$

(20)

Combining (19) and (20) and using (18) gives $2L$ equations that can be solved for $\Delta x_{\ell }$ and $d_{\ell }$ as functions of $x$ . Equation (16) can then be solved for $x$ .

For $w_{\ell }=1/L$ , (20) implies the equimarginal principle and (19) and (20) yield the efficiency condition for a public good (the marginal cost of control equals the sum of the marginal benefits of reduced net damages). With unequal weights, (20) implies that the marginal cost of emission reduction should be lower – and $\mathrm{\Delta }{x}_{\ell }$ smaller – where $w_{\ell }>1/L$ .

If (19) and (20) can be solved for $\mathrm{\Delta }{x}_{\ell }>0$ at some locations, but at others

$$\begin{array}{c}\begin{array}{c}{w}_{\ell }{C}_{\ell }^{\prime }\left(0\right)\ge -\frac{1}{\mathrm{rx}}[1-{\left(\frac{x}{\hat{X}}\right)}^{r/g}]\sum _{\ell =1}^L{w}_{\ell }{B}_{\ell }+\frac{2}{\mathrm{rx}}[1-{\left(\frac{x}{\hat{X}}\right)}^{r/g}]\sum _{\ell =1}^L{w}_{\ell }{A}_{\ell }\left(d_{\ell }\right)K_{1\ell }\hfill \\ \multicolumn{1}{c}{+\frac{2K_2}{\mathrm{rx}}\{\ln \left(\frac{x}{\hat{X}}\right)+[\frac{g}{r}+\hat{X}][1-{\left(\frac{x}{\hat{X}}\right)}^{r/g}]\}\sum _{\ell =1}^L{w}_{\ell }{A}_{\ell }\left(d_{\ell }\right)}\end{array}\hfill \end{array}$$

(21)

locations in the latter group will all have $\mathrm{\Delta }{x}_{\ell }=0$ . Thus, high emission reduction costs and a large $w_{\ell }$ would tend to make $\mathrm{\Delta }{x}_{\ell }=0$ optimal. Conversely, locations with $C^{\prime }\left(0\right)<0$ are more likely to require $\mathrm{\Delta }{x}_{\ell }>0$ . In other words, “no regrets” policies delivering net benefits apart from lower CO₂ emissions should be the highest priority.

The right side of (21) is identical across locations. If it is negative because, for example, at many locations $B_{\ell }$ is high or ${\overline{n}}_{\ell }<y_{1\ell }$ making $K_{1\ell }<0$ , then $\Delta x_{\ell }=0$ wherever $C^{\prime }\left(0\right)>0$ . Smaller losses $A_{\ell }\left(d_{\ell }\right)$ at many locations would also favor $\mathrm{\Delta }{x}_{\ell }=0$ at all locations where $C^{\prime }\left(0\right)>0$ . In turn, high annual variability of natural temperature anomalies, ${\sigma }_{\ell }^2$ , and a large difference $y_{1\ell }-{\overline{n}}_{\ell }$ would increase $K_{0\ell }$ , favoring higher $d_{\ell }$ and smaller losses $A_{\ell }\left(d_{\ell }\right)$ . For a given $d_{\ell }$ , $A_{\ell }\left(d_{\ell }\right)$ will also be lower where public defensive measures are more effective.

4.2.2. Non-cooperative Outcome

Now assume a Nash equilibrium where each government chooses its policies taking policies chosen elsewhere as given. Denoting the expected aggregate CO₂ level $x$ from (16) by the function $Z(X_0,\mathrm{\Delta }{x}_1,\mathrm{\Delta }{x}_2,\dots ,\mathrm{\Delta }{x}_L)$ , policy in location $s$ would solve:

$$\begin{array}{c}\begin{array}{c}\underset{\mathrm{\Delta }{x}_s,d_s}{\max }\frac{B_s}{r}\{\frac{g}{r}[1-{\left(\frac{Z}{\hat{X}}\right)}^{r/g}]+Z\}-\frac{D_s\left(d_s\right)}{r}-C_s\left(\mathrm{\Delta }{x}_s\right)-I_s\left(d_s\right)+{\lambda }_s\mathrm{\Delta }{x}_s\hfill \\ \multicolumn{1}{c}{-\frac{A_s\left(d_s\right)}{r}\{K_{0s}-\frac{2g{K}_2\ln \hat{X}}{r}{\left(\frac{Z}{\hat{X}}\right)}^{r/g}+2(K_{1s}+\frac{g}{r}{K}_2)(\ln Z+\mathrm{gP}{V}_0)+K_2{\left(Z\right)}^2\}}\end{array}\hfill \end{array}$$

(22)

The first order necessary condition for an interior choice of $\Delta x_s>0$ is:

$$\begin{array}{c}\begin{array}{c}{C}_s^{\prime }\left(\mathrm{\Delta }{x}_s\right)=-\frac{B_s}{\mathrm{rZ}}[1-{\left(\frac{Z}{\hat{X}}\right)}^{r/g}]+\frac{2}{\mathrm{rZ}}[1-{\left(\frac{Z}{\hat{X}}\right)}^{r/g}]A_s\left(d_s\right)K_{1s}\hfill \\ \multicolumn{1}{c}{+\frac{2K_2}{\mathrm{rZ}}\{\ln \left(\frac{Z}{\hat{X}}\right)+[\frac{g}{r}+\hat{X}][1-{\left(\frac{Z}{\hat{X}}\right)}^{r/g}]\}{A}_s\left(d_s\right)}\end{array}\hfill \end{array}$$

(23)

while the first order condition for $d_s$ is equation (18) with $\ell =s$ and $x=Z$ . The fact that the equations for $d_s$ are otherwise identical, and the absence of $w_{\ell }$ from (18), reflects the observation in Section 3.3 that taking public defensive actions does not require a global agreement. Equations (23), together with the re-written (18) will give $2L$ equations to be solved simultaneously for $\{\Delta x_s,d_s{\}}_{s=1}^L$ .

Comparing (23) to (19) and (20) clarifies the collective action problem for climate policy. Where (23) has $B_s,A_s\left(d_s\right)K_{1s}$ , and $A_s\left(d_s\right)$ for just location $s$ , (19) has weighted averages $\sum _{\ell =1}^L{w}_{\ell }{B}_{\ell },\sum _{\ell =1}^L{w}_{\ell }{A}_{\ell }\left(d_{\ell }\right)K_{1\ell }$ , and $\sum _{\ell =1}^L{w}_{\ell }{A}_{\ell }\left(d_{\ell }\right)$ across all locations. Furthermore, (23) includes the full marginal cost at $s$ while it is multiplied by $w_s<1$ in (20). Since emission reductions are viewed as less effective, the marginal cost curve appears steeper in the non-cooperative case.

4.3. Technology Policy

Hastening the transition by subsidizing R&D into alternative energy technology could also reduce CO₂ accumulation. To succeed, the policy must anticipate what alternative energy technology can displace fossil fuels en masse. A successful policy would lower energy costs starting from an earlier transition date, encouraging developing countries expected to dominate impending energy demand growth to adopt it voluntarily. Although the policy could be implemented without international cooperation, learning externalities could preclude an efficient outcome if such cooperation is absent.

To investigate the policy, return to the simple global model but now assume that a subsidy with cost $S(\hat{X}-\hat{x})$ (as a fraction of GDP) can reduce $\hat{X}$ to $\hat{x}$ , with $S^{\prime }\left(0\right)=0$ . Now $X_0$ is exogenous and $d$ and $\hat{x}$ are chosen to maximize:

$$\begin{array}{c}\begin{array}{c}\mathcal{L}=\frac{B}{r}\{\frac{g}{r}[1-{\left(\frac{X_0}{\hat{x}}\right)}^{r/g}]+\ln X_0\}-\frac{D\left(d\right)}{r}-I\left(d\right)-S(\hat{X}-\hat{x})\hfill \\ \multicolumn{1}{c}{-\frac{A\left(d\right)}{r}\{K_0-\frac{2g{K}_2\ln \hat{x}}{r}{\left(\frac{X_0}{\hat{x}}\right)}^{r/g}+2(K_1+\frac{g}{r}{K}_2)(\ln X_0+\frac{g}{r}[1-{\left(\frac{X_0}{\hat{x}}\right)}^{r/g}])}\\ \multicolumn{1}{c}{+K_2{\left(\ln X_0\right)}^2\}+\lambda (\hat{X}-\hat{x})}\end{array}\hfill \end{array}$$

(24)

The first order necessary conditions for $\hat{x}$ and $d$ can be written as:

$$\lambda =\frac{1}{r\hat{x}}{\left(\frac{X_0}{\hat{x}}\right)}^{r/g}\{B-2A\left(d\right)[K_1+K_2\ln \hat{x}]\}+S^{\prime }(\hat{X}-\hat{x})\ge 0,\mathrm{with}\hat{X}\ge \hat{x},\lambda (\hat{X}-\hat{x})=0$$

(25)

$$\begin{array}{c}\begin{array}{c}-\frac{A^{\prime }\left(d\right)}{r}\{K_0-\frac{2g{K}_2\ln \hat{x}}{r}{\left(\frac{X_0}{\hat{x}}\right)}^{r/g}+2(K_1+\frac{g}{r}{K}_2)(\ln X_0+\frac{g}{r}[1-{\left(\frac{X_0}{\hat{x}}\right)}^{r/g}])\hfill \\ \multicolumn{1}{c}{+K_2{\left(\ln X_0\right)}^2\}-\frac{D^{\prime }\left(d\right)}{r}=I^{\prime }\left(d\right)}\end{array}\hfill \end{array}$$

(26)

From (25), and noting that (26) becomes (13) with $x=X_0$ when $\hat{x}=\hat{X}$ , choosing $\hat{x}<\hat{X}$ will not be maximizing if:

$$B>2A\left(d\left(X_0\right)\right)[K_1+K_2\ln \hat{X}]$$

(27)

The additional term multiplying $2A\left(d\left(X_0\right)\right)$ in the corresponding bound (15) on $B$ for not choosing $x<X_0$ can be written

$$K_2\{\frac{g}{r}+\ln \left(\frac{X_0}{\hat{X}}\right){[1-{\left(\frac{X_0}{\hat{X}}\right)}^{r/g}]}^{-1}\}$$

(28)

which is negative for $X_0<\hat{X}$ , but can be shown (using L’Hôpital’s Rule) to converge to zero as $X_0\uparrow \hat{X}$ . Thus, for $X_0<\hat{X}$ , there will be a range of values of $B$ for which it will be maximizing to reduce $\hat{x}$ below $\hat{X}$ while it is not maximizing to reduce $x$ below $X_0$ . When $r>g$ , discounting offsets growth in CO₂, making immediate reductions less important, and when $\hat{X}>>X_0$ , it is more valuable to reduce permanent peak CO₂ than interim amounts.

In the interior case for the choice of $\hat{x}<\hat{X}$ , $\lambda =0$ and (25) implies $\hat{x}$ solves:

$$r\hat{x}{\left(\frac{\hat{x}}{X_0}\right)}^{r/g}{S}^{\prime }(\hat{X}-\hat{x})=2A\left(d\right)[K_1+K_2\mathrm{In}\hat{x}]-B$$

(29)

for $d$ simultaneously solving (26). External benefits $B$ , or $K_1<0$ , are opportunity costs of reducing $\hat{x}$ . The multiplier of $S^{\prime }$ on the left of (29) will be large, implying the deviation of $\hat{x}$ from $\hat{X}$ will be small, when $\hat{x}>>X_0$ and $r>>g$ .

5. Comparison to IAMs

Weyant (2017) notes that the main benefit-cost IAMs used to calculate optimal climate policies are the Dynamic Integrated Climate-Economy (DICE) model (Nordhaus 2014), the Framework for Uncertainty, Negotiation, and Distribution (FUND) model (Anthoff and Tol 2013), and the Policy Analysis of the Greenhouse Effect (PAGE) model (Hope 2011). He observes that the FUND model delivers the lowest net cost of CO₂ emissions in part because it assumes a greater ability of those affected by climate change to adapt.

de Bruin et al. (2009) claim that most IAMs consider only induced (adjustment to the new climate represented in the model by transition costs and lags) and implicit (incorporated in the functional relationships) adaptation to climate change. Since these adaptations are of a “private nature,” they are assumed to be optimal and already incorporated into the damage function. de Bruin et al. (2009) modified the DICE model damage function to distinguish gross from net damages. The difference is the benefits of ecological, social, and economic changes in response to climate change minus the costs incurred. They find adaptation to be more important for reducing the costs of climate change in earlier periods, but mitigation becomes more important later.

In a more recent paper, Estrada et al. (2019) modified the damage function to make it a function not of the difference between GSTA and an unchanging base value but rather the difference between GSTA and an adjusting value $a_t$ to which the system is adapted at $t$ , scaled by a “coping range” ${\sigma }_t$ of the system that also adjusts over time. They further split $a_t$ into the sum of planned ${\varphi }_1$ and autonomous (spontaneous and reactive) adaptation ${\varphi }_2$ . The costs of autonomous adaptation are private and an increasing function of GSTA, while it also becomes less effective as GSTA increases. Planned adaptation ${\varphi }_1$ is a policy variable that has a cost that is a power function of ${\varphi }_1$ .

The distinction between autonomous and planned adaptation is partly related to the one we make between self-insurance measures (that lower costs ex-post) and self-protection measures (that reduce expected costs ex-ante). However, assuming planned adaptation is a policy variable also makes it akin to our category of public adaptation measures. As in most benefit-cost IAMs, we assume that private adaptive measures are implemented optimally and their effects are already netted out in the damages function. Defensive measures $d$ in our model are public measures alone.

Similarly, the explicit direct benefits (10) of CO₂ recognized in our model are only the external benefits. Private benefits are already netted out in the damages function. Allowing external benefits from CO₂ accumulation is another difference between our model and current IAMs. Since these are external benefits, they are likely to be captured efficiently only if incorporated into official policy actions.

In our model, adaptation moderates damages from changes in GSTA by reducing the coefficient $A\left(d\right)$ . This is analogous to how adaptation reduces the damage from GSTA departures from a base value in the IAMs. However, $A\left(d\right)$ in our model multiplies the expected damages. As a result, public defensive measures also protect against uncertainties in the effect of emissions on accumulation ( ${\omega }^2$ ) and natural temperature changes ( ${\sigma }^2$ ). These are reflected in the term $K_0$ , which affects the choice of $d$ but does not directly affect the choice of $x$ . In addition, uncertainty in the effect of ${\mathrm{CO}}_2$ accumulation on temperature ( ${\nu }^2$ ) directly affects the choice of $d$ and $x$ via the terms $K_0$ and $K_2$ .

In contrast to existing IAMs, we also recognize (via the term $D$ ) that public adaptation measures can reduce damages apart from those caused by changes in GSTA. Historical distributions of weather events at most locations provide ample opportunity to experience extreme outcomes. In addition, many public self-insurance measures are likely to reduce the costs of other disasters that have nothing to do with adverse weather events. Current IAMs ignore all these benefits of public defensive measures.

The decentralized model in Section 4.2 highlights another critical difference between our model and current IAMs. Each jurisdiction independently chooses public defensive measures to address local conditions. By contrast, emission control is a single global policy likely to be plagued by severe free-rider problems.

Finally, by including endogenous energy transition, the model can analyze policies to lower the costs of technologies that can displace fossil fuels en masse. They are shown to have some advantages relative to policies aimed at reducing current CO₂ accumulation.

6. Concluding Comments

Curry (2023) credits Rayner and Prins (2007) with characterizing climate change as a “wicked” problem. Curry elaborates, “Dealing with wicked problems promotes a focus on considering problems from multiple perspectives, designing programs that accommodate complexity and ambiguity, management strategies that account for crises and surprises, improving policy and evaluation capabilities, and strengthening the collaborative capacities of the policy system.” She contrasts this characterization with a narrow framing of climate change and even extreme weather as being caused overwhelmingly by excess CO₂ in the atmosphere, leading to the simple policy prescription of reducing CO₂ emissions.

The model developed in this paper recognizes that climate policy has to allow for multiple uncertainties, sources of extreme weather apart from an enhanced greenhouse effect, direct external benefits of CO₂, public defensive measures, geographical diversity, and significant interactions between climate and energy policies. The links between CO₂ emissions and accumulation, accumulation and temperatures, and temperatures and expected damages are all uncertain. Furthermore, random forces apart from CO₂ accumulation affect temperatures and other climate characteristics. The model shows that while public defensive measures insure against all these sources of uncertainty, emissions control insures against only one source of changes in climates. Some public defensive measures also lessen the costs of a range of non-weather shocks. Direct external benefits of CO₂, or energy security benefits from fossil fuel use, can make emission reductions less attractive. Such benefits do not directly affect the case for public defensive measures. For example, the model showed that a reduction in the discount rate can reduce the optimal level of emissions reduction. Part of the reason is that it increases the present value of direct external benefits of CO₂. While a lower discount rate also increases the present value of damages from higher CO₂ levels, increased public defensive measures can offset these while simultaneously reducing the present value of damages not associated with changes in CO₂.

Another advantage of public defensive measures is that they are local policies that can be implemented without requiring others to agree. They also can be tailored to the risks most relevant to each country. By contrast, since emission reductions are a global public good, choosing them requires explicit negotiations that are plagued by formidable free-riding problems. The model showed how myriad local realities would affect optimal emission control under a global agreement. For example, positive net benefits of CO₂ or a higher desired GSTA at just some locations make emissions reductions at all locations less desirable.

Finally, Section 4.3 examined a policy of hastening the transition by subsidizing alternative energy technology R&D. To succeed, subsidies must anticipate what alternative technology can displace fossil fuels en masse without compulsion. A successful policy would have several advantages relative to reducing current emissions. Lowering the cost of the backstop will give lower energy costs starting from an earlier transition date and continuing thereafter. The new technology would be adopted voluntarily. The model showed that under some mild assumptions about cost and benefit functions, positive R&D subsidies are optimal in more circumstances than current emission reductions.