Damage

The relationship between global warming and its impact on economy is driven by the damage function. We have implemented two different equations for the damage model: * Standard DICE: the DICE model equation [^1] * Tipping point: a more drastic version with damage acceleration. You can choose between them by changing the input $tipping_point$: if $tipping_point = False$ then the standard DICE is used and if $tipping_point = True$ the tipping point version is used.

Time Step

The time step $t$ in each equation represents the period we are looking at. In the inputs we initialize the data with 2020 information. The user can choose the year end and the duration of the period (in years) by changing the parameters $year\, end$ and $time \,step$. For example, for a year start at 2020, year end in 2100 and a duration of time step of 5 years we have $t \, \epsilon \,[0, 16]$.

Standard DICE Damage

In DICE[^1] model the damage function gives a relationship between temperature increase and economic damage. The fraction of output loss due to climate change $\Omega_t$ at $t$ is: $$\Omega_t = \pi_1 T_{t\,AT}+ \pi_2 T_{t\,AT}^\varepsilon$$ where $\pi_1$ and $\pi_2$ are $damage_int$ and $damage_quad$ in the inputs. $T_{t\,AT}$ is the atmospheric temperature increase (see Temperature change model documentation). $\varepsilon$ ($damag_expo$) represents the form of the relationship between temperature and economic damage. We set the default value to $\varepsilon = 2$, such that the we have a quadratic relationship. It can be modified by changing the input $damag_expo$.
Total damage in trillions dollars is then: $$Damages_t = \Omega _t. Y_t$$ with $Y_t$ the gross economic output in trillion dollars (see macroeconomics documentation). The form of the damage fraction in the latest version of DICE (2017) is different between the documentation[^1] and the code[^3]. We chose to implement the equation from the code which is the one used to obtain the results.

Tipping point

We use another equation for the damage model from Weitzman (2009)[^2] based on the assumption that once temperatures increase above a given point, damages may accelerate. In this version damages drastically increase after a temperature increase of 6°C. We have then: $$D_t = (\frac{T_{t\,AT}}{20.46})^2 + (\frac{T_{t\,AT}}{6.081})^{6.754}$$ and $$\Omega_t = \frac{D_t}{1 + D_t}$$ . Then as before we have the total economics damage in trillions dollars:
$$Damages_t = \Omega _t. Y_t$$

Damage to productivity

See the documentation of macroeconomics model.

References and notes

[^1]: Nordhaus, W. D. (2017). Revisiting the social cost of carbon. Proceedings of the National Academy of Sciences, 114(7), 1518-1523.

[^2]: Weitzman, M. L. (2009). On modeling and interpreting the economics of catastrophic climate change. The Review of Economics and Statistics, 91(1), 1-19.

[^3]: Version of the code we use is "DICE-2016R-091916ap.gms" available at https://sites.google.com/site/williamdnordhaus/dice-rice. In the documentation (see [^1] ) $\Omega_t = \frac{1}{1+\pi_1 T_{t\,AT}+ \pi_2 T_{t\,AT}^\varepsilon}$.