3 Stochastic Simulation¶

StocasticTrajectories_simulate.py is used for simulating paths with shock term.

StocasticTrajectories_plot.py plots figures using stochastic paths. We will show how each part of code plot what figures.

Simulation Variables Summary¶

State Variables

``states`` (hist): An array containing the simulated paths of the state variables over time. Each state includes:
- ``K``: Log of capital stock \(\log K_t\).
- ``Y``: Temperature anomaly \(Y_t\).
- ``L``: Log of technology level \(\log R_t\).

Control Variables

``i``: Investment \(i^k_t\) in levels, representing the amount of investment over time.
``e``: Emission \(\mathcal{E}_t\), representing emissions over time.
``x``: R&D investment \(i^r_t\) in levels.

Economic Variables¶

``c``: Consumption over time, computed based on output, investment, and R&D expenditures.

\[C_t = \alpha K_t \left (1 - \phi_0\left(A^b_t\right)^{\phi_1} \right) - I_t^k - I_t^r\]

``dmg``: Damage function over time, representing the impact of temperature anomaly on the economy.

\[N_t= e^{ \lambda_1 + \lambda_2 y_t }\]

Value Functions¶

``vt``: Value function \(\hat{V}\) over time before technology jump.
``v_post_techt``: Value function \(\hat{V}^L\) over time after technology jump.

Adjustment Functions¶

``ht``: Uncertainty adjustment \(h_y\) for temperature anomaly.
``hkt``: Uncertainty adjustment \(h_k\) for capital.
``hjt``: Uncertainty adjustment \(h_r\) for knowledge capital.
``gt_dmg``: Damage jump probability adjustments \(g^\ell\) over time for different damage models.
``gt_tech``: Technology jump probability adjustment \(f^L\) over time.

Probability Variables¶

``distorted_damage_prob``: Distorted probability of damage jumps over time \(J^\ell \cdot g^\ell\).
``distorted_tech_prob``: Distorted probability of technology jumps over time \(J^L \cdot f^L\).
``true_tech_prob``: True probability of technology jumps over time \(J^\ell\).
``true_damage_prob``: True probability of damage jumps over time \(J^L\).
``TA``: Temperature anomaly \(Y_t\) over time.

Relative Entropy Variables¶

``RelativeEntropy_hk``: Relative entropy related to capital uncertainty adjustment.

\[\frac{1}{2} \times \xi_k \times h_k^2\]

``RelativeEntropy_hY``: Relative entropy related to temperature anomaly uncertainty adjustment.

\[\frac{1}{2} \times \xi_c \times h_t^2\]

``RelativeEntropy_hj``: Relative entropy related to knowledge capital uncertainty adjustment.

\[\frac{1}{2} \times \xi_j \times h_j^2\]

``RelativeEntropy_TechJump``: Relative entropy related to technology jumps.

\[\xi_j \times J^L \left(1 - f + f \times \log(f)\right)\]

``RelativeEntropy_DamageJump``: Relative entropy related to damage jumps.

\[\xi_d \times \sum_\ell J^\ell \left(1 - g^\ell + g^\ell \times \log(g^\ell)\right)\]

3.1 Sample Simulation¶

from src.plot import plot_simulated_stoc_path_full2_selected

plot_simulated_stoc_path_full2_selected("RD_Plot", "Figure 22: R&D Investment as Percentage of GDP","%", [0,10])

plot_simulated_stoc_path_full2_selected("e", """Figure 23: Simulated Pathways of Emissions""","", [6,25])

plot_simulated_stoc_path_full2_selected("LogSVRD_Plot", """Figure 24: Simulated Pathways of the Social Value of R&D  <br> Exponential of vertical axis gives economic value in units of consumption""","", [4,10])

plot_simulated_stoc_path_full2_selected("LogSCGW_Plot", """Figure 25: Simulated Pathways of the Social Cost of Global Warming  <br> Exponential of vertical axis gives economic value in units of consumption""","", [6,14])

plot_simulated_stoc_path_full2_selected("TA", """Figure 26: Simulated Pathways of Temperature Anomaly""","", [0,3])

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

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

plot_simulated_stoc_path_full2_selected("distorted_tech_prob", "Figure 27: Distorted Probability of a Technology Change Jump","", [0,1])

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

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

plot_simulated_stoc_path_full2_selected("distorted_damage_prob", "Figure 28: Distorted Probability of a Damage Jump","", [0,1])

plot_simulated_stoc_path_full2_selected("gamma3", "Figure 29: Damage Realization","", [0,1/3])

3.2 Stochastic Simulation with More Pathways¶

from src.plot import plot_simulated_stoc_path_full2_split2


plot_simulated_stoc_path_full2_split2("RD_Plot", "Figure 30: R&D Investment as Percentage of GDP","%",
                                      [0,10])

plot_simulated_stoc_path_full2_split2("e", """Figure 31: Simulated Pathways of Emissions""","", [6,25])

plot_simulated_stoc_path_full2_split2("LogSVRD_Plot", """Figure 32: Simulated Pathways of the Social Value of R&D  <br> Exponential of vertical axis gives economic value in units of consumption""","", [6,25])

plot_simulated_stoc_path_full2_split2("LogSCGW_Plot", """Figure 33: Simulated Pathways of the Social Cost of Global Warming  <br> Exponential of vertical axis gives economic value in units of consumption""","", [6,14])

plot_simulated_stoc_path_full2_split2("TA", """Figure 34: Simulated Pathways of Temperature Anomaly""","", [0,3])

plot_simulated_stoc_path_full2_split2("distorted_tech_prob", "Figure 35: Distorted Probability of a Technology Change Jump","", [0,1])

plot_simulated_stoc_path_full2_split2("distorted_damage_prob", "Figure 36: Distorted Probability of a Damage Jump","", [0,1])

plot_simulated_stoc_path_full2_split2("gamma3", "Figure 37: Damage Realization","", [0,1/3])