def newPlot(title, Y, k):
ax = fig.add_subplot(1, 3, k, projection='3d')
ax.plot_surface(x1,
x2,
Y,
rstride=1,
cstride=1,
cmap=cm.coolwarm,
linewidth=0,
antialiased=False)
ax.set_xlabel('$x_1$')
ax.set_ylabel('$x_2$')
ax.set_zlabel('$y$')
plt.title(title)
nrep = 10000
sinit = np.full(nrep, pbar)
iinit = 0
data = model.simulate(T,sinit,iinit, seed=945)
# Print Ergodic Moments
frm = '\t{:<10s} = {:5.2f}'
print('\nErgodic Means')
print(frm.format('Price', data['unit profit'].mean()))
print(frm.format('Age', data.i.mean()))
print('\nErgodic Standard Deviations')
print(frm.format('Price', data['unit profit'].std()))
print(frm.format('Age', data.i.std()))
# Plot Simulated and Expected Continuous State Path
print(demo.qplot('time', 'unit profit', '_rep',
data=data[data['_rep'] < 3],
geom='line',
main='Simulated and Expected Price',
ylab='Net Unit Profit',
xlab='Period'))
# Plot Expected Discrete State Path
data[['time', 'i']].groupby('time').mean().plot(legend=False)
plt.title('Expected Machine Age')
plt.xlabel('Period')
plt.ylabel('Age')
plt.show()
def subfig(k, x, y, xlim, ylim, title):
plt.subplot(2, 2, k)
plt.plot(x, y)
plt.xlim(xlim)
plt.ylim(ylim)
plt.title(title)
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')
plt.xlabel('Year')
plt.ylabel('Reservoir Level')
# Plot Simulated Policy Path
D = data[data['_rep'] < 3][['time', 'released', '_rep']]
D.pivot('time', '_rep', 'released').plot(legend=False, lw=1)
data.groupby('time')['released'].mean().plot(color='k', linestyle='--')
plt.title('Simulated and Expected Irrigation')
plt.xlabel('Year')
plt.ylabel('Irrigation')
# Print Steady-State and Ergodic Moments
ff = '\t%15s = %5.2f'
D = data[data['time'] == T]
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()
# Plot Chebychev Collocation Residual and Approximation Error
plt.figure(figsize=[12, 6])
demo.subplot(1, 2, 1, 'Chebychev Collocation Residual\nand Approximation Error', 'Wealth', 'Residual/Error')
plt.plot(Wealth, np.c_[S.resid, v - vtrue], Wealth, np.zeros_like(Wealth), 'k--')
plt.legend(['Residual','Error'], loc='lower right')
# Plot Linear-Quadratic Approximation Error
demo.subplot(1, 2, 2, 'Linear-Quadratic Approximation Error', 'Wealth', 'Error')
plt.plot(Wealth, vlq - vtrue)
# Plot State and Policy Paths
opts = dict(spec='r*', offset=(-5, -5), fs=11, ha='right')
data[['Wealth', 'Investment']].plot()
plt.title('State and Policy Paths')
demo.annotate(T, sstar, 'steady-state wealth\n = %.2f' % sstar, **opts)
demo.annotate(T, kstar, 'steady-state investment\n = %.2f' % kstar, **opts)
plt.xlabel('Period')
plt.ylabel('Wealth/Investment')
plt.xlim([0, T + 0.5])
# Print Steady-State
print('\n\nSteady-State')
print(' Wealth = %5.4f' % sstar)
print(' Investment = %5.4f' % kstar)
plt.show()
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')
plt.xlabel('Year')
plt.ylabel('Reservoir Level')
# Plot Simulated Policy Path
D = data[data['_rep'] < 3][['time', 'released', '_rep']]
D.pivot('time', '_rep', 'released').plot(legend=False, lw=1)
data.groupby('time')['released'].mean().plot(color='k', linestyle='--')
plt.title('Simulated and Expected Irrigation')
plt.xlabel('Year')
plt.ylabel('Irrigation')
# Print Steady-State and Ergodic Moments
ff = '\t%15s = %5.2f'
D = data[data['time'] == T]
# 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()
X2.shape = nplot
fig = plt.figure(figsize=[15, 6])
ax = fig.add_subplot(1, 2, 1, projection='3d')
ax.plot_surface(X1,
X2,
error,
rstride=1,
cstride=1,
cmap=cm.coolwarm,
linewidth=0,
antialiased=False)
ax.set_xlabel('$x_1$')
ax.set_ylabel('$x_2$')
ax.set_zlabel('error')
plt.title('Chebychev Approximation Error')
# The plot indicates that an order 11 by 11 Chebychev approximation scheme
# produces approximation errors no bigger in magnitude than 1x10^-10.
# Let us repeat the approximation exercise, this time constructing an order
# 21 by 21 cubic spline approximation scheme:
n = [21, 21] # order of approximation
S = BasisSpline(n, a, b, f=f)
yapp = S(X) # approximant values at grid nodes
error = (yapp - yact).reshape(nplot)
ax = fig.add_subplot(1, 2, 2, projection='3d')
ax.plot_surface(X1,
X2,
error,
rstride=1,
yc = C(x) # values
dc = C(x, 1) # first derivative
sc = C(x, 2) # second derivative
# Construct and evaluate cubic spline interpolant
S = BasisSpline(n, a, b, f=f) # chose basis functions
ys = S(x) # values
ds = S(x, 1) # first derivative
ss = S(x, 2) # second derivative
# Plot function approximation error
plt.figure()
plt.subplot(2, 1, 1),
plt.plot(x, y - yc[0])
plt.ylabel('Chebychev')
plt.title('Function Approximation Error')
plt.subplot(2, 1, 2)
plt.plot(x, y - ys[0])
plt.ylabel('Cubic Spline')
plt.xlabel('x')
# Plot first derivative approximation error
plt.figure()
plt.subplot(2, 1, 1),
plt.plot(x, d - dc[0])
plt.ylabel('Chebychev')
plt.title('First Derivative Approximation Error')
plt.subplot(2, 1, 2)
plt.plot(x, d - ds[0], 'm')
if abs(i - j) > 1:
AA[i,j] = 0
n = np.hstack((np.arange(50, 250, 50), np.arange(300, 1100, 100)))
ratio = np.empty(n.size)
for k in range(n.size):
A = AA[:n[k], :n[k]]
b = bb[:n[k]]
tt = tic()
for i in range(100):
x = solve(A, b)
toc1 = toc(tt)
S = csc_matrix(A)
tt = tic()
for i in range(100):
x = spsolve(S, b)
toc2 = toc(tt)
ratio[k] = toc2 / toc1
# Plot effort ratio
plt.figure(figsize=[6, 6])
plt.plot(n, ratio)
plt.xlabel('n')
plt.ylabel('Ratio')
plt.title('Ratio of Sparse Solve Time to Full Solve Time')
plt.show()
q1 = quad(lambda x: np.abs(f(x)-g(x)) ** p1, a, b)[0] ** (1 / p1)
q2 = quad(lambda x: np.abs(f(x)-g(x)) ** p2, a, b)[0] ** (1 / p2)
print('\nCompute function metrics')
print('\tnorm 1 = {:6.4f}, norm 2 = {:6.4f}'.format(q1, q2))
# Illustrate function metrics
x = np.linspace(a, b, 200)
plt.figure(figsize=[12, 4])
plt.subplot(1, 2, 1)
plt.plot([0, 2], [0, 0], 'k:', linewidth=4)
plt.plot(x, f(x) - g(x), 'b', linewidth=4, label='f - g')
plt.xlabel('x')
plt.ylabel('y')
plt.xticks([0, 1, 2])
plt.yticks([-1, 0, 1, 2, 3])
plt.title('f - g')
plt.subplot(1, 2, 2)
plt.plot(x, np.abs(f(x) - g(x)), 'b', linewidth=4, label='f - g')
plt.xlabel('x')
plt.ylabel('y')
plt.xticks([0, 1, 2])
plt.yticks([0, 1, 2, 3])
plt.title('|f - g|')
plt.show()
# Demonstrate Pythagorean Theorem
a, b = -1, 1
f = lambda x: 2 * x**2 - 1
g = lambda x: 4 * x**3 -3*x
# 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)
data.groupby('time')['wealth'].mean().plot(color='k', linestyle='--')
plt.title('Simulated and Expected Wealth')
plt.xlabel('Period')
plt.ylabel('Wealth')
# Plot Simulated Policy Path
subdata = data[data['_rep'] < 3][['time', 'investment', '_rep']]
subdata.pivot(index='time', columns='_rep', values='investment').plot(legend=False, lw=1)
plt.plot(data.groupby('time')['investment'].mean(), 'k-')
plt.title('Simulated and Expected Investment')
plt.xlabel('Period')
plt.ylabel('Investment')
# Print Steady-State and Ergodic Moments
return np.r_[d.flatten(), s_0 - s0, s_T]
storage = NLP(resid, F.c.flatten(), tnodes, T, n, F, r, k, eta, s0)
c = storage.broyden(print=True)
F.c = np.reshape(c, (2, n))
nplot = 501
t = np.linspace(0, T, nplot)
(p, s), (dp, ds) = F(t, [[0, 1]])
res_p = dp - r * p - k
res_s = ds + p**-eta
plt.figure()
plt.subplot(2, 1, 1)
plt.plot(t, res_p)
plt.title('Residuals')
plt.ylabel('d(price) residual')
plt.subplot(2, 1, 2)
plt.plot(t, res_s)
plt.xlabel('time')
plt.ylabel('d(storage) residual')
plt.figure()
plt.subplot(2, 1, 1)
plt.plot(t, p)
plt.title('Solution')
plt.ylabel('Price')
plt.subplot(2, 1, 2)
plt.plot(t, s)
dc = C(x, 1) # first derivative
sc = C(x, 2) # second derivative
# Construct and evaluate cubic spline interpolant
S = BasisSpline(n, a, b, f=f) # chose basis functions
ys = S(x) # values
ds = S(x, 1) # first derivative
ss = S(x, 2) # second derivative
# Plot function approximation error
plt.figure()
plt.subplot(2, 1, 1),
plt.plot(x, y - yc[0])
plt.ylabel('Chebychev')
plt.title('Function Approximation Error')
plt.subplot(2, 1, 2)
plt.plot(x, y - ys[0])
plt.ylabel('Cubic Spline')
plt.xlabel('x')
# Plot first derivative approximation error
plt.figure()
plt.subplot(2, 1, 1),
plt.plot(x, d - dc[0])
plt.ylabel('Chebychev')
plt.title('First Derivative Approximation Error')
plt.subplot(2, 1, 2)
return np.r_[d.flatten(), s_0 - s0, s_T]
storage = NLP(resid, F.c.flatten(), tnodes, T, n, F, r, k, eta, s0)
c = storage.broyden(print=True)
F.c = np.reshape(c, (2, n))
nplot = 501
t = np.linspace(0, T, nplot)
(p, s), (dp, ds) = F(t, [[0, 1]])
res_p = dp - r * p - k
res_s = ds + p ** -eta
plt.figure()
plt.subplot(2, 1, 1)
plt.plot(t, res_p)
plt.title('Residuals')
plt.ylabel('d(price) residual')
plt.subplot(2, 1, 2)
plt.plot(t, res_s)
plt.xlabel('time')
plt.ylabel('d(storage) residual')
plt.figure()
plt.subplot(2, 1, 1)
plt.plot(t, p)
plt.title('Solution')
plt.ylabel('Price')
plt.subplot(2, 1, 2)
## 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)
data.groupby('time')['wealth'].mean().plot(color='k', linestyle='--')
plt.title('Simulated and Expected Wealth')
plt.xlabel('Period')
plt.ylabel('Wealth')
# Plot Simulated Policy Path
subdata = data[data['_rep'] < 3][['time', 'investment', '_rep']]
subdata.pivot(index='time', columns='_rep',
values='investment').plot(legend=False, lw=1)
plt.plot(data.groupby('time')['investment'].mean(), 'k-')
plt.title('Simulated and Expected Investment')
plt.xlabel('Period')
plt.ylabel('Investment')
# Print Steady-State and Ergodic Moments
ff = '\t%15s = %5.2f'
def subfig(k, x, y, xlim, ylim, title):
plt.subplot(2, 2, k)
plt.plot(x, y)
plt.xlim(xlim)
plt.ylim(ylim)
plt.title(title)