3 Uncertain environmental and economic damages¶
Damage functions¶
We posit a damage process, \(N_t = \{ N_t : t\ge 0\}\) to capture negative externalities on society imposed by carbon emissions. The reciprocal of damages, \({\frac 1 {N_t}}\), diminishes the productive capacity of the economy because of the impact of climate change. We follow much of climate economics literature by presuming that the process \(N\) reflects, in part, the outcome of a damage function evaluated at the temperature anomaly process. Importantly, we use a family of damage functions in place of a single function. Our construction of the alternative damage functions is similar to Barnett et al. with specifications motivated in part by prior contributions. Importantly, we modify their damage specifications in three ways:
we entertain more damage functions, including ones that are more extreme;
we allow for damage function steepness to emerge at an ex ante unknown temperature anomaly threshold;
we presume that ex post this uncertainty is resolved;
We consider a specification under which there is a temperature anomaly threshold after which the damage function could be much more curved. This curvature in the “tail” of the damage function is only revealed to decision-makers when a Poisson event is triggered. As our model is highly stylized, the damages captured by the Poisson event are meant to capture more than just the economic consequences of a narrowly defined temperature movements. Temperature changes are allowed to trigger other forms of climate change that in turn can spill over into the macroeconomy.
In our computational implementation, we use a piecewise log-quadratic function for mapping how temperature changes induced by emissions alter economic opportunities. The Poisson intensity governing the jump probability is an increasing function of the temperature anomaly. We specify it so that the Poisson event is triggered prior to the anomaly hitting an upper threshold \({\overline y}\). Construct a process
where \(\tau\) is the date of a Poisson event. Notice that \({\overline Y}_\tau = {\overline y}\). The damages are given by
where:
In our illustration, we consider 20 damage specifications, \(i.e.\ M = 20\). The \(\gamma_3\)’s are 20 equally spaced values between 0 and 1/3.
The parameter values are as follows:
Parameter |
Value |
---|---|
\(\underline{y}\) |
1.5 |
\(\bar y\) |
2 |
\(\gamma_1\) |
0.00017675 |
\(\gamma_2\) |
0.0044 |
\(\gamma_3^m\) |
\(\frac{0.33 (m - 1)}{19}, m=1, 2, \dots, 20\) |
For motivation of choosing 1.5 and 2 degrees as thresholds, see Remark 4.1.
from src.plots import plot4, plot3
plot4()
Intensity function¶
Where the values for \(r_1\) and \(r_2\) are as follows:
Parameter |
Value |
---|---|
\(r_1\) |
1.5 |
\(r_2\) |
2.5 |
# Intensity function
plot3()