2 Distorted Probability

After solving HJB equations, we display consequences for distorting probabilities. Run below code to plot distorted climate model distribution. ```bash bash ./conduction/ZeroShockTrajectories_simulate.sh “false” “false” “false” “true” bash ./conduction/ZeroShockTrajectories_plot.sh “false” “false” “false” “true”

2.1 Distorted climate and damage model distributions

In constructing distorted climate and damage model, we solves:

\[\min_{q,\, \int_\Theta q(\theta) dP_t(\theta) = 1} \hspace{.3cm} \chi \int_\Theta q(\theta) \log q(\theta) d P_t(\theta)\]

subject to

\[\begin{equation} \int_\Theta \mu[X_t,\Phi^*(X_t)] q(\theta) dP_t(\theta) = {\bar h}(X_t) \end{equation}\]

where \(\Phi^*\) is robustly optimal solution and \({\bar h}\) is deduced from a preference specification with misspecification aversion. We proceed this way to maintain comparability across channels in our analysis of robustness. The minimizing solution takes the form of

\[q^*_t(\theta) \hspace{.2cm} \propto \hspace{.2cm} \exp\left[ - \zeta \mu(X_t, a^*(X_t) \right]\]

and \(\zeta\) is chosen so that the constraint is satisfied.

from src.plot7 import plot_climatehist
plot_climatehist("Figure 4: Distorted climate model distribution")

from src.plot import plot_gammahist

plot_gammahist("""Figure 7: Distorted Climate Model Distribution""")

2.2 Distorted Jump Probability

Distorted jump probability

\[{\widetilde E}\left[ \exp\left( - \int_0^t \left[ \sum_{\ell=1}^{L } {\mathcal J}^{\ell}(X_u) g^{\ell*}(X_u) \right]du \right)\left[ \sum_{\ell=1}^{L } {\mathcal J}^{\ell}(X_u) g^{\ell*}(X_u) \right]\mid X_0 = x\right]\]

from src.plot import Distorted_total_prob

Distorted_total_prob()

Distorted technology jump probability is calculated by line 536. It follows equation

\[{\widetilde E}\left[ \exp\left( - \int_0^t \left[ {\mathcal J}^{L}(X_u) g^{L*}(X_u) \right]du \right){\mathcal J}^{L}(X_u) g^{L*}(X_u) \mid X_0 = x\right].\]

from src.plot import Distorted_tech_jump_prob

Distorted_tech_jump_prob()

Distorted damage jump probability is caculated in line 543. It follows below equation

\[{\widetilde E}\left[ \exp\left( - \int_0^t \left[ \sum_{\ell=1}^{L-1} {\mathcal J}^{\ell}(X_u) g^{\ell*}(X_u) \right]du \right)\left[ \sum_{\ell=1}^{L-1} {\mathcal J}^{\ell}(X_u) g^{\ell*}(X_u) \right]\mid X_0 = x\right]\]

from src.plot import Distorted_damage_jump_prob

Distorted_damage_jump_prob()

2.3 Optimal Controls

from src.plot import Optimal_Control_Capital

Optimal_Control_Capital()

from src.plot import Optimal_Control_RD

Optimal_Control_RD()

from src.plot import Optimal_Control_Emission

Optimal_Control_Emission()