# Plot Optimal Policy
demo.figure('Optimal Investment Policy', 'Wealth', 'Investment')
plt.plot(s, k.T)
# Plot Value Function
demo.figure('Value Function', 'Wealth', 'Value')
plt.plot(s, v.T)
# Plot Shadow Price Function
demo.figure('Shadow Price Function', 'Wealth', 'Shadow Price')
plt.plot(s, growth.Value(s, order=1).T)
# Plot Residual
demo.figure('Bellman Equation Residual', 'Wealth', 'Residual')
plt.plot(s, resid.T)
plt.hlines(0, smin, smax, 'k', '--')
## SIMULATION
# Simulate Model
T = 20
nrep = 50000
sinit = np.full((1, nrep), smin)
data = growth.simulate(T, sinit)
# Plot Simulated State Path
subdata = data[data['_rep'] < 3][['time', 'wealth', '_rep']]
subdata.pivot(index='time', columns='_rep', values='wealth').plot(legend=False,
lw=1)
# Plot Optimal Policy
demo.figure('Optimal Extraction', 'Ore Stock', 'Ore Extracted')
plt.plot(s, q.T)
# Plot Value Function
demo.figure('Value Function', 'Ore Stock', 'Value')
plt.plot(s, v.T)
# Plot Shadow Price Function
demo.figure('Shadow Price Function', 'Ore Stock', 'Shadow Price')
plt.plot(s, model.Value(s, 1))
# Plot Residual
demo.figure('Bellman Equation Residual', 'Ore Stock', 'Residual')
plt.plot(s, resid.T)
plt.hlines(0, 0, smax, 'k', '--')
## SIMULATION
# Simulate Model
T = 20
data = model.simulate(T, smax)
# Plot State and Policy Paths
data[['stock', 'extracted']].plot()
plt.title('State and Policy Paths')
plt.legend(['Stock', 'Extraction'])
plt.hlines(sstar, 0, T, 'k', '--')
plt.xlabel('Period')
plt.ylabel('Stock / Extraction')
# Plot Optimal Policy
demo.figure('Optimal Irrigation Policy', 'Reservoir Level', 'Irrigation')
plt.plot(s, x.T)
# Plot Value Function
demo.figure('Value Function', 'Reservoir Level', 'Value')
plt.plot(s, v.T)
# Plot Shadow Price Function
demo.figure('Shadow Price Function', 'Reservoir Level', 'Shadow Price')
plt.plot(s, model.Value(s, 1).T)
# Plot Residual
demo.figure('Bellman Equation Residual', 'Reservoir Level', 'Residual')
plt.plot(s,resid.T)
plt.hlines(0, smin, smax, 'k', '--')
## SIMULATION
# Simulate Model
T = 30
nrep = 100000
sinit = np.full((1, nrep), smin)
data = model.simulate(T, sinit, seed=945)
# Plot Simulated State Path
D = data[data['_rep'] < 3][['time', 'reservoir', '_rep']]
D.pivot(index='time', columns='_rep', values='reservoir').plot(legend=False, lw=1)
data.groupby('time')['reservoir'].mean().plot(color='k', linestyle='--')
plt.title('Simulated and Expected Reservoir Level')
# In[8]:
figs = []
figs.append(demo.figure("Runge's Function", '', 'y'))
plt.plot(x, y)
plt.text(-0.8, 0.8, r'$y = \frac{1}{1+25x^2}$', fontsize=18)
plt.xticks=[]
# In[9]:
figs.append(demo.figure("Runge's Function $11^{th}$-Degree\nPolynomial Approximation Error.",'x', 'Error'))
plt.hlines(0, a, b, 'gray', '--')
plt.plot(x, errcheb[4], label='Chebychev Nodes')
plt.plot(x, errunif[4], label='Uniform Nodes')
plt.legend(loc='upper center')
# Plot approximation error per degree of approximation
# In[10]:
figs.append(demo.figure("Log10 Polynomial Approximation Error for Runge's Function",'', 'Log10 Error'))
plt.plot(n, nrmcheb, label='Chebychev Nodes')
plt.plot(n, nrmunif, label='Uniform Nodes')
plt.legend(loc='upper left')
plt.xticks=[]
q = Q(p)
return p + q / (-3.5 * p**(-4.5)) - np.sqrt(q) - q**2
# Approximation structure
n, a, b = 21, 0.5, 2.5
Q = BasisChebyshev(n, a, b)
c0 = np.zeros(n)
c0[0] = 2
p = Q.nodes
# Solve for effective supply function
monopoly = NLP(resid)
Q.c = monopoly.broyden(c0)
# Setup plot
nplot = 1000
p = nodeunif(nplot, a, b)
rplot = resid(Q.c)
# Plot effective supply
demo.figure("Monopolist's Effective Supply Curve", 'Quantity', 'Price')
plt.plot(Q(p), p)
# Plot residual
demo.figure('Functional Equation Residual', 'Price', 'Residual')
plt.hlines(0, a, b, 'k', '--')
plt.plot(p, rplot)
plt.show()
# Plot Optimal Policy
demo.figure('Optimal Extraction', 'Ore Stock', 'Ore Extracted')
plt.plot(s, q.T)
# Plot Value Function
demo.figure('Value Function', 'Ore Stock', 'Value')
plt.plot(s, v.T)
# Plot Shadow Price Function
demo.figure('Shadow Price Function', 'Ore Stock', 'Shadow Price')
plt.plot(s, model.Value(s, 1))
# Plot Residual
demo.figure('Bellman Equation Residual', 'Ore Stock', 'Residual')
plt.plot(s, resid.T)
plt.hlines(0, 0, smax,'k', '--')
## SIMULATION
# Simulate Model
T = 20
data = model.simulate(T, smax)
# Plot State and Policy Paths
data[['stock', 'extracted']].plot()
plt.title('State and Policy Paths')
plt.legend(['Stock','Extraction'])
plt.hlines(sstar, 0, T, 'k', '--')
plt.xlabel('Period')
plt.ylabel('Stock / Extraction')
# In[8]:
figs = []
figs.append(demo.figure("Runge's Function", '', 'y'))
plt.plot(x, y)
plt.text(-0.8, 0.8, r'$y = \frac{1}{1+25x^2}$', fontsize=18)
plt.xticks = []
# In[9]:
figs.append(
demo.figure(
"Runge's Function $11^{th}$-Degree\nPolynomial Approximation Error.",
'x', 'Error'))
plt.hlines(0, a, b, 'gray', '--')
plt.plot(x, errcheb[4], label='Chebychev Nodes')
plt.plot(x, errunif[4], label='Uniform Nodes')
plt.legend(loc='upper center')
# Plot approximation error per degree of approximation
# In[10]:
figs.append(
demo.figure("Log10 Polynomial Approximation Error for Runge's Function",
'', 'Log10 Error'))
plt.plot(n, nrmcheb, label='Chebychev Nodes')
plt.plot(n, nrmunif, label='Uniform Nodes')
plt.legend(loc='upper left')
plt.xticks = []
# Print Ergodic Moments
ff = '\t{:12s} = {:5.2f}'
print('\nErgodic Means')
print(ff.format('Wage', data['wage'].mean()))
print(ff.format('Employment', (data['i'] == 'employed').mean()))
print('\nErgodic Standard Deviations')
print(ff.format('Wage', data['wage'].std()))
print(ff.format('Employment', (data['i'] == 'employed').std()))
# Plot Expected Discrete State Path
subdata = data[['time', 'i']]
subdata['i'] = subdata['i'] == 'employed'
subdata.groupby('time').mean().plot(legend=False)
plt.title('Probability of Employment')
plt.xlabel('Period')
plt.ylabel('Probability')
# Plot Simulated and Expected Continuous State Path
subdata = data[data['_rep'] < 3][['time', 'wage', '_rep']]
subdata.pivot(index='time', columns='_rep', values='wage').plot(legend=False,
lw=1)
plt.hlines(data['wage'].mean(), 0, T)
plt.title('Simulated and Expected Wage')
plt.xlabel('Period')
plt.ylabel('Wage')
plt.show()
