from compecon.tools import nodeunif from mpl_toolkits.mplot3d import Axes3D from matplotlib import cm # Univariate Taylor approximation x = np.linspace(-1, 1, 100) y = (x + 1) * np.exp(2 * x) y1 = 1 + 3 * x y2 = 1 + 3 * x + 8 * x**2 plt.figure(figsize=[6, 6]) plt.plot(x, y, 'k', linewidth=3, label='Function') plt.plot(x, y1, 'b', linewidth=3, label='1st order approximation') plt.plot(x, y2, 'r', linewidth=3, label='2nd order approximation') plt.legend() plt.xticks([-1, 0, 1]) plt.show() ## Bivariate Taylor approximation nplot = [101, 101] a = [0, -1] b = [2, 1] x1, x2 = nodeunif(nplot, a, b) x1.shape = nplot x2.shape = nplot y = np.exp(x2) * x1**2 y1 = 2 * x1 - x2 - 1 y2 = x1**2 - 2 * x1 * x2 + 0.5 * x2**2 + x2
S = BasisSpline(n, a, b, f=ff)
L = BasisSpline(n, a, b, k=1, f=ff)
# Compute actual and fitted values on grid
y = ff(x) # actual
yc = C(x) # Chebychev approximant
ys = S(x) # cubic spline approximant
yl = L(x) # linear spline approximant
# Plot function approximations
plt.figure()
ymin = np.floor(y.min())
ymax = np.ceil(y.max())
xlim = [a, b]
ylim = [-0.2, 1.2] if ifunc == 2 else [ymin, ymax]
subfig(1, x, y, xlim, ylim, 'Function')
subfig(2, x, yc, xlim, ylim, 'Chebyshev')
subfig(3, x, ys, xlim, ylim, 'Cubic Spline')
subfig(4, x, yl, xlim, ylim, 'Linear Spline')
# Plot function approximation error
plt.figure()
plt.plot(x, np.c_[yc - y, ys - y], linewidth=3)
plt.xlabel('x')
plt.ylabel('Approximation Error')
plt.legend(['Chebychev Polynomial', 'Cubic Spline'])
# pltlegend('boxoff')
plt.show()
model = DPmodel(basis,
reward,
transition,
discount=delta,
j=['keep', 'clear cut'])
model.solve()
resid, sr, vr = model.residuals()
# Plot Action-Contingent Value Functions
demo.figure('Action-Contingent Value Functions', 'Biomass', 'Value of Stand')
plt.plot(sr, vr.T)
plt.legend(model.labels.j, loc='upper center')
# Compute and Plot Optimal Harvesting Stock Level
scrit = np.interp(0.0, vr[1] - vr[0], sr)
vcrit = np.interp(scrit, sr, vr[0])
demo.annotate(scrit,
vcrit,
'$s^* = {:.2f}$'.format(scrit),
'wo', (-5, 5),
fs=12)
print('Optimal Biomass Harvesting Level = {:5.2f}'.format(scrit))
# Plot Residual
demo.figure('Bellman Equation Residual', 'Biomass', 'Percent Residual')
plt.plot(sr, 100 * resid.T / vr.max(0).T)
vlq, plq = growth_model.Value(S.Wealth, order)
klq = growth_model.Policy(S.Wealth)
# ============ Compute Analytic Solution
vtrue = vstar + b * (np.log(S.Wealth) - np.log(sstar))
# ============== Make plots:
Wealth = S.Wealth.T
# Plot Optimal Policy
demo.figure('Optimal Investment Policy', 'Wealth', 'Investment')
plt.plot(Wealth, np.c_[k, klq])
demo.annotate(sstar, kstar,'$s^*$ = %.2f\n$k^*$ = %.2f' % (sstar, kstar), 'bo', (10, -7))
plt.legend(['Chebychev Collocation','L-Q Approximation'], loc = 'upper left')
# Plot Value Function
demo.figure('Value Function', 'Wealth', 'Value')
plt.plot(Wealth, np.c_[v, vlq])
demo.annotate(sstar, vstar,'$s^*$ = %.2f\n$V^*$ = %.2f' % (sstar, vstar),'bo', (10, -7))
plt.legend(['Chebychev Collocation','L-Q Approximation'], loc= 'upper left')
# Plot Shadow Price Function
demo.figure('Shadow Price Function', 'Wealth', 'Shadow Price')
plt.plot(Wealth, np.c_[pr, plq])
demo.annotate(sstar, pstar,'$s^*$ = %.2f\n$\lambda^*$ = %.2f' % (sstar, pstar), 'bo', (10, 7))
plt.legend(['Chebychev Collocation','L-Q Approximation'])
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')
plt.show()
plt.plot(s, v.T)
# Plot Shadow Price Function
demo.figure('Shadow Price Function', 'Stock', 'Shadow Price')
plt.plot(s, model.Value(s, 1).T)
# Plot Residual
demo.figure('Bellman Equation Residual', 'Stock', 'Residual')
plt.plot(s, resid.T)
plt.hlines(0, smin, smax, 'k', '--')
## SIMULATION
# Simulate Model
T = 15
data = model.simulate(T, smin)
print(data)
# Plot State and Policy Paths
opts = dict(spec='r*', offset=(0, -5), fs=11, ha='right')
data[['available', 'harvest']].plot()
demo.annotate(T, sstar, 'steady-state stock = %.2f' % sstar, **opts)
demo.annotate(T, qstar, 'steady-state harvest = %.2f' % qstar, **opts)
plt.xlim([0, T + 0.25])
plt.title('State and Policy Paths')
plt.xlabel('Period')
plt.ylabel('Stock / Harvest')
plt.legend(['Stock', 'Harvest'], loc='right')
plt.show()
# Plot Shadow Price Function
demo.figure('Shadow Price Function', 'Stock', 'Shadow Price')
plt.plot(s, model.Value(s, 1).T)
# Plot Residual
demo.figure('Bellman Equation Residual','Stock', 'Residual')
plt.plot(s, resid.T)
plt.hlines(0, smin, smax, 'k', '--')
## SIMULATION
# Simulate Model
T = 15
data = model.simulate(T, smin)
print(data)
# Plot State and Policy Paths
opts = dict(spec='r*', offset=(0,-5), fs=11, ha='right')
data[['available', 'harvest']].plot()
demo.annotate(T, sstar, 'steady-state stock = %.2f' % sstar, **opts)
demo.annotate(T, qstar, 'steady-state harvest = %.2f' % qstar, **opts)
plt.xlim([0, T + 0.25])
plt.title('State and Policy Paths')
plt.xlabel('Period')
plt.ylabel('Stock / Harvest')
plt.legend(['Stock','Harvest'], loc='right')
plt.show()
## Analysis
# Display Optimal Policy
xtemp = model.policy.reshape((n1, n2))
header = '{:^8s} {:^8s} {:^8s} {:^8s}'.format('Age', 'Lo', 'Med', 'Hi')
print('Optimal Policy')
print(header)
print(*('{:^8d} {:^8s} {:^8s} {:^8s}\n'.format(s, *X[x])
for s, x in zip(s1, xtemp)))
# Plot Value Function
demo.figure('Optimal Replacement Value', 'Age', 'Optimal Value (thousands)')
plt.plot(s1, model.value.reshape((n1, n2)) / 1000)
plt.legend(['Low', 'Med', 'Hi'])
# Compute Steady-State distribution
pi = model.markov().reshape((n1, n2))
# Display Steady-State distribution
print(' Invariant Distribution ')
print(header)
print(*('{:^8d} {:8.3f} {:8.3f} {:8.3f}\n'.format(s, *x)
for s, x in zip(s1, pi)))
# Compute Steady-State Mean Cow Age and Productivity
pi = pi.flatten()
avgage = np.dot(pi.T, S1)
avgpri = np.dot(pi.T, S2)
print('Steady-state Age = {:8.2f}'.format(avgage))
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=[]
# In[11]:
## Analysis
# Display Optimal Policy
xtemp = model.policy.reshape((n1, n2))
header = '{:^8s} {:^8s} {:^8s} {:^8s}'.format('Age','Lo', 'Med', 'Hi')
print('Optimal Policy')
print(header)
print(*('{:^8d} {:^8s} {:^8s} {:^8s}\n'.format(s, *X[x]) for s, x in zip(s1, xtemp)))
# Plot Value Function
demo.figure('Optimal Replacement Value', 'Age', 'Optimal Value (thousands)')
plt.plot(s1, model.value.reshape((n1,n2)) / 1000)
plt.legend(['Low','Med','Hi'])
# Compute Steady-State distribution
pi = model.markov().reshape((n1, n2))
# Display Steady-State distribution
print(' Invariant Distribution ')
print(header)
print(*('{:^8d} {:8.3f} {:8.3f} {:8.3f}\n'.format(s, *x) for s, x in zip(s1, pi)))
# Compute Steady-State Mean Cow Age and Productivity
pi = pi.flatten()
avgage = np.dot(pi.T, S1)
# 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')
plt.show()
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 = []
# In[11]:
print('Critical Search Wage = {:5.1f}'.format(wcrit0))
wcrit1 = np.interp(0, vr[1, 1] - vr[1, 0], sr)
vcrit1 = np.interp(wcrit1, sr, vr[1, 0])
print('Critical Quit Wage = {:5.1f}'.format(wcrit1))
# Plot Action-Contingent Value Function - Unemployed
demo.figure('Action-Contingent Value, Unemployed', 'Wage', 'Value')
plt.plot(sr, vr[0].T)
demo.annotate(wcrit0,
vcrit0,
'$w^*_0 = {:.1f}$'.format(wcrit0),
'wo', (5, -5),
fs=12)
plt.legend(['Do Not Search', 'Search'], loc='upper left')
# Plot Action-Contingent Value Function - Unemployed
demo.figure('Action-Contingent Value, Employed', 'Wage', 'Value')
plt.plot(sr, vr[1].T)
demo.annotate(wcrit1,
vcrit1,
'$w^*_0 = {:.1f}$'.format(wcrit1),
'wo', (5, -5),
fs=12)
plt.legend(['Quit', 'Work'], loc='upper left')
# Plot Residual
demo.figure('Bellman Equation Residual', 'Wage', 'Percent Residual')
L = BasisSpline(n, a, b, k=1, f=ff)
# Compute actual and fitted values on grid
y = ff(x) # actual
yc = C(x) # Chebychev approximant
ys = S(x) # cubic spline approximant
yl = L(x) # linear spline approximant
# Plot function approximations
plt.figure()
ymin = np.floor(y.min())
ymax = np.ceil(y.max())
xlim = [a, b]
ylim = [-0.2, 1.2] if ifunc==2 else [ymin, ymax]
subfig(1, x, y, xlim, ylim, 'Function')
subfig(2, x, yc, xlim, ylim, 'Chebyshev')
subfig(3, x, ys, xlim, ylim, 'Cubic Spline')
subfig(4, x, yl, xlim, ylim, 'Linear Spline')
# Plot function approximation error
plt.figure()
plt.plot(x, np.c_[yc - y, ys - y], linewidth=3)
plt.xlabel('x')
plt.ylabel('Approximation Error')
plt.legend(['Chebychev Polynomial','Cubic Spline'])
# pltlegend('boxoff')
plt.show()