# Carbon pricing and planetary boundaries

Sep 17, 2020

### Model components

The results of this paper are derived from a model that is built around the economic sectors outlined as the most important drivers of planetary pressures in Supplementary Table 1. This includes production sectors that have an important direct effect on the ESPs or that have important links to such sectors. They may be linked by using output from such sectors as inputs, providing inputs to such sectors, competing for inputs with such sectors or providing outputs that serve as substitutes for the output from those sectors. The resulting set of included production sectors are: agriculture (producing food and biofuel), energy services, fossil-fuel extraction, renewable energy (other than biofuel), fertilizer production, phosphate extraction, water supply, fisheries, and industrial manufacturing. The demand for final consumption goods is derived from the maximization of households’ utility. Since we have economic policies in the model, we are implicitly assuming some government entity that imposes these policies, but since we consider the policies exogenous (not, e.g., determined to optimize some objective) we do not explicitly model the government.

We solve the model as a competitive equilibrium where we assume that all agents maximize their respective objectives while taking prices as given (prices are given from the perspective of the individual agent, but are endogenously determined by aggregate supply and demand). We then analyze changes in the endogenously determined model variables in response to an assumed exogenous change in economic policy.

In the model, competition for resources thus leads to a number of important trade-offs. These arise from three main sources including, alternative uses of the output of a sector (e.g., output from the agricultural sector can be used as food or biofuels), sectors competing for the use of inputs (e.g., land can be used for agriculture, forestry or maintained as undisturbed natural land) or from inputs being substitutes or complements in production or consumption (e.g., nitrogen and phosphorus preferably being used in fixed proportions).

The production sectors are modeled either by using an explicit production function or by a production cost function. A production function is specified for agriculture, energy services, fertilizer production, fisheries, timber production and industrial manufacturing sectors since their factor inputs are directly connected to one or more ESPs (see the previous section on “Economic drivers of planetary pressures”), thus making their input substitutability important. For all sectors except agriculture, we use one level constant elasticity of substitution (CES) functions. For agriculture, we use a nested CES function (see below). Sectors whose production processes are of less importance, are represented by a production cost function. These sectors include phosphate, water, fossil fuel, and renewable energy. Also, in many sectors, certain inputs e.g., labor and capital, are economically important but their explicit modeling is not directly relevant for our analysis (i.e., of negligible importance to the ESPs). To account for these inputs, we include an aggregate input, which we refer to as other inputs, in all production sectors except energy services and assume that these are supplied with a given sector-specific price elasticity of supply. The possibility of adjusting these other inputs leads to decreased use in sectors where their marginal value decreases and increased use in sectors where their marginal value increases, and thus to some extent captures the possibility to move inputs between sectors in response to changing economic conditions.

We will now present the model sectors in more detail. A list of model quantities, their prices and uses can be found in Table 1 (different uses of a quantity are denoted by subscripts).

The agricultural sector uses inputs land (LA), fertilizers (P), water (W), energy services (({{mathcal{E}}}_{A})) and other inputs (MA) as inputs to produce output that can be used for food or biofuels. Producers maximize their profit, taking prices as given. Their profit maximization problem is

$$mathop{max }limits_{{L}_{A},P,W,{{mathcal{E}}}_{A},{M}_{A}}{p}_{A}Aleft({L}_{A},P,W,{{mathcal{E}}}_{A},{M}_{A}right)-{p}_{L}{c}_{A}({L}_{A}){L}_{A}\ -, {p}_{P}P-{p}_{W}W-{p}_{{mathcal{E}}}{{mathcal{E}}}_{A}-{p}_{{M}_{A}}{M}_{A},$$

(1)

where cA(LA) captures the cost of converting land to agricultural land. The agricultural production function is a CES function between land and non-land inputs, where non-land inputs are aggregated using a CES function.

The energy-services sector combines energy from different sources into a bundle of energy services (({mathcal{E}})). The different sources are biofuels (AB), fossil fuels (({E}_{{mathcal{E}}})) and renewables (R). The producers in this sector solve the profit maximization problem

$$mathop{max }limits_{{A}_{B},{E}_{{mathcal{E}}},R}{p}_{{mathcal{E}}}{mathcal{E}}({A}_{B},{E}_{{mathcal{E}}},R)-{p}_{A}{A}_{B}-{p}_{E}{E}_{{mathcal{E}}}-{p}_{R}R.$$

(2)

We model production of fertilizers (P) as using fossil fuel (EP), phosphate (({mathcal{P}})) and other inputs (MP). The use of fossil fuel is intended to capture the fossil-fuel (more specifically natural-gas) intensive production of the nitrogen component of fertilizers. We thus treat fossil fuel use in fertilizer production as a proxy for nitrogen. The profit maximization problem of fertilizer producers is

$$mathop{max }limits_{{E}_{P},{mathcal{P}},{M}_{P}}{p}_{P}Pleft({E}_{P},{mathcal{P}},{M}_{P}right)-{p}_{E}{E}_{P}-{p}_{{mathcal{P}}}{mathcal{P}}-{p}_{{M}_{P}}{M}_{P}.$$

(3)

For timber production (T) we only consider the input land (LT) and other inputs (MT). The producers then solve the maximization problem

$$mathop{max }limits_{{L}_{T},{M}_{T}}{p}_{T}T({L}_{T},{M}_{T})-{p}_{L}{c}_{T}({L}_{T}){L}_{T}-{p}_{{M}_{T}}{M}_{T},$$

(4)

where cT is a cost of converting (e.g., clearing) land for forestry.

Industrial manufacturing (Y) requires energy (({{mathcal{E}}}_{Y})) and other inputs (MY). While we refer to this sector as manufacturing, the substitutability between energy and other inputs is chosen to match that of the economy as a whole. The substitutability thus reflects not only the manufacturing sector but also the service sector that has a significantly lower energy intensity but is economically important. The maximization problem of the representative producer is

$$max {p}_{Y}Yleft({{mathcal{E}}}_{Y},{M}_{Y}right)-{p}_{{mathcal{E}}}{{mathcal{E}}}_{Y}-{p}_{{M}_{Y}}{M}_{Y}.$$

(5)

The fisheries sector uses inputs fossil fuel (EF) and other inputs (MF). The producers solve the maximization problem

$$mathop{max }limits_{{E}_{F},{M}_{F}}{p}_{F}F({E}_{F},{M}_{F})-{p}_{E}{E}_{F}-{p}_{{M}_{F}}{M}_{F}.$$

(6)

Extraction of fossil fuel (E) is modeled by assuming a gross extraction cost (gE) that increases with increased extraction (gE(E) thus gives the total cost of extracting quantity E). We assume that the tax on fossil fuels (a percentage tax τE) is paid by the firms that extract and sell it. Extraction firms solve the profit maximization problem

$$mathop{max }limits_{E}frac{{p}_{E}}{1+{tau }_{E}}E-{g}_{E}(E).$$

(7)

The sectors phosphate (({mathcal{P}})), water (W), renewable energy (other than biofuels) (R) and the other inputs (MA, MF, MP, MT, and MY) are similarly represented by a production or extraction cost and the profit-maximization problem of the producers are given by

$$mathop{max }limits_{X}{p}_{X}X-{g}_{X}(X) ,, {rm{for}} ,, Xin {{mathcal{P}},W,R,{M}_{A},{M}_{F},{M}_{P},{M}_{T},{M}_{Y}}.$$

(8)

We have now described the maximization problems underlying decisions made by all producers. The representative household also solves a maximization problem, maximizing the utility derived from consumption. The households’ preferences are represented by utility function U and the utility-maximization problem, subject to the income being I, is given by

$$mathop{max }limits_{{A}_{{mathcal{F}}},F,Y,{L}_{U},T} Uleft({mathcal{F}}left({A}_{{mathcal{F}}},Fright),tilde{{mathcal{F}}}left(Y,{L}_{U},Tright)right)\ {rm{s}}.{rm{t}}. ,, {p}_{A}{A}_{{mathcal{F}}}+{p}_{F}F+{p}_{Y}Y+{p}_{L}{L}_{U}+{p}_{T}Tle I.$$

(9)

This specification has divided consumption into two levels. While this division is not necessary at this level of generality, it clarifies the assumed substitutabilities between goods. We assume greater substitutability within than between categories. The upper level consists of food (({mathcal{F}})) and non-food ((tilde{{mathcal{F}}})) goods, with the former category consisting of food from agriculture and from fisheries, and the latter of manufactured goods, natural land and timber. The inclusion of natural land is intended to capture various ways in which households’ demand for natural lands lead to land being kept from other uses, e.g., preservation of land as national parks. We assume that timber is consumed directly by the households.

This completes the description of the modeling of all decision-making agents in the model. In addition to conditions derived from these maximization problems, we must also specify market-clearing conditions that make sure that supplied and demanded quantities add up.

For land (L), the total supply is assumed to be fixed:

$$L={L}_{A}+{L}_{T}+{L}_{U}.$$

(10)

The remaining market-clearing conditions are for agricultural production

$$A={A}_{{mathcal{F}}}+{A}_{B},$$

(11)

fossil fuel

$$E={E}_{{mathcal{E}}}+{E}_{F}+{E}_{P}$$

(12)

and energy services

$${mathcal{E}}={{mathcal{E}}}_{A}+{{mathcal{E}}}_{Y}.$$

(13)

In summary, production functions, market-clearing conditions, budget constraints and first-order conditions from the maximization problems of representative agents provide us with 41 equilibrium conditions pinning down the 41 endogenous prices and quantities. The full set of equilibrium conditions are available in the Supplementary Methods.

### Solution Approach

We note a few features of our model, some of which have already been mentioned: there are no explicit externalities; policies are applied exogenously; all sectors are assumed to be competitive; market clearing determines the equilibrium. In this context, we can work with the decentralized equilibrium, which may be analyzed by considering the first order conditions. In our model, there are 41 unknown prices and quantities in the model, determined by 41 equilibrium conditions. Being exogenous, policies represent parameters that are known in advance; denote a generic “policy” pertaining to any one ESP by τ. Let Xi denote the generic ith variable, an endogenous price or quantity. The jth equilibrium condition can then generally be written as:

$${G}_{j}left({X}_{1},ldots ,{X}_{41};tau right)=0.$$

(14)

This system of equations implicitly define all resulting equilibrium quantities and prices as functions of the policy i.e. ({X}_{i}={X}_{i}left(tau right)).

There are now two solution approaches: the first is to solve the set of resulting non-linear equations (and thereby obtain all the equilibrium values); the second is to trace out marginal changes in the equilibrium values in response to a change in the policy, τ. The latter approach can be illustrated by considering the total derivative of the equilibrium conditions with respect to the policy. This leads to a system of equations, with the jth equation being

$$mathop{sum }limits_{i}^{41}left[frac{{X}_{i}}{{G}_{j}}frac{partial {G}_{j}}{partial {X}_{i}}hat{{X}_{i}}right]=-frac{1}{{G}_{j}}frac{partial {G}_{j}}{partial tau },$$

(15)

where

$$hat{{X}_{i}}equiv frac{1}{{X}_{i}}frac{d{X}_{i}}{dtau }$$

(16)

is the relative change in variable Xi. These can be interpreted as a linear approximation of the percentage change in the variable induced by a one percentage point increase in the fossil fuel tax. Assume, for instance, that we get ({hat{X}}_{i}=2) and consider a one percentage point increase in the tax rate, ΔτE = 0.01. We would then get (frac{1}{{X}_{i}}Delta {X}_{i}approx {hat{X}}_{i}Delta {tau }_{E}=0.02). Hence, a one percentage point increase in the tax induces a two percent increase in the quantity. The result is a system of 41 equations in 41 unknowns, the (hat{{X}_{i}}), and is most useful because of linearity in the unknowns. Indeed this approach can be viewed as linear approximation of the equilibrium response to a change in the policy parameter. The required empirical parameter values needed for numerical computations are fewer, easier to find, and easier to interpret. Furthermore, if considering changes in other parameters of the model (e.g., changes in other policies) only the right-hand side of (15) needs to be changed.

### Data and parametrization

We parameterize the model based partly on data extracted directly from the widely-used GTAP database, described below, and partly on empirical estimates from various sources in the literature. As described above, we mainly need three types of values: quantity shares, expenditure shares and elasticities of various kinds. In total, we need 39 empirical estimates to run the model. In our computations, we set the initial carbon price equal to zero. In reality there are various forms of carbon prices. It is difficult to get a precise measure of all these, but the global average is likely a relatively small negative price. For our analysis, this makes little difference. Assuming a different initial price would scale all results somewhat since the effect of a one percentage point increase in the price would, relatively speaking, be smaller or larger depending on the initial price. All other parameter values that we use are empirically derived based on the current effective carbon price. In the following section, we provide tables with parameter values and their sources.

The first type of parameter that occurs are quantity shares. By quantity share ({Q}_{X,{X}_{Z}}) we mean the share of total quantity X used in a specific sector Z. The full set of values, including their sources are given in Table 2. The exceptions are the quantity shares of fossil fuel going to different sectors and the share of agricultural production going to food or biofuel. These were derived as follows.

Total energy consumption in 2011 was 12,225 Mtoe30. Out of this, 10624 Mtoe came from fossil fuel related sources. Fertilizer production uses about 1.2% of total energy supply and almost all of this comes from fossil fuels31. Hence we assume that the share of fossil fuels going to fertilizer production is ({Q}_{E,{E}_{P}}=frac{12,225}{10,624}times 1.2 % approx 1.4 %). For fisheries production, we assume a global fuel consumption of 40 billion litre’s of fuel32. Assuming that this is mostly diesel, this corresponds to 40 Mtoe of fossil fuel or ({Q}_{E,{E}_{F}}=frac{40}{10,624}approx 0.4 %) of total fossil fuel use. Finally we assume the remaining fossil fuels are used in energy production i.e., 98.2%.

In order to compute the share of agricultural production going to bioufuels we used data underlying the FAO Agricultural Outlook report 2016–202533. For each major agricultural commodity (e.g., wheat, maize, rice, etc.) we computed the share of agricultural production used for biofuels and then computed a weighted sum using the fraction of land used to harvest a specific commodity as weight. This resulted in a quantity share ({Q}_{A,{A}_{B}}approx 3.8 %).

Agriculture accounts for only a relatively small proportion of total final energy demand in both industrialized and developing countries. In OECD countries, for example, around 3–5% of total final energy consumption is used directly in the agriculture sector, while for developing countries, the equivalent figure is likely slightly higher in the a range of 4–8% of total final commercial energy use34. Based on these estimates, we concluded that ({Q}_{{mathcal{E}},{{mathcal{E}}}_{A}}) = 5% constitutes a reasonable baseline.

The second type of that occurs in our equilibrium conditions are expenditure shares. The expenditure share ({Gamma }_{X}^{Z}) of input X in sector Z is the share of total spending on inputs in sector Z that goes to X. To pin down these at the global level, we employed the GTAP database15. More specifically, we used the GTAP data set corresponding to the year 2014, for 141 countries and 57 sectors. The GTAP database is a unique global economic data set constructed by collating and reconciling data on national input-output tables, international trade, production, consumption, and macro-economic data sets from various international data sources. This has further been extended by ref. 35 to include renewable energy commodities, based on several energy data sources, including the International Energy Agency (IEA) data set and the World Bank data set. Furthermore, ref. 36 has extended this even further to include water as an endowment, used in both agricultural and other sectors. Finally, we have a data set in which we can derive the shares of labor, capital, land, water, and several other inputs in producing all commodities. Some inputs, such as fertilizers are not separately identified in this data set, but they are subsumed in broader GTAP sectors such as chemicals, rubber, and plastics. Therefore, we make broad reasonable assumptions to derive the shares of such granular-level inputs; for example, we assume that most of agricultural consumption of output from the GTAP sectors chemicals, rubber, and plastics are fertilizers and pesticides. For all production sectors except energy services, we assign the residual expenditure share, remaining when all inputs of direct interest have been accounted for, to other inputs M. The details are given below and summarized in Table 3.

Agriculture. Our agricultural production function distinguishes between land and non-land inputs (with “other inputs” in the non-land category). The expenditure share of land is 19.2%. The expenditure shares of fertilizers, water, energy, and other inputs are 6.43%, 1.93%, 3.33%, and 71.1%, respectively. Their respective shares out of non-land inputs are their total shares divided by the total non-land share. This means that ({Gamma }_{{L}_{A}}^{A}=0.192), ({Gamma }_{{tilde{L}}_{A}}^{A}=0.808), ({Gamma }_{P}^{{tilde{L}}_{A}}=frac{0.0643}{0.808}=0.0796), ({Gamma }_{W}^{{tilde{L}}_{A}}=frac{0.0193}{0.808}=0.0239), ({Gamma }_{{{mathcal{E}}}_{A}}^{{tilde{L}}_{A}}=frac{0.0643}{0.808}=0.0412), and ({Gamma }_{{M}_{A}}^{{tilde{L}}_{A}}=frac{0.711}{0.808}=0.880).

Energy services. The expenditure shares of biofuels, fossil fuels and renewables are 0.37%, 94.33%, and 5.30% respectively. That is ({Gamma }_{{A}_{B}}^{{mathcal{E}}}=0.0037), ({Gamma }_{{E}_{{mathcal{E}}}}^{{mathcal{E}}}=0.9433), and ({Gamma }_{R}^{{mathcal{E}}}=0.0530).

Utility. The expenditure shares of food from agriculture, fish, manufactured goods, recreational land use, and timber are 11.93%, 0.42%, 86.86%, 0.15%, and 0.65%. This gives expenditure share of food ({Gamma }_{{mathcal{F}}}^{U}=0.1235) and expenditure share of non-food goods ({Gamma }_{tilde{{mathcal{F}}}}^{U}=0.8765). The within-category expenditure shares are ({Gamma }_{{A}_{{mathcal{F}}}}^{{mathcal{F}}}=frac{11.93}{12.35}=0.9660), ({Gamma }_{F}^{{mathcal{F}}}=frac{0.42}{12.35}=0.0340), ({Gamma }_{Y}^{tilde{{mathcal{F}}}}=frac{86.86}{87.65}=0.9910), ({Gamma }_{{L}_{U}}^{tilde{{mathcal{F}}}}=frac{0.15}{87.65}=0.001711), and ({Gamma }_{T}^{tilde{{mathcal{F}}}}=frac{0.65}{87.65}=0.007416).

Timber. The expenditure shares of land and other inputs are 37.48% and 62.52%, respectively. That is ({Gamma }_{{L}_{T}}^{T}=0.3748) and ({Gamma }_{{M}_{T}}^{T}=0.6252).

Composite goods. The expenditure shares of energy services and other inputs are 6.38% and 93.62%, respectively. That is, ({Gamma }_{{{mathcal{E}}}_{Y}}^{Y}=0.0638) and ({Gamma }_{{M}_{Y}}^{Y}=0.9362).

Fertilizers. The expenditure share of energy is 10.95%. The factor share of phosphate is assumed to be a share ({xi }_{{mathcal{P}}}=0.5) out of the factor share of non energy intermediates 62.53%. That is ({Gamma }_{{E}_{P}}^{P}=0.1095) and ({Gamma }_{{mathcal{P}}}^{P}=0.5* 0.6253=0.3127). this leaves the expenditure share or other inputs as ({Gamma }_{{M}_{P}}^{P}=0.5778).

Finally, we need several estimates of elasticities, including the elasticity of substitution, price elasticity of supply and elasticities of conversion costs. For the majority of parameters, we were able to track down estimates from the literature which are presented together with their corresponding reference in Table 4. Where the uncertainty in the estimates were high we employed a wide band for the sensitivity analysis. The parameters that are varied in the sensitivity analysis are indicated as [min, max, and mean] with mean being the baseline values.

### Numerical results

The full sets of changes in our model quantities and prices resulting from the two policies are presented in Table 5.

We now describe the mapping from changes in model variables to effects on ESPs. For the model variables freshwater (W), natural land-use (LU), phosphate (({mathcal{P}})), and nitrogen (assumed to be proportional to fossil fuel use in fertilizer production EP), there is a simple one-to-one mapping with model variables. For climate change, ocean acidification, biodiversity loss and aerosol loading, however, the mapping is more complicated. For climate change and ocean acidification, we measure the change in pressure on both ESPs as the net change in CO2 emissions. For biosphere integrity, we measure changes in pressure as a change in threats to endangered species (more details on this are given below). We measure aerosol loading as changes in aerosol optical depth. For chemical pollution and ozone depletion, we map pressures to contributing sectors, but do not make any quantitative analysis of the net effects.

Climate change and ocean acidification—are both driven by carbon emissions and we use these emissions as our proxy for the pressures inflicted on these boundaries. To translate changes in model variables into changes in emissions, we use data from refs. 37,38. From the figure on page 2 of ref. 37 we get the percentage contribution of carbon dioxide emissions per sector outlined in the report. Using these percentages we can thus recover the amount of actual carbon emissions in gigaton carbondioxide (GtCO2) per year connected to a specific variable in our model.

Using this approach, we start by looking at the energy-related emissions that, according to ref. 37, account for a total of 66.5%. Multiplying by the aggregate total emissions in 2005 (44.15 GtCO2) we get 29,36 GtCO2. Next, we allocate these energy-related emissions to the energy service production sector, fossil fuel extraction, and emissions from fertilizer production. From ref. 37 we have that 6.4% (2.826 GtCO2 eq) of the total energy-related emissions is due to extraction processes. Based on ref. 38, fertilizer production is estimated to cause emissions of 0.575 GtCO2 eq. Hence we can split the total energy-related emission of 29.36 GtCO2 based on these percentages. This implies that 25,960 GtCO2 will be connected to the energy services output in our model, 2.826 GtCO2 is attributed to the fossil fuel extraction process and 0.575 GtCO2 is connected to fertilizer production.

The other emission-related variables in our model are more straightforward. Emissions from industrial processes in ref. 37 are assigned to manufacturing in our model (In total 4.6% = 2.031 GtCO2). Emission from land-use change are assigned to the change in natural land in our model (12.2% = 5.387 GtCO2). Emissions from agriculture are assigned to the total agricultural production variable (13.8% = 6.093 GtCO2). For the fisheries sector,39 estimate carbon dioxide emissions to be ~0.14 GtCO2.

Using these assignments as a status quo, we can calculate the total policy impact by simply multiplying the percentage change in our model variables resulting from the policy by the status quo emission levels. In total, our model variables cover ~97.4% of the emissions outlined in ref. 37. The results of this exercise, in terms of percentage changes to each planetary pressure, is outlined in Supplementary Table 2 for the carbon tax policy and Supplementary Table 3 for the combined carbon tax and biofuel tax policy.

To summarize, we find that a 1% increase in the carbon tax leads to a reduction in carbon dioxide emissions by −0.25% or −0.11 GtCO2 yr−1, which is what we use as an indicator of the change in pressure accrued to the climate change and ocean acidification boundary. For the combination of carbon and biofuel tax, the change is −0.26% or −0.12 GtCO2 yr−1.

Biodiversity loss—is a notoriously difficult task to assess at a global scale. Studies that quantify terrestrial biodiversity losses resulting from the environmental pressures of human activities typically focus on land-related impacts40,41. There are, however, multiple other environmental pressures causing loss of biodiversity that are not related to land-use42. In ref. 3 the global extinction rate is used as one way of quantifying this boundary (defined as extinctions per million species-years). Here, we will make use of the IUCN Red List of Threatened Species to derive a measure of biodiversity loss. The Red List identifies not only the species that have been confirmed to have gone extinct but also the species that are currently threatened and, if pressures remain, may become extinct in the future.43 identify the drivers behind the prevalent threats to the species on the Red List in a comprehensive assessment of more than 8000 species. These drivers can be directly identified as variables in our model. In ref. 43 there is overlap between threats in the sense that multiple activities can pose threats to a given species. We refer to a decrease in an activity posing a threat to a certain number of species as a decrease in threats. Without knowing the overlap between threats, we can not translate this into changes in number of threatened species. Therefore, we use the change in threats as our measure. Agricultural activity poses threats to 5295 species, which is the largest number of threats. The second-largest threat comes from logging, which threatens 4049 species, and we assign this to timber production in our model. Apart from those, we make the following assignments. Pollution from agriculture threatens 1523 of the species and this is assigned to fertilizer production. Over exploitation (fishing), threatening 1118 species, is assigned to fisheries production. Energy production (oil and gas) and renewable energy production account for threats to 56 species, which we assign to fossil fuel extraction and renewables. Finally, threats from urban development (industrial), pollution (except agriculture), human disturbance (work), transport, energy production (mining) summed to 3573 which we assign to manufacturing. There are also significant biodiversity effects of climate change, which threatens 1688 of the species. In this analysis, we abstract from the effects of changes in one ESP on other ESPs (unless the ESP is directly captured by a model variable). We can note, however, that including the effects of climate change would lead to larger decreases of biodiversity loss.

Hence, having connected the categories of threats to species by driver in ref. 43 to our model variables, we can measure the biodiversity impact of a policy by assessing whether the number of threats increases or declines as a result of the policy. For example, if the agricultural production increases by 1% as a result of a policy in our model then this would increase the number of threats from agricultural activity by 52.95 (0.01 × 5296).

The results, in terms of percentage change to the number of threatened species, are outlined in the column labeled Biodiv. in Supplementary Table 2 for the carbon tax policy and Supplementary Table 3 for the combined carbon tax and biofuel subsidy removal. To summarize, this implies that the total number of threats decline by 0.018% for the carbon tax and by 0.011% for the combined carbon and biofuel tax.

Finally, it should be noted that there are indeed several caveats to our approach for assessing biodiversity loss. First, it should be noted that this measure of biodiversity loss is just a proxy for true biodiversity loss. Future work would benefit from assessing the drivers of the actual rate of species loss as defined in e.g., ref. 1. Furthermore, we have taken the description of threats in43 and mapped them to our model variables. For instance, all threats assigned to agriculture in ref. 43 are assigned to agricultural production in our model. Perhaps some part of these threats come from land use change associated with agriculture rather than agriculture as such. In that case they should be mapped to our land use variables. We do not have a proper basis for such reassignment and, therefore, stick close to their assignment. Qualitatively, this distinction could matter for the carbon tax in isolation, but will not be important for the carbon tax combined with biofuel policy.

Aerosol loading—is proxied following3 which use aerosol optical depth (AOD) as an indicative measure of planetary pressure. To determine how AOD changes as a result of policy, we use data from three sources44,45,46. The impact is calculated as follows. First, we calculate a global average estimate of AOD from the main regional anthropogenic sources (sulfur (0.0392), black carbon (0.0003) and organic carbon (0.0011)) provided by ref. 44. Second, we use data from ref. 45 to calculate the share of global aerosol contributing emissions for each of these respective sources (sulfur (3.6%), black carbon (32%) and organic carbon (63%)) that stem specifically from biomass burning (assuming that approximately 90% of biomass burning emissions result from land-use change46). Third, using these estimates, we can calculate the amount of global AOD which ought to be attributed to emissions from fossil fuel and biofuels (0.038) and biomass burning (0.0022). These estimates are then connected to the model variables fossil fuel consumption (in energy services, fertilizer production and fisheries), biofuel production and change in natural land. In total, a 1% carbon tax leads to a 0.0136% (−5.5 × 10−6) decline in AOD and the combined carbon tax and biofuel policy leads to a decline of 0.014% (−5.7 × 10−6) (further details can be found in Supplementary Tables 2 and Table 3).

Stratospheric ozone depletion and chemical pollution—are not directly quantified in terms of their effects on the boundaries. Stratospheric ozone depletion increases with N2O emissions from agricultural production, fossil fuel use, manufacturing, and biofuels. For an increase in the carbon tax all these activities except for biofuel use decreases. The net effect is thus potentially ambiguous. If the carbon tax increase is complemented with a decrease in biofuel subsidies, all relevant variables decrease and we conclude that the net effect is a decrease in the pressure.

Chemical pollution. Chemical pollution increases in manufacturing, extracted fossil fuels, total agricultural production, agricultural production for food, fossil fuel use in fertilizer production, and fossil fuel use in energy services production. All these activities decrease with a carbon tax, with or without a biofuel policy. Hence, we conclude that chemical pollution will decrease in both cases.

For the remaining boundaries—the impacts are easier to assess since they are directly tied to specific model variables. First, the impact on the biogeochemical flows is assigned to the model variables phosphate and fossil-fuel use in fertilizer production. While the former is self-explanatory, the latter is used as a proxy for nitrogen, which relies almost entirely on fossil fuels in its production. For phosphorus, we translate the change into Gg P yr−1 using the value for current flows (mined and applied to erodible soils) from ref. 3:  −0.000068 × 14,000 ≈ −0l9 Gg P yr−1 for the carbon tax and  −0.0005 × 14,000 ≈ −7 Gg P yr−1 for the combined carbon and biofuel policy. For nitrogen, we translate the change into TgN yr−1 using ref. 3: −0.0013 × 150 ≈ −0.2 TgN yr−1 for the carbon tax and   −0.0018 × 150 ≈ −0.28 TgN yr−1 for the combined carbon and biofuel policy. Second, for the land-system boundary, we rely on the model variable natural land use as an indicator of the direction this boundary is moving in. This is translated into MHa using the average (between high and low value) for “natural forests” in ref. 47:  −0.00014 × 3507 ≈ −0.5 MHa for the carbon tax and 0.00043 × 3507 ≈ 1.5 MHa for the combined carbon and biofuel policy. Third, the freshwater boundary is directly tied to the water variable in our model. We translate this into km3 yr−1 using the value for current use in ref. 3, we the reduction is given by 2600 × 0.00009 ≈ 0.24 km3 yr−1 for the carbon tax policy and  −2600 × 0.00036 ≈ −0.93 km3 yr−1 combined carbon tax and biofuel policy.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.