# Damage to GDP🔗

In En-ROADS, economic growth can be reduced from what it would otherwise be, due to the effects of climate change on human activity. En-ROADS uses temperature change as a proxy for the multiple effects of shifting patterns of temperature, rainfall, disease, etc., that might affect the economy. Economists refer to these effects as the "damage function" and measure the net present value of potential damage as the social cost of carbon.

## Literature on damage function🔗

In the scientific literature, aggregate economic impact of climate change is expressed as a fraction of ‘annual income’, global GDP or GDP per capita. It is formulated as an increasing function of global mean temperature change from preindustrial times. Extensive research into the literature shows the vast disparity between estimates of damage at varying temperature changes. See Damage Function References.

We assessed the very low estimates (Nordhaus, 2007, 2013, and 2016; Weitzman, 2012), ranging from 1% at 2°C, 2-3% at 3°C, and 4-9% at 4°C, and 6-25% at 5°C, to be unrealistic.

The four sources we deemed most credible and covering a range of rates of increasing damage with increasing temperature change are:

- Burke
*et al.*(2018) - Burke
*et al.*(2015) - Dietz and Stern (2015)
- Howard and Sterner (2017)

Burke et al. (2015) estimate the macro impacts of climate change from micro impacts based on an extensive empirical study (e.g. daily temperature effect on labor productivity per person scaled up to annual and global). They conclude that, taking nonlinearities into account, the damage is much higher than the earlier estimates, which is 21% of GDP per capita by 2100 on average. Wealthy countries are not unaffected. Their estimates take different responses by countries into account. In the ‘pooled response’ formulation, rich and poor countries are assumed to respond identically to the temperature change. Short run estimates account for 1 year of temperature, whereas long run estimates account for 5 years of temperature change.

In their 2018 study where they focus on the impact of mitigation targets, they estimate 15%–25% loss in GDP per capita by 2100 for 2.5–3°C warming, and more than 30% for 4°C. Their damage function is widely used in recent studies that analyze the social cost of carbon (Ricke et al., 2018; Taconet et al., 2020; Glanemann et al., 2020). Dietz and Stern follow the formulation of Weitzman (2012), yet assume 50% damage at 4°C.

Through a meta-analysis, Howard and Sterner (2017) determined quadratic equations to define the damage function with varying assumptions:

- Preferred model for non-catastrophic damage
- Preferred model for total (non-catastrophic plus catastrophic) damages
- Preferred model for total damages plus productivity

## Modelling the damage function in En-ROADS🔗

The literature has a variety of damage function forms and values. In En-ROADS, we would like to capture all these, and to allow users explore a wider variety of damage values while keeping the model robust. Accordingly, there are five options in the model, four presets using the equations to reflect the chosen literature and one to customize the damage function with a logistic equation with user specified parameters. There is also an additional option to turn off the damage entirely.

For each source, the model uses the exact formula given, or determined if not provided, to capture the preset.
Burke *et al* (2015 and 2018) do not define a damage function but instead show curves of damage vs. temperature change.
Accordingly, we digitized the graphs and assessed regression analyses Ω with cubic, quadratic, and linear equations for Ω = Damage function = 1-1/(1+D).
Cubic regression, i.e., Ω = 1-1/(1+𝛼*T+𝛽*T2+𝛾*T𝛿)), best captures the fit for all relevant temperatures.
Unlike Dietz and Stern (2015) and Burke *et al* (2015 and 2018), Howard and Sterner (2017) define Ω = D as noted below.

Burke et al, 2018 SR Pooled

𝛼 = 0.3079; 𝛽 = -0.0532; 𝛾 = 0.004; 𝛿 = 3

Burke et al, 2015 LR Pooled

𝛼 = 0.3074; 𝛽 = 0.0144; 𝛾 = 0.0168; 𝛿 = 3

Dietz and Stern, 2015

𝛼 = 0; 𝛽 = 1/18.82; 𝛾 = 4𝛿; 𝛿 = 6.754

Howard and Sterner, 2017

Ω = D “Preferred model for total damages plus productivity”

Ω = 1.145 * T2

For the customized damage function, we use a logistic function formulation with three parameters, *L*, *k* and *x _{0}*, where

*L*is the maximum damage,

*k*refers to the steepness of the damage curve and

*x*is the inflection point.

_{0}$$ \tag 1 D(t) = \frac{L}{1+e^{-k(T(t)-x_0)}} $$

This allows for a function form that captures the damage function shapes and values presented in the literature and allows parameterization based on easily understandable user inputs (sliders) such as “the damage % at 2°C warming” and/or the “maximum damage” saturates at the maximum damage value entered by the users or at 100% so that the damage and GDP values are kept in realistic ranges for extreme temperatures.

## Social Cost of Carbon🔗

Social cost of carbon (SCC) is the marginal cost of emitting one extra tone of CO_{2} in a given year.
It is a commonly used metric in US administration and climate policy debate.
En-ROADS shows the SCC in the present year (i.e. 2023) calculated according to the emission trajectory in the baseline scenario, and the subsequent economic damage of this emission trajectory which depends on the user inputs for the damage function, Social Discount Rate, and climate sensitivity assumptions.

To calculate SCC in En-ROADS, we adopt the approach followed by United States Interagency Working Group (IWG) (Greenstone et al., 2013), which calculated the SCC values used by the US government.
This approach involved simulating the integrated assessment models until 2300, since atmospheric CO_{2} has a very long lifetime and the economic damages from today’s emissions are observed for centuries.
Therefore, even though the normal time horizon of En-ROADS is until 2100, for SCC calculation it is extended until 2300.
In other words, all scenarios displayed by En-ROADS cover the horizon through 2100, yet SCC is calculated based on two additional simulations run upon demand (when users click on the SCC table on UI) through 2300.
For the post-2100 period in these simulations to 2300, we make the following assumptions following IWG:

- IWG assumes that population growth rate declines linearly after 2100, reaching zero in the year 2200, hence a stable population after 2100. In the En-ROADS population stabilizes by 2100 already in the baseline scenario.
- GDP per capita growth rate is assumed to decline linearly after 2100, from whatever value it takes in 2100 based on user inputs and damage, reaching zero in the year 2300.
- The rate of decline in the Carbon intensity of GDP (CO
_{2}emissions from energy / GDP) between 2090 and 2100 is maintained from 2100 through 2300. To formulate this assumption,- We calculate the average rate of change of the Carbon intensity of GDP in 2090-2100.
- We compute the Post-2100 carbon intensity of GDP according to this new constant rate of change.
- We calculate the post-2100 CO
_{2}emissions from energy are as the multiplication of this Post-2100 carbon intensity of GDP * Global GDP.

- Net land use CO
_{2}emissions (LULUCF net emissions) are assumed to decline linearly after 2100, from any value they take in a scenario in 2100, reaching zero in the year 2200. - Non-CO
_{2}GHG emissions (that of CH_{4}, N_{2}O, SF_{6}, PFC and HFC) are assumed to follow the same rate of change as CO_{2}emissions. In other words, the post-2100 trajectory of all these GHG gases are set to follow the trajectory of CO_{2}.

With these assumptions for the 2100-2300 period, SCC is calculated with the following three main steps:

**Step 1: Run a baseline damage scenario through 2300 and calculate the present value of damage**

With any user-set assumptions for the economic impact of temperature rise (damage function) and other climate system assumptions, the damage, i.e. the percentage of global GDP loss (D) is calculated as in Equation 1 above.
From there, annual Global GDP Loss (L) is calculated as the corresponding fraction of Global GDP (Gross World Product, GWP), Equation 2.
These losses over time are discounted to the present year with the variable Present Value of Global GDP Loss (PVL) based on the user-set Social Discount Rate (r) as in Equation 3, where t_{p} is the present year.
Present Value of Cumulative Damage until time is the accumulation of PVL as denoted in Equation 4 where t_{0} and t_{f} are the initial and final time, respectively, i.e. 1990 and 2300.

$$ \begin{align} \tag 2 L(t) &= D(t) \cdot GWP(t) \\[10pt] \tag 3 PVL(t) &= L(t) \cdot \frac{1}{(1+r) ^ {MAX\{0,t-t_p\}}} \\[6pt] \tag 4 CPVL(t) &= \int_{t_0}^{t_f} PVL(t)\,dt \end{align} $$

**Step 2: Run an emission shock scenario through 2300 and calculate the present value of damage**

The same scenario as in Step 1 is simulated with an additional 1 Gton of CO_{2} emissions in the present year.
In other words, the trajectory of CO_{2} emissions is perturbed with the pulse of 1 GtonCO_{2} yr^{-1} in the present year.

**Step 3: Calculate SCC as the marginal damage between the two simulations**

The difference between the Present Value of Cumulative Damage by 2300 in the two simulations yields the social cost of carbon. This formulation is denoted in Equation 5:

$$ \tag 5 SCC(t_p) = \frac{CPVL^2(t_f) - CPVL^1(t_f)}{e} $$

CPVL^{1}(t_{f}) = *Present Value of Cumulative Damage* in the baseline damage scenario in the final time (2300)

CPVL^{2}(t_{f}) = *Present Value of Cumulative Damage* in the emission shock scenario in the final time (2300)

e = amount of the emission shock (1 GtonCO_{2} yr^{-1})

## Damage Function References🔗

- IPCC. (2014). Summary for policymakers. in Climate Change 2014: Impacts, Adaptation, and Vulnerability. Part A: Global and Sectoral Aspects. Contribution of Working Group II to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change (eds. Field, C.B., V.R. Barros, D.J. Dokken, K.J. Mach, M.D. Mastrandrea, T.E. Bilir, M. Chatterjee, K.L. Ebi, Y.O. Estrada, R.C. Genova, B. Girma, E.S. Kissel, A.N. Levy, S. MacCracken, & P.R. Mastrandrea, and L.L. White) 1–32 Cambridge University Press.
- Nordhaus, W. D. (2007). Accompanying notes and documentation on development of DICE-2007 model: Notes on DICE-2007. v8 of September 21, 2007. N. Hav. CT Yale Univ.
- Ackerman, F. & Stanton, E. (2012). Climate risks and carbon prices: Revising the social cost of carbon. Econ. Open-Access Open-Assess. E-J. 6, 10.
- Nordhaus, W. & Sztorc, P. (2013). DICE 2013R: Introduction and user’s manual.
- Tol, R. S. (2009). The economic effects of climate change. J. Econ. Perspect. 23, 29–51.
- Nordhaus, W. D. (2017). Revisiting the social cost of carbon. Proc. Natl. Acad. Sci. 114, 1518–1523.
- Keen, S. (2020). The appallingly bad neoclassical economics of climate change. Globalizations 1–29.
- Weitzman, M. L. (2012). GHG targets as insurance against catastrophic climate damages. J. Public Econ. Theory 14, 221–244.
- Hanemann, W. M. (2008). What is the economic cost of climate change?
- Dietz, S. & Stern, N. (2015). Endogenous growth, convexity of damage and climate risk: how Nordhaus’ framework supports deep cuts in carbon emissions. Econ. J. 125, 574–620.
- Burke, M., Hsiang, S. M. & Miguel, E. (2015). Global non-linear effect of temperature on economic production. Nature 527, 235–239.
- Burke, M., Davis, W. M. & Diffenbaugh, N. S. (2018). Large potential reduction in economic damages under UN mitigation targets. Nature 557, 549–553.
- Ricke, K., Drouet, L., Caldeira, K. & Tavoni, M. Country-level social cost of carbon. Nat. Clim. Change 8, 895–900 (2018).
- Taconet, N., Méjean, A. & Guivarch, C. (2020). Influence of climate change impacts and mitigation costs on inequality between countries. Clim. Change 1–20.
- Glanemann, N., Willner, S. N. & Levermann, A. (2020). Paris Climate Agreement passes the cost-benefit test. Nat. Commun. 11, 1–11.
- Greenstone, M., Kopits, E. & Wolverton, A. (2013). Developing a Social Cost of Carbon for US Regulatory Analysis: A Methodology and Interpretation. Review of Environmental Economics and Policy 7, 23–46.