### Damages from climate change

The results in webDICE, with the exception of the social costs of carbon do not depend on the discount rate. Instead, webDICE calculates the change in consumption in each period, translates that to a change in utility via the utility function and sums up the changes in utility.

The ‘pure rate of time preference’ is subtly different than the discount rate. The discount rate applies to money or, in webDICE, things money can buy like consumption. It is like the interest rate you get when you put money in the bank or that you pay when you borrow money. The pure rate of time preference is part of the process of adding up the utility of people to determine how well society is doing. It applies directly to utility, not to money. It is a way of computing the present value of the utility of people who live in the future. more

The implied discount rate in the model is $r=\eta g+\rho$ where $\eta$ is the elasticity of the marginal utility of consumption, $g$ is the growth rate of consumption, and $\rho$ is the pure rate of time preference. webDICE does not use the implied discount rate except for calculating the social cost of carbon (link).

Climate change will cause many different sorts of harms, including changes to rainfall patterns, sea level rise, extreme weather, extinction of many species, and so forth. webDICE represents all of these harms ina single aggregated fashion by assuming that usual output is reduced from what it otherwise would have been because of climate change. In the default mode, it assumes that as temperatures increase, the harms from climate change increase and that the increment of harm grows. For example, there will be harms from a 2°C temperature increase. The harms will be larger from a 3°C increase. Moreover, the increase in harms will be greater when we go from 4°C to 5°C than from 2°C to 3°C. The default mode assumes that harms go up with the square of temperature increases, but you can choose to increase or reduce this rate.

In advanced mode, you can also choose from three alternative ways climate change might harm society. The first ‘incomensurable environmental goods’ is to adjust damages to take into account that climate change might destroy things that are hard to replace. For example, climate change might not generally harm the economy very much but severely hurt agriculture. We cannot easily replace the food with need with other other items such as televisions. If you pick this mode, webDICE assumes that substitution away from certain goods is expensive (but not entirely possible) and therefore a given level of level of temperature increase will often lead to greater harms.

The second ‘tipping point’ assumes that once temperatures reach 6°C that harms accelerate — for example, because ecosystems might collapse or ice sheets might irreversibly melt.

The final type of harm is ‘fraction of productivity’. In this case, climate change reduces both output and also reduces the growth of productivity in the economy. You can choose which fraction of harms reduce output as in the default mode, and which fraction reduces productivity. In the default choice, 5% of the harms are applied to productivity and the remaining 95% are applied exactly as in the normal damages mode. The harms for ‘fraction of productivity’ damages can be very extreme if you choose a medium of high level of harms to apply to productivity, often leading to an economy below the subsistence level.

We do not know how climate change will harm people or the extent to which it will. The choices are intended only to allow you to consider some possibilities. The following figure illustrates the effect choice of damage function will have on the output of the model. Each of the following pathways uses the default webDICE parameter choices:

The default DICE damage function approximates a polynomial relationship between temperature and economic damages due to climate change. That is, the fraction of output lost in each period increases to the user-defined power of the increase in temperature. The default setting of the model specifies a quadratic relationship between temperature and economic damage, however the user can define this relationship anywhere between linear and quartic. The choice of this parameter also affects the appropriate alternative damage function. In the default setting of DICE damages, the model is calibrated so that 1.7% of GDP is lost when temperature increases by 2.5°C.

*webDICE supports four different damages functions*

#### Default

The default DICE damage function approximates a polynomial relationship between temperature and economic damages due to climate change. That is, the fraction of output lost in each period increases to the $\epsilon$ power of the increase in temperature. The functional form is modeled: \[ \Omega(t)=\frac{1}{1+\pi_{2}T_{AT}(t)^{\varepsilon}}. \] The change in atmospheric temperature, $T_{AT},$ is guided by other assumptions in the model. Users can choose $\epsilon$.

The exponent on temperature change, $\epsilon,$ represents the form of the relationship between an increasing temperature and damages. The default setting represents a quadratic relationship, that is $\epsilon=2.$. Users can choose to vary this relationship from linear to quartic (i.e. in the range $[1,4]$). This coefficient calibrates the model so that 1.7% of GDP is lost when temperature increases by 2.5°C in the quadratic form. The users choice of $\epsilon$ applies to the other appropriate alternative damage functions.

#### Incomensurable environmental goods

The default damage function implicitly assumes that different types of consumption goods can be readily substituted for one another. For example, if agriculture represents 24% of GDP, a 1% loss to agriculture would correspond with a 0.24% loss in GDP and a 100% loss in the agricultural sector would yield only a 24% drop in global GDP. This specification of the damage function yields limited damages because it implicitly assumes that other goods can be substituted for food.

The ‘environmental goods’ damage function modifies the damage function to allow some goods to have limited substitutability. As the price of those goods rise a growing portion of GDP will be devoted to that good.

This damage function is based on a model by Sterner and Persson (2008) [6] as modified by Weitzman. Sterner and Persson include relative price effects in a modified version of DICE by setting utility to be a CES function of the consumption of material goods and of an unspecified environmental good: \[ U\left(c(t)\right)=\frac{\left(\left[\left(1-b\right)\cdot c(t)^{1-1/\sigma}+b\cdot E^{1-1/\sigma}\right]^{\sigma/(1-\sigma}\right)^{1-\alpha}}{1-\alpha}, \] where $c(t)$ and $E$ represent the consumption of two goods that will be affected differently by climate change, the elasticity of substitution between them is given by $\sigma$, and $b$ sets the share of overall consumption of the environmental good $E$. Sterner and Persson assume that only the consumption of the environmental good will be affected by changes in temperature, so that $E=E_{0}/\left(1+\alpha T_{AT}(t)^{2}\right)$, where $E_{0}$ is the level of consumption of the environmental good in 2005 and $\alpha$ is a constant used for calibration.

The default damage function is set in terms of loss of consumption. To compare this to the Sterner and Persson damages, which reduce utility, set the elasticity of marginal utility, $\alpha=2$. The default DICE utility function can be translated to the following disutility function (which determines utility lost as a result of climate damages in each time period): \[ U(c(t))=-[\frac{1}{c(t)}\times(1+\pi_{2M}T_{AT}^{\epsilon})]. \] If $\eta=2$ and $\sigma=1/2,$, the Sterner and Persson CES utility function is equivalent to the disutility function: \[ U(c(t))=-[\frac{1}{c(t)}+\left(1+\pi_{2A}T^{2}\right)]. \] where $\pi_{2A}=ab/[(1-b)E_{0}]$.

The only difference between these two disutility functions is the replacement of multiplication in the standard DICE utility function with addition (as well as the value of the coefficient on temperature change to allow for identical calibration). Weitzman argues that no reasonable prior allows us to distinguish these two functional forms.

To translate the Sterner and Persson damage function into DICE, we compute the consumption that would produce equivalent utility using the CRRA utility function used in DICE. The resulting damage adjusted consumption is \[ \Omega(t)=\frac{1}{1+c(t)\times\pi_{2A}T_{AT}^{\epsilon}}. \]

Compared to the default damages, this damage function includes a factor $c(t)$ in the denominator. From here, one can determine output and therefore damages from climate change to output. By setting multiplicative and additive damages equal to each other in the period when $T_{AT}\approx2.5^{\circ}$, webDICE approximates the same calibration as the default damage function. Using this calibration, the model applies damages to consumption and uses the same process as above to determine output and damages in each period.

Much like Sterner and Persson, the user will find that employing this damage function in the standard DICE model will yield ‘a far more stringent emissions policy than Nordhaus found with his multiplicive’ form damage function. Note that the Sterner and Persson utility function allows one to choose the elasticity across the two types of goods. In the translation used here, this is set so that $\sigma=1/2$ and cannot be modified.

#### Tipping Point

Because of possible positive feedbacks in the climate system, once temperatures increase above a given point, damages may accelerate. For example, if warming is such that methane is released from the permafrost, this will increase warming, in turn causing more methane release. The tipping point damage function allows for this possibility.

webDICE uses the form of a tipping point damage function proposed by Martin Weitzman. With this damage function, damages drastically increase after temperatures have increased by around 6°C.

Because of possible positive feedbacks in the climate system, once temperatures increase above a given point, damages may accelerate. For example, if warming is such that methane is released from the permafrost, this will increase warming, in turn causing more methane release. The tipping point damage function allows for this possibility.

webDICE uses the form of a tipping point damage function proposed by Martin Weitzman. With this damage function, damages drastically increase after temperatures have increased by around 6°C.

\[ \Omega=\frac{1}{1+(\frac{T_{AT}}{20.46})^{2}+(\frac{T_{AT}}{6.081})^{6.754}}. \]

#### Fraction of productivity

This ‘damages to productivity growth’ specification of the damage function, derived from Moyer et al. [7], directs a portion of damages from climate change to the growth of total factor productivity (TFP). This implies that climate change may not only affect what we produce but how we produce it.

As discussed in section 1.3 total factor productivity (TFP), $A(t)$, in default DICE grows according to: \[ A(t)=\frac{A(t-1)}{1-A_{g}(t-1)}. \]

Given the growth rate of $A(t)$, this damage function then allows a user-defined fraction of damages, $f$, to reduce the level of TFP instead of all damages being applied solely to output. More succinctly, this specification is defining a new path for TFP, which for purposes of this discussion we will call $A^{\ast}(t)$: \[ A^{\ast}(t)=(1-f\cdot\Omega(t))\times\frac{A^{\ast}(t-1)}{1-A_{g}(t-1)}, \] where $A(0)=A^{\ast}(0)$. The remainder of damages is applied to output as in the default damage function. Damages to output, $\Omega_{Y}(t)$, is given by: \[ \Omega_{Y}(t)=1-\frac{(1-\Omega(t))}{[1-f\cdot(1-\Omega(t))]}. \]

Output in a given time period is equal to: \[ Y(t)=\Omega_{Y}(t)\cdot A^{\ast}(t)L(t)^{1-\gamma}K(t)^{\gamma} \] This specification yields the same single period loss in consumption as the default DICE damage function but will significantly lower long term growth because productivity is reduce by climate change.

[6] Thomas Sterner and U. Martin Persson, ‘An Even Sterner Review’, Review of Environmental Economics and Policy 2 (2008): 61-76, doi: 10.1093/reep/rem024

[7] Moyer, Elisabeth J. and Woolley, Mark D. and Glotter, Michael and Weisbach, David A., Climate Impacts on Economic Growth as Drivers of Uncertainty in the Social Cost of Carbon (July 31,2013). RDCEP Working Paper No. 13-02. Available at SSRN: http://ssrn.com/abstract=2312770 or http://dx.doi.org/10.2139/ssrn.2312770