import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
L_0 = 1
n = 0.01
L = [L_0]
for t in range(200):
L = L + [L[t]*np.exp(n)]
solow_df = pd.DataFrame()
solow_df['L'] = L
solow_df.L.plot()
plt.show()
E_0 = 1
g = 0.015
E = [E_0]
for t in range(200):
E = E + [E[t]*np.exp(g)]
solow_df['E'] = E
solow_df.plot()
plt.show()
(2.1.2) $ Y = \kappa^\theta E L $
(2.2.2) $ \ln(Y) = \theta\ln(\kappa) + \ln(L) + \ln(E) $
(2.2.3) $ \frac{1}{Y}\frac{dY}{dt} = g_Y = \theta \left( \frac{1}{\kappa}\frac{d\kappa}{dt} \right) + \frac{1}{L}\frac{dL}{dt} + \frac{1}{E}\frac{dE}{dt} $
(2.1.14) $ \frac{dK}{dt} = sY - \delta K = \left( \frac{s}{\kappa} - \delta \right)K $
(2.2.5) $ \frac{1}{K}\frac{dK}{dt} = g_{K} = \frac{s}{\kappa} - \delta $
$ g_Y = \theta g_{\kappa} + n + g $
So the proportional rate of growth of capital-intensity $ \kappa $ is:
(2.2.8) $ \frac{1}{\kappa}\frac{d\kappa}{dt} = g_\kappa = g_K - g_Y = \frac{s}{\kappa} - \delta - \left( \frac{\theta}{1+\theta} \right)\left( \frac{s}{\kappa} - \delta \right) - \frac{n + g}{1+\theta} $
(2.2.9) $ \frac{1}{\kappa}\frac{d\kappa}{dt} = \frac{s/\kappa - (n+g+\delta)}{1+\theta} $
κ_0 = 8
κ = [κ_0]
s = 0.20
δ = 0.025
θ = 1
for t in range(200):
κ = κ + [κ[t]*(1 + (s/κ[t] - (n+g+δ))/(1+θ))]
solow_df['κ'] = κ
solow_df.κ.plot()
plt.show()