Science

Science

1. Resolution

FUND 3.9 is defined for 16 regions, specified in Table R. The model runs from 1950 to 3000 in time-steps of a year.

2. Population and income

Population and per capita income follow exogenous scenarios. There are five standard scenarios, specified in Tables P and Y. The FUND scenario is based on the EMF14 Standardised Scenario, and lies somewhere in between the IS92a and IS92f scenarios (Leggett et al., 1992). The other scenarios follow the SRES A1B, A2, B1 and B2 scenarios (Nakicenovic and Swart, 2001), as implemented in the IMAGE model (IMAGE Team, 2001).

We assume that all regions are in a steady state after the year 2300. For the years 2301-3000 per capita income growth rates are constant and equal to the values of the year 2300, while population does not change.

3. Emission, abatement and costs

3.1. Carbon dioxide (CO₂)

Carbon dioxide emissions are calculated on the basis of the Kaya identity:

\[M_{t,r}=\frac{M_{t,r}}{E_{t,r}}\frac{E_{t,r}}{Y_{t,r}}\frac{Y_{t,r}}{P_{t,r}}P_{t,r}=\psi_{t,r}\varphi_{t,r}Y_{t,r} \tag{CO1.1}\]

where $M$ denotes emissions, $E$ denote energy use, $Y$ denotes GDP and $P$ denotes population; $t$ is the index for time, $r$ for region. The carbon intensity of energy use, and the energy intensity of production follow from:

\[\psi_{t,r} = g_{t - 1,r}^{\psi}\psi_{t - 1,r} - \alpha_{t - 1,r}\tau_{t - 1,r} \tag{CO2.2}\]

and

\[\varphi_{t,r} = g_{t - 1,r}^{\varphi}\varphi_{t - 1,r} - \alpha_{t - 1,r}\tau_{t - 1,r} \tag{CO2.3}\]

where $\tau$ is policy intervention and $\alpha$ is a parameter. The exogenous growth rates $g$ are referred to as the Autonomous Energy Efficiency Improvement (AEEI) and the Autonomous Carbon Efficiency Improvement (ACEI). See Tables AEEI and ACEI for the five alternative scenarios (values for the years 2301-3000 again equal the values for the year 2300). Policy also affects emissions via

\[M_{t,r} = \left( \psi_{t,r} - \chi_{t,r}^{\psi} \right)\left( \varphi_{t,r} - \chi_{t,r}^{\varphi} \right)Y_{t,r} \tag{CO2.1}\]
\[\chi_{t,r}^{\psi} = \kappa_{\psi}\chi_{t - 1,r} + \left( 1 - \alpha_{t - 1,r} \right)\tau_{t - 1,r}^{\psi} \tag{CO2.4}\]

and

\[\chi_{t,r}^{\varphi} = \kappa_{\varphi}\chi_{t - 1,r} + \left( 1 - \alpha_{t - 1,r} \right)\tau_{t - 1,r}^{\varphi} \tag{CO2.5}\]

Thus, the variable $0 < \alpha < 1$ governs which part of emission reduction is permanent (reducing carbon and energy intensities at all future times) and which part of emission reduction is temporary (reducing current energy consumptions and carbon emissions), fading at a rate of $0 < \kappa < 1$. In the base case, $\kappa_{\psi} = \kappa_{\varphi} = 0.9$ and

\[\alpha_{t,r} = 1 - \frac{\tau_{t,r}/100}{1 + \tau_{t,r}/100} \tag{CO2.6}\]

So that $\alpha = 0.5$ if $\tau = \$100/\text{tC}$. One may interpret the difference between permanent and temporary emission reduction as affecting commercial technologies and capital stocks, respectively. The emission reduction module is a reduced form way of modelling that part of the emission reduction fades away after the policy intervention is reversed, but that another part remains through technological lock-in. Learning effects are described below. The parameters of the model are chosen so that FUND roughly resembles the behaviour of other models, particularly those of the Energy Modeling Forum (Weyant, 2004; Weyant et al., 2006).

The costs of emission reduction $C$ are given by

\[\frac{C_{t,r}}{Y_{t,r}} = \frac{\beta_{t,r}\tau_{t,r}^{2}}{H_{t,r}H_{t}^{g}} \tag{CO2.7}\]

$H$ denotes the stock of knowledge. Equation (CO2.6) gives the costs of emission reduction in a particular year for emission reduction in that year. In combination with Equations (CO2.2)-(CO2.5), emission reduction is cheaper if smeared out over a longer time period. The parameter $\beta$ follows from

\[\beta_{t,r} = 0.784 - 0.084\sqrt{\frac{M_{t,r}}{Y_{t,r}} - \operatorname{}\frac{M_{t,s}}{Y_{t,s}}} \tag{CO2.8}\]

That is, emission reduction is relatively expensive for the region that has the lowest emission intensity. The calibration is such that a 10% emission reduction cut in 2003 would cost 1.57% (1.38%) of GDP of the least (most) carbon-intensive region; this is calibrated to Hourcade et al. (1996, 2001). An 80% (85%) emission reduction would completely ruin the economy. Later emission reductions are cheaper by Equations (CO2.7) and (CO2.8). Emission reduction is relatively cheap for regions with high emission intensities. The thought is that emission reduction is cheap in countries that use a lot of energy and rely heavily on fossil fuels, while other countries use less energy and less fossil fuels and are therefore closer to the technological frontier of emission abatement. For relatively small emission reduction, the costs in FUND correspond closely to those reported by other top-down models, but for higher emission reduction, FUND finds higher costs, because FUND does not include backstop technologies, that is, a carbon-free energy supply that is available in unlimited quantities at fixed average costs.

The regional and global knowledge stocks follow from

\[H_{t,r} = H_{t - 1,r}\sqrt{1 + \gamma_{R}\tau_{t - 1,r}} \tag{CO2.9}\]

and

\[H_{t}^{G} = H_{t - 1}^{G}\sqrt{1 + \gamma_{G}\tau_{t,r}} \tag{CO2.10}\]

Knowledge accumulates with emission abatement. More knowledge implies lower emission reduction costs. The parameters $\gamma$ determine which part of the knowledge is kept within the region, and which part spills over to other regions as well. In the base case, $\gamma_{R} = 0.9$ and $\gamma_{G} = 0.1$. The model is similar in structure and numbers to that of Goulder and Schneider (1999) and Goulder and Mathai (2000).

Emissions from land use change and deforestation are exogenous, and cannot be mitigated. Numbers are found in Tables CO2F, again for five alternative scenarios.

3.2. Methane (CH₄)

Methane emissions are exogenous, specified in Table CH4 (emissions for the years 2301-3000 are equal to emissions in the year 2300). There is a single scenario only, based on IS92a (Leggett et al., 1992). The costs of emission reduction are quadratic. Table OC specifies the parameters, which are calibrated to USEPA (2003).

3.3. Nitrous oxide (N₂O)

Nitrous oxide emissions are exogenous, specified in Table N2O (emissions for the years 2301-3000 are equal to emissions in the year 2300). There is a single scenario only, based on IS92a (Leggett et al., 1992). The costs of emission reduction are quadratic. Table OC specifies the parameters, which are calibrated to USEPA (2003).

3.4. Sulfurhexafluoride (SF₆)

SF₆ emissions are linear in GDP and GDP per capita. Table SF6 gives the parameters. The numbers for 1990 and 1995 are estimated from IEA data (http://data.iea.org/ieastore/product.asp?dept_id=101&pf_id=305). There is no option to reduce SF₆ emissions.

3.5. Dynamic Biosphere

Emissions from the terrestrial biosphere follow

\[E_{t}^{B} = \beta\left( T_{t} - T_{2010} \right)\frac{B_{t}}{B_{\mathrm{\max}}} \tag{DB.1}\]

with

\[B_{t} = B_{t - 1} - E_{t - 1}^{B} \tag{DB.2}\]

where

The model is calibrated to the review of (Denman et al. 2007). Emissions from the terrestrial biosphere before the year 2010 are zero.

4. Atmosphere and climate

4.1. Concentrations

Methane, nitrous oxide and sulphur hexafluoride are taken up in the atmosphere, and then geometrically depleted:

\[C_{t} = C_{t - 1} + \alpha E_{t} - \beta\left( C_{t - 1} - C_{\text{pre}} \right) \tag{C.1}\]

where $C$ denotes concentration, $E$ emissions, $t$ year, and $\text{pre}$ pre-industrial. Table C displays the parameters $\alpha$ and $\beta$ for all gases. Parameters are taken from Forster et al. (2007).

The atmospheric concentration of carbon dioxide follows from a five-box model:

\[Box_{i,t} = \rho_{i}Box_{i,t - 1} + 0.000471\alpha_{i}E_{t} \tag{C.2a}\]

with

\[C_{t} = \sum_{i = 1}^{5}{\alpha_{i}\text{Bo}x_{i,t}} \tag{C.2b}\]

where $\alpha_{i}$ denotes the fraction of emissions $E$ (in million metric tonnes of carbon) that is allocated to $Box_{i}$ (0.13, 0.20, 0.32, 0.25 and 0.10, respectively) and $\rho$ the decay-rate of the boxes ($\rho = exp( - \frac{1}{\mathrm{\text{lifetime}}})$, with life-times infinity, 363, 74, 17 and 2 years, respectively). The model is due to Maier-Reimer and Hasselmann (1987), its parameters are due to Hammitt et al. (1992). Thus, 13% of total emissions remains forever in the atmosphere, while 10% is—on average—removed in two years. Carbon dioxide concentrations are measured in parts per million by volume.

4.2. Radiative forcing

Radiative forcing is specified as follows:

\[\begin{aligned} RF_{t} = &5.35\ln\frac{\text{CO}2_{t}}{275}\\ &+ 0.036 \times 1.4\left( \sqrt{\text{CH}4_{t}} - \sqrt{790} \right) + 0.12\left( \sqrt{N2O_{t}} - \sqrt{285} \right) \\ &-0.47\ln\left( 1 + 2.01 \times 10^{- 5}\text{CH}4_{t}^{0.75}285^{0.75} + 5.31 \times 10^{- 15}\text{CH}4_{t}^{2.52}285^{1.52} \right) \\ &- 0.47\ln\left( 1 + 2.01 + 10^{- 5}790^{0.75}N2O_{t}^{0.75} + 5.31 \times 10^{- 15}790^{2.52}N2O_{t}^{1.52} \right) \\ &+ 2 \times 0.47\ln\left( 1 + 2.01 \times 10^{- 5}790^{0.75}285^{0.75} + 5.31 \times 10^{- 15}790^{2.52}285^{1.52} \right) \\ &+ 0.00052\left( \text{SF}6_{t} - 0.04 \right) + rfSO2_{t} \end{aligned} \tag{C.3}\]

Parameters are taken from Ramaswamy et al. (2001) and Forster et al. (2007) for the indirect effect of methane on tropospheric ozone. Radiative forcing from SO2 at time t ($\text{rfSO}2_{t}$) is exogenous; the FUND scenario uses the forcing from RCP85 and the SRES scenarios use the forcing as interpreted by IMAGE 2.2.

4.3. Temperature and sea level rise

The global mean temperature $T$ is governed by a geometric build-up to its equilibrium (determined by radiative forcing $RF$). In the base case, global mean temperature $T$ rises in equilibrium by 3.0°C for a doubling of carbon dioxide equivalents, so:

\[T_{t} = \left( 1 - \frac{1}{\varphi} \right)T_{t - 1} + \frac{1}{\varphi}\frac{CS}{5.35\ln 2}RF_{t} \tag{C.4}\]

where $CS$ is climate sensitivity, set to 3.0 (with a gamma distribution with shape=6.48 and scale=0.55). $\varphi$ is the e-folding time and set to

\[\varphi = \max\left( \alpha + \beta^{l}CS + \beta^{q}CS^{2},1 \right) \tag{C.5}\]

where $\alpha$ is set to -42.7, $\beta^{l}$ is set to 29.1 and $\beta^{q}$ is set to 0.001, such that the best guess e-folding time for a climate sensitvitiy of 3.0 is 44 years.

Regional temperature is derived by multiplying the global mean temperature by a fixed factor (see Table RT) which corresponds to the spatial climate change pattern averaged over 14 GCMs (Mendelsohn et al. 2000).

Global mean sea level is also geometric, with its equilibrium level determined by the temperature and a life-time of 500 years:

\[S_{t} = \left( 1 - \frac{1}{\rho} \right)S_{t - 1} + \gamma\frac{1}{\rho}T_{t} \tag{C.6}\]

where $\rho = 500$ (with a triangular distribution bounded by 250 and 1000) is the e-folding time. $\gamma = 2$ (with a gamma distribution with shape=6 and scale=0.4) is sea-level sensitivity to temperature.

Temperature and sea level are calibrated to the best guess temperature and sea level for the IS92a scenario of Kattenberg et al. (1996).

Code

The code for the climate dynamics component can be found at https://github.com/fund-model/MimiFUND.jl/blob/master/src/components/ClimateDynamicsComponent.jl.

The calibration code for the climate dynamics component can be found at https://github.com/fund-model/MimiFUND.jl/blob/master/calibration/climate/fundcscalibration.ipynb.

5. Impacts

5.1. Agriculture

The impacts of climate change on agriculture at time $t$ in region $r$ are split into three parts: impacts due to the rate of climate change $A_{t,r}^{r}$; impacts due to the level of climate change $A_{t,r}^{l}$; and impacts from carbon dioxide fertilisation $A_{t,r}^{f}$:

\[A_{t,r} = A_{t,r}^{r} + A_{t,r}^{l} + A_{t,r}^{f} \tag{A.1}\]

The first part (rate) is always negative: As farmers have imperfect foresight and are locked into production practices, climate change implies that farmers are maladapted. Faster climate change means greater damages. The third part (fertilization) is always positive. CO₂ fertilization means that plants grow faster and use less water. The second part (level) can be positive or negative. There is an optimal climate for agriculture. If climate change moves a region closer to (away from) the optimum, impacts are positive (negative); and impacts are smaller nearer to the optimum.

For the impact of the rate of climate change (i.e., the annual change of climate) on agriculture, the assumed model is:

\[A_{t,r}^{r} = \alpha_{r}\left( \frac{\Delta T_{t}}{0.04} \right)^{\beta} + \left( 1 - \frac{1}{\rho} \right)A_{t - 1,r}^{r} \tag{A.2}\]

where

The model for the impact due to the level of climate change since 1990 is:

\[A_{t,r}^{l} = \delta_{r}^{l}T_{t} + \delta_{r}^{q}T_{t}^{2} \tag{A.3}\]

where

CO₂ fertilisation has a positive, but saturating effect on agriculture, specified by

\[A_{t,r}^{f} = \gamma_{r}\ln\frac{\text{CO}2_{t}}{275} \tag{A.4}\]

where

The parameters in Table A are calibrated, following the procedure described in Tol (2002a), to the results of Kane et al. (1992), Reilly et al. (1994), Morita et al. (1994), Fischer et al. (1996), and Tsigas et al. (1996). These studies all use a global computable general equilibrium model, and report results with and without adaptation, and with and without CO₂ fertilisation. The regional results from these studies are assumed to hold for each country in the respective regions. They are averaged over the studies and the climate scenarios for each country, and aggregated to the FUND regions. The standard deviations in Table A follow from the spread between studies and scenarios. Equation (A.4) follows from the difference in results with and without CO2 fertilization. Equation (A.3) follows from the results with full adaptation. Equation (A.2) follows from the difference in results with and without adaptation.

Equations (A.1)-(A.4) express the impact of climate change as a percentage of agricultural production. In order to express this as a percentage of income, we need to know the share of agricultural production in total income. This is assumed to fall with per capita income, that is,

\[\frac{\text{GA}P_{t,r}}{Y_{t,r}} = \frac{\text{GA}P_{1990,r}}{Y_{1990,r}}\left( \frac{y_{1990,r}}{y_{t,r}} \right)^{\epsilon} \tag{A.5}\]

where

Code

The code for the agricultural impacts component can be found at https://github.com/fund-model/MimiFUND.jl/blob/master/src/components/ImpactAgricultureComponent.jl.

5.2. Forestry

The model is:

\[F_{t,r} = \alpha_{r}\left( \frac{y_{t,r}}{y_{1990,r}} \right)^{\epsilon}\left( 0.5\left( \frac{T_{t}}{1.0} \right)^{\beta} + 0.5\gamma\ln\left( \frac{\text{CO}2_{t}}{275} \right) \right) \tag{F.1}\]

where

The parameter $\alpha$ is estimated as the average of the estimates by Perez-Garcia et al. (1995) and Sohngen et al. (2001). Perez-Garcia et al. (1995) present results for four different climate scenarios and two management scenarios, while Sohngen et al. (2001) use two different climate scenario and two alternative ecological scenarios. The results are mapped to the FUND regions assuming that the impact is uniform elative to GDP. The impact is averaged within the study results, and then the weighted average between the two studies is computed and shown in Table EFW. The standard deviation follows.

5.3. Water resources

The impact of climate change on water resources follows:

\[W_{t,r} = \min\left\{ \alpha_{r}Y_{1990,r}\left( 1 - \tau \right)^{t - 2000}\left( \frac{y_{t,r}}{y_{1990,r}} \right)^{\beta}\left( \frac{P_{t,r}}{P_{1990,r}} \right)^{\eta}\left( \frac{T_{t}}{1.0} \right)^{\gamma},\frac{Y_{t,r}}{10} \right\} \tag{W.1}\]

where

These parameters are from calibrating FUND to the results of Downing et al. (1995, 1996).

5.4. Energy consumption

For space heating, the model is:

\[SH_{t,r} = \frac{\alpha_{r}Y_{1990,r}\frac{\operatorname{atan}T_{t}}{\operatorname{atan}{1.0}}\left( \frac{y_{t,r}}{y_{1990,r}} \right)^{\epsilon}\left( \frac{P_{t,r}}{P_{1990,r}} \right)}{\prod_{s = 1990}^{t}{\text{AEE}I_{s,r}}} \tag{E.1}\]

where

These parameters are from calibrating FUND to the results of Downing et al. (1995, 1996). Savings on space heating are assumed to saturate. The income elasticity of heating demand is taken from Hodgson and Miller (1995, cited in Downing et al., 1996), and estimated for the . Space heating demand is linear in the number of people for want of scenarios of number of households and house sizes. Energy efficiency improvements in space heating are assumed to be equal to the average energy efficiency improvements in the economy.

For space cooling, the model is:

\[SC_{t,r} = \frac{\alpha_{r}Y_{1990,r}\left( \frac{T_{t}}{1.0} \right)^{\beta}\left( \frac{y_{t,r}}{y_{1990,r}} \right)^{\epsilon}\left( \frac{P_{t,r}}{P_{1990,r}} \right)}{\prod_{s = 1990}^{t}{\text{AEE}I_{s,r}}} \tag{E.2}\]

where

These parameters are from calibrating FUND to the results of Downing et al. (1995, 1996). Space cooling is assumed to be more than linear in temperature because cooling demand accelerates as it gets warmer. The income elasticity of cooling demand is taken from Hodgson and Miller (1995, cited in Downing et al., 1996), and estimated for the . Space cooling demand is linear in the number of people for want of scenarios of number of households and house sizes. Energy efficiency improvements in space cooling are assumed to be equal to the average energy efficiency improvements in the economy.

5.5. Sea level rise

Table SLR shows the accumulated loss of drylands and wetlands for a one metre rise in sea level. The data are taken from Hoozemans et al. (1993), supplemented by data from Bijlsma et al. (1995), Leatherman and Nicholls (1995) and Nicholls and Leatherman (1995), following the procedures of Tol (2002a).

Potential cumulative dryland loss without protection is assumed to be a function of sea level rise:

\[{\overset{\overline{}}{\text{CD}}}_{t,r} = \min\left\lbrack \delta_{r}S_{t}^{\gamma_{r}},\zeta_{r} \right\rbrack \tag{SLR.1}\]

where

Potential dryland loss in the current year without protection is given by potential cumulative dryland loss without protection minus actual cumulative dryland lost in previous years:

\[{\overset{\overline{}}{D}}_{t,r} = {\overset{\overline{}}{\text{CD}}}_{t,r} - CD_{t - 1,r} \tag{SLR.2}\]

where

Actual dryland loss in the current year depends on the level of protection:

\[D_{t,r} = \left( 1 - P_{t,r} \right){\overset{\overline{}}{D}}_{t,r} \tag{SLR.3}\]

where

Actual cumulative dryland loss is given by:

\[\text{CD}_{t,r} = CD_{t - 1,r} + D_{t,r} \tag{SLR.4}\]

where

The value of dryland is assumed to be linear in income density ($/km^2^):

\[VD_{t,r} = \varphi\left( \frac{\frac{Y_{t,r}}{A_{t,r}}}{YA_{0}} \right)^{\epsilon} \tag{SLR.5}\]

where

Wetland loss is assumed to be a linear function of sea level rise:

\[W_{t,r} = \omega_{r}^{S}\Delta S_{t} + \omega_{r}^{M}P_{t,r}\Delta S_{t} \tag{SLR.6}\]

where

Cumulative wetland loss is given by

\[W_{t,r}^{C} = \min\left( W_{t - 1,r}^{C} + W_{t - 1,r},W_{r}^{M} \right) \tag{SLR.7}\]

where

Wetland loss (SLR.6) goes to zero if all wetland threatened by sea-level rise in a region is lost.

Wetland value is assumed to increase with income and population density, and fall with wetland size:

\[VW_{t,r} = \alpha\left( \frac{y_{t,r}}{y_{0}} \right)^{\beta}\left( \frac{d_{t,r}}{d_{0}} \right)^{\gamma}\left( \frac{W_{1990,r} - W_{t,r}^{C}}{W_{1990,r}} \right)^{\delta} \tag{SLR.8}\]

where

If dryland gets lost, the people living there are forced to move. The number of forced migrants follows from the amount of land lost and the average population density in the region. The value of this is set at 3 (1.5,&gt;0) times the regional per capita income per migrant (Tol, 1995). In the receiving country, costs equal 40% (20%,&gt;0) of per capita income per migrant (Cline, 1992).

Table SLR displays the annual costs of fully protecting all coasts against a one metre sea level rise in a hundred years time. If sea level would rise slower, annual costs are assumed to be proportionally lower; that is, costs of coastal protection are linear in sea level rise. The level of protection, that is, the share of the coastline protected, is based on a cost-benefit analysis:

\[P_{t,r} = \max\left\{ 0,1 - \frac{1}{2}\left( \frac{\mathrm{\text{NPV}}VP_{t,r} + \mathrm{\text{NPV}}VW_{t,r}}{\mathrm{\text{NPV}}VD_{t,r}} \right) \right\} \tag{SLR.9}\]

where

Equation (SLR.9) is due to Fankhauser (1994). See below.

Table SLR reports average costs per year over the next century. $\mathrm{\text{NPV}}\text{VP}$ is calculated assuming annual costs to be constant. This is based on the following. Firstly, the coastal protection decision makers anticipate a linear sea level rise. Secondly, coastal protection entails large infrastructural works which last for decades. Thirdly, the considered costs are direct investments only, and technologies for coastal protection are mature. Throughout the analysis, a pure rate of time preference, $\rho$, of 1% per year is used. The actual discount rate lies thus 1% above the growth rate of the economy, $g$. The net present costs of protection $PC$ equal

\[\mathrm{\text{NPV}}VP_{t,r} = \sum_{s = t}^{\infty}{\left( \frac{1}{1 + \rho + \eta g_{t,r}} \right)^{s - t}\pi_{r}\Delta S_{t}} = \frac{1 + \rho + \eta g_{t,r}}{\rho + \eta g_{t,r}}\pi_{r}\Delta S_{t} \tag{SLR.10}\]

where

$\mathrm{\text{NPV}}\text{VW}$ is the net present value of the wetlands lost due to full coastal protection. Wetland values are assumed to rise in line with Equation (SLR.8). All growth rates and the rate of wetland loss are as in the current year. The net present costs of wetland loss $\text{WL}$ follow from

\[\mathrm{\text{NPV}}VW_{t,r} = \sum_{s = t}^{\infty}{W_{t,r}VW_{s,r}\left( \frac{1}{1 + \rho + \eta g_{t,r}} \right)^{s - t}} = W_{t,r}VW_{t,r}\frac{1 + \rho + \eta g_{t,r}}{\rho + \eta g_{t,r} - \beta g_{t,r} - \gamma p_{t,r} - \delta w_{t,r}} \tag{SLR.11}\]

where

$\mathrm{\text{NPV}}\text{VD}$ denotes the net present value of the dryland lost if no protection takes place. Land values are assumed to rise at the rate of income growth. All growth rates and the rate of wetland loss are as in the current year. The net present costs of dryland loss are

\[\mathrm{\text{NPV}}VD_{t,r} = \sum_{s = t}^{\infty}{{\overset{\overline{}}{D}}_{t,r}VD_{t,r}\left( \frac{1 + \epsilon d_{t,r}}{1 + \rho + \eta g_{t,r}} \right)^{s - t}} = {\overset{\overline{}}{D}}_{t,r}VD_{t,r}\frac{1 + \rho + \eta g_{t,r}}{\rho + \eta g_{t,r} - \epsilon d_{t,r}} \tag{SLR.12}\]

where

Protection levels are bounded between 0 and 1.

5.6. Ecosystems

Tol (2002a) assesses the impact of climate change on ecosystems, biodiversity, species, landscape etcetera based on the "warm-glow" effect. Essentially, the value, which people are assumed to place on such impacts, are independent of any real change in ecosystems, of the location and time of the presumed change, etcetera – although the probability of detection of impacts by the “general public” is increasing in the rate of warming. This value is specified as

\[E_{t,r} = \alpha P_{t,r}\frac{\frac{y_{t,r}}{y_{r}^{b}}}{1 + \frac{y_{t,r}}{y_{r}^{b}}}\frac{\frac{\Delta T_{t}}{\tau}}{1 + \frac{\Delta T_{t}}{\tau}}\left( 1 - \sigma + \sigma\frac{B_{0}}{B_{t}} \right) \tag{E.1}\]

where

The number of species follows

\[B_{t} = \max\left\{ \frac{B_{0}}{100},B_{t - 1}\left( 1 - \rho - \gamma\frac{\Delta T^{2}}{\tau^{2}} \right) \right\} \tag{E.2}\]

where

These parameters are expert guesses. The number of species is assumed to be constant until the year 2000 at 14,000,000 species.

5.7. Human health: Diarrhoea

The number of additional diarrhoea deaths $D_{t,r}^{d}$ in region $r$ and time $t$ is given by

\[D_{t,r}^{d} = \mu_{r}^{d}P_{t,r}\left( \frac{y_{t,r}}{y_{1990,r}} \right)^{\epsilon}\left( \frac{T_{t,r}}{T_{\mathrm{pre - industrial},r}} \right)^{\eta} \tag{HD.1}\]

where

Equation (HD.1), specifically parameters $\epsilon$ and $\eta$, was estimated based on the WHO Global Burden of Diseases data (http://www.who.int/health_topics/global_burden_of_disease/en/). Diarrhoea morbidity has the same equation as mortality, but with $\epsilon$=-0.42 (0.12) and $\eta$=0.70 (0.26); base morbidity is given in Table HD, column 4. Table HD gives impact estimates, ignoring economic and population growth.

See section 5.12. for a description of the valuation of mortality and morbidity.

5.8. Human health: Vector-borne diseases

The number of additional deaths from vector-borne diseases, $D_{t,r}^{v}$ is given by:

\[D_{t,r}^{v} = D_{1990,r}^{v}\alpha_{r}^{v}T_{t}^{\beta}\left( \frac{y_{t,r}}{y_{1990,r}} \right)^{\gamma} \tag{HV}\]

where

See section 5.12. for a description of the valuation of mortality and morbidity. Morbidity is proportional to mortality, using the factor specified in Table HM.

5.9. Human health: Cardiovascular and respiratory mortality

Cardiovascular and respiratory disorders are worsened by both extreme cold and extreme hot weather. Martens (1998) assesses the increase in mortality for 17 countries. Tol (2002a) extrapolates these findings to all other countries, based on formulae of the shape:

\[D^{c} = \alpha^{c} + \beta^{c}T_{B} \tag{HC.1}\]

where

Equation (HC.1) is specified for populations above and below 65 years of age for cardiovascular disorders. Cardiovascular mortality is affected by both heat and cold. In the case of heat, $T_{B}$ denotes the average temperature of the warmest month. In the case of cold, $T_{B}$ denotes the average temperature of the coldest month. Respiratory mortality is not age-specific.

Equation (HC.1) is readily extrapolated. With warming, the baseline temperature $T_{B}$ changes. If this change is proportional to the change in the global mean temperature, the equation becomes quadratic. Summing country-specific quadratic functions results in quadratic functions for the regions:

\[D_{t,r}^{c} = \alpha_{r}^{c}T_{t} + \beta_{r}^{c}T_{t}^{2} \tag{HC.2}\]

where

One problem with (HC.2) is that it is a non-linear extrapolation based on a data-set that is limited to 17 countries and, more importantly, a single climate change scenario. A global warming of 1°C leads to changes in cardiovascular and respiratory mortality in the order of magnitude of 1% of baseline mortality due to such disorders. Per cause, the total change in mortality is restricted to a maximum of 5% of baseline mortality, an expert guess. This restriction is binding. Baseline cardiovascular and respiratory mortality derives from the share of the population above 65 in the total population.

If the fraction of people over 65 increases by 1%, cardiovascular mortality increases by 0.0259% (0.0096%). For respiratory mortality, the change is 0.0016% (0.0005%). These parameters are estimated from the variation in population above 65 and cardiovascular and respiratory mortality over the nine regions in 1990, using data from http://www.who.int/health_topics/global_burden_of_disease/en/.

Mortality as in equations (HC.1) and (HC.2) is expressed as a fraction of population size. Cardiovascular mortality, however, is separately specified for younger and older people. In 1990, the per capita income elasticity of the share of the population over 65 is 0.25 (0.08). This is estimated using data from http://earthtrends.wri.org

Heat-related mortality is assumed to be limited to urban populations. Urbanisation is a function of per capita income and population density:

\[U_{t,r} = \frac{\alpha\sqrt{y_{t,r}} + \beta\sqrt{PD_{t,r}}}{1 + \alpha\sqrt{y_{t,r}} + \beta\sqrt{PD_{t,r}}} \tag{HC.3}\]

where

See section 5.12. for a description of the valuation of mortality and morbidity. Morbidity is proportional to mortality, using the factor specified in Table HM.

5.10. Extreme weather: Tropical storms

The economic damage $TD$ due to an increase in the intensity of tropical storms (hurricanes, typhoons) follows

\[TD_{t,r} = \alpha_{r}Y_{t,r}\left( \frac{y_{t,r}}{y_{1990,r}} \right)^{\epsilon}\left\lbrack \left( 1 + \delta T_{t,r} \right)^{\gamma} - 1 \right\rbrack \tag{TS.1}\]

where

The mortality $\text{TM}$ due to an increase in the intensity of tropical storms (hurricanes, typhoons) follows

\[TM_{t,r} = \beta_{r}P_{t,r}\left( \frac{y_{t,r}}{y_{1990,r}} \right)^{\eta}\left\lbrack \left( 1 + \delta T_{t,r} \right)^{\gamma} - 1 \right\rbrack \tag{TS.2}\]

where

See section 5.12. for a description of the valuation of mortality and morbidity.

5.11. Extreme weather: Extratropical storms

The economic damage due to an increase in the intensity of extratropical storms follows the equation below:

\[\text{ET}D_{t,r} = \alpha_{r}Y_{t,r}\left( \frac{y_{t,r}}{y_{1990,r}} \right)^{\epsilon}\delta_{r}\left\lbrack \left( \frac{C_{CO2,t}}{C_{CO2,pre}} \right)^{\gamma} - 1 \right\rbrack \tag{ETS.1}\]

where

\[\text{ET}M_{t,r} = \beta_{r}P_{t,r}\left( \frac{y_{t,r}}{y_{1990,r}} \right)^{\varphi}\delta_{r}\left\lbrack \left( \frac{C_{CO2,t}}{C_{CO2,pre}} \right)^{\gamma} - 1 \right\rbrack \tag{EST.2}\]

where

See section 5.12. for a description of the valuation of mortality and morbidity.

5.12. Mortality and Morbidity

The value of a statistical life is given by

\[\text{VS}L_{t,r} = \alpha\left( \frac{y_{t,r}}{y_{0}} \right)^{\epsilon} \tag{MM.1}\]

where

This calibration results in a best guess value of a statistical life that is 200 times per capita income (Cline, 1992).

The value of a year of morbidity is given by

\[VM_{t,r} = \beta\left( \frac{y_{t,r}}{y_{0}} \right)^{\eta} \tag{MM.2}\]

where

This calibration results in a best guess value of a year of morbidity that is 0.8 times per capita income (Navrud, 2001).

Acknowledgements

We thank Adriana Ciccone for helpful comments on this documentation.

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