for k in range(n // 2):
xii = xi[(2 * k):(2 * k + 3)]
xiii = linspace(xii[0], xii[2], 125)
p = fitquad(xii)
plt.fill_between(xiii, p(xiii), color='yellow')
if k == 0:
plt.plot(xiii,
p(xiii),
'r--',
label=r'$\tilde{f}_{%d}(x)$' % (n + 1))
else:
plt.plot(xiii, p(xiii), 'r--')
plt.vlines(xi, 0, f(xi), 'k', linestyle=':')
plt.hlines(0, xmin - 0.1, xmax + 0.1, 'k', linewidth=2)
plt.xlim(xmin - 0.1, xmax + 0.1)
xtl = ['$x_{%d}$' % i for i in range(n + 1)]
xtl[0] += '=a'
xtl[n] += '=b'
plt.xticks(xi, xtl)
plt.yticks([0], ['0'])
plt.legend()
return fig
# In[6]:
figs = [plot_simp(n) for n in [2, 4, 8]]
demo.savefig(figs)
n = 1000
x = np.linspace(a, b, n)
r = resid(F.c)
# ### Plot function inverse
# In[6]:
fig1 = demo.figure('Implicit Function', 'x', 'f(x)')
plt.plot(x, F(x))
# ### Plot residual
# In[7]:
fig2 = demo.figure('Functional Equation Residual', 'x', 'Residual')
plt.hlines(0, a, b, 'k', '--')
plt.plot(x, r)
# In[8]:
demo.savefig([fig1, fig2])
B = BasisChebyshev(n,a,b) basisplot(x, B.Phi(x).T,'Chebychev Polynomial Basis Functions on [0,1]') nodeplot(B.nodes,'Chebychev Nodes on [0,1]') # ### % Plot linear spline basis functions and nodes # In[8]: L = BasisSpline(n,a,b,k=1) basisplot(x, L.Phi(x).T.toarray(),'Linear Spline Basis Functions on [0,1]') nodeplot(L.nodes,'Linear Spline Standard Nodes on [0,1]') # ### % Plot cubic spline basis functions and nodes # In[9]: C = BasisSpline(n,a,b,k=3) basisplot(x, C.Phi(x).T.toarray(),'Cubic Spline Basis Functions on [0,1]') nodeplot(C.nodes,'Cubic Spline Standard Nodes on [0,1]') # In[10]: demo.savefig(figures)
v += w[k] * np.maximum(K - np.exp(pnext), delta * Value(pnext))
Value[:] = v
pcrit[t + 1] = f.broyden(pcrit[t])
# ### Print Critical Exercise Price 300 Periods to Expiration
# In[9]:
print('Critical Price = %5.2f' % np.exp(pcrit[-1]))
# ### Plot Critical Exercise Prices
# In[10]:
fig1 = demo.figure('American Put Option Optimal Exercise Boundary',
'Periods Remaining Until Expiration',
'Exercise Price')
plt.plot(np.exp(pcrit))
# In[ ]:
demo.savefig([fig1])
results.set_index(['Integral', 'Nodes n'], inplace=True)
results
# ## Plot the functions
# In[7]:
a, b, n = -1, 1, 301
x = np.linspace(a, b, n)
plt.figure(figsize=[10,4])
demo.subplot(1, 2, 1, '$e^{-x}$','','',[a,b],[0,f(a)])
plt.plot(x, f(x), linewidth=3)
plt.xticks([-1,0,1])
plt.yticks([0])
demo.subplot(1, 2, 2, '$\sqrt{|x|}$','','',[a,b],[0,g(a)])
plt.plot(x, g(x), linewidth=3)
plt.xticks([-1,0,1])
plt.yticks([0])
# In[8]:
results.to_latex('figures/demqua03.tex', escape=False, float_format='%.1f')
demo.savefig([plt.gcf()])
print('\nErgodic Standard Deviations')
print(ff.format('Wage', data['wage'].std()))
print(ff.format('Employment', (data['i'] == 'employed').std()))
# ### Plot Expected Discrete State Path
# In[15]:
data.head()
# In[16]:
data['ii'] = data['i'] == 'employed'
fig3 = demo.figure('Probability of Employment', 'Period', 'Probability')
plt.plot(data[['ii', 'time']].groupby('time').mean())
# ### Plot Simulated and Expected Continuous State Path
# In[17]:
subdata = data[data['_rep'].isin(range(3))]
fig4 = demo.figure('Simulated and Expected Wage', 'Period', 'Wage')
plt.plot(subdata.pivot('time', '_rep', 'wage'))
plt.plot(data[['time', 'wage']].groupby('time').mean(), 'k--', label='mean')
# In[18]:
demo.savefig([fig1, fig2, fig3, fig4])
# Next, for each possible price shock, we compute next period log-price by adding the shock to current log-prices (the nodes of the Value object). Then, we use each next-period price to compute the expected value of an option with one-period to maturity (save the values in ```v```). We update the value function to reflect the new time-to-maturity and use ```broyden``` to solve for the critical value. We repeat this procedure until we reach the $T=300$ horizon.
# In[8]:
for t in range(T):
v = np.zeros((1, n))
for k in range(m):
pnext = Value.nodes + e[k]
v += w[k] * np.maximum(K - np.exp(pnext), delta * Value(pnext))
Value[:] = v
pcrit[t + 1] = f.broyden(pcrit[t])
# ### Print Critical Exercise Price 300 Periods to Expiration
# In[9]:
print('Critical Price = %5.2f' % np.exp(pcrit[-1]))
# ### Plot Critical Exercise Prices
# In[10]:
fig1 = demo.figure('American Put Option Optimal Exercise Boundary',
'Periods Remaining Until Expiration', 'Exercise Price')
plt.plot(np.exp(pcrit))
# In[ ]:
demo.savefig([fig1])
plt.plot(x, f(x), linewidth=3, label=r'$f(x)$')
for k in range(n//2):
xii = xi[(2*k):(2*k+3)]
xiii = linspace(xii[0], xii[2], 125)
p = fitquad(xii)
plt.fill_between(xiii, p(xiii), color='yellow')
if k==0:
plt.plot(xiii, p(xiii),'r--', label=r'$\tilde{f}_{%d}(x)$' % (n+1))
else:
plt.plot(xiii, p(xiii),'r--')
plt.vlines(xi, 0, f(xi),'k', linestyle=':')
plt.hlines(0,xmin-0.1, xmax+0.1,'k',linewidth=2)
plt.xlim(xmin-0.1, xmax+0.1)
xtl = ['$x_{%d}$' % i for i in range(n+1)]
xtl[0] += '=a'
xtl[n] += '=b'
plt.xticks(xi, xtl)
plt.yticks([0],['0'])
plt.legend()
return fig
# In[6]:
figs = [plot_simp(n) for n in [2, 4, 8]]
demo.savefig(figs)
data.head()
# In[18]:
subdata = data[data['_rep'].isin(range(3))]
# In[19]:
fig3 = demo.figure('Simulated and Expected Price', 'Period','Net Unit Profit')
demo.qplot('time','unit profit','_rep',subdata)
plt.plot(data[['time','unit profit']].groupby('time').mean(),'k--',label='mean')
plt.legend()
# ### Plot Expected Discrete State Path
# In[20]:
fig4 = demo.figure('Expected Machine Age', 'Period','Age')
demo.qplot('time','age','_rep',subdata)
plt.plot(data[['time','age']].groupby('time').mean(),'k--',label='mean')
plt.legend()
demo.savefig([fig1,fig2,fig3,fig4])
Phi = np.array([x**j for j in np.arange(n)]) basisplot(x, Phi, 'Monomial Basis Functions on [0,1]') # ### % Plot Chebychev basis functions and nodes # In[7]: B = BasisChebyshev(n, a, b) basisplot(x, B.Phi(x).T, 'Chebychev Polynomial Basis Functions on [0,1]') nodeplot(B.nodes, 'Chebychev Nodes on [0,1]') # ### % Plot linear spline basis functions and nodes # In[8]: L = BasisSpline(n, a, b, k=1) basisplot(x, L.Phi(x).T.toarray(), 'Linear Spline Basis Functions on [0,1]') nodeplot(L.nodes, 'Linear Spline Standard Nodes on [0,1]') # ### % Plot cubic spline basis functions and nodes # In[9]: C = BasisSpline(n, a, b, k=3) basisplot(x, C.Phi(x).T.toarray(), 'Cubic Spline Basis Functions on [0,1]') nodeplot(C.nodes, 'Cubic Spline Standard Nodes on [0,1]') # In[10]: demo.savefig(figures)
# ## Two-sided finite difference derivative
# In[5]:
d2 = deriv_error(x - h, x + h)
e2 = np.log10(eps**(1 / 3))
# ## Plot finite difference derivatives
# In[6]:
ylim = [-15, 5]
xlim = [-15, 0]
lcolor = [z['color'] for z in plt.rcParams['axes.prop_cycle']]
demo.figure('Error in Numerical Derivatives', '$\log_{10}(h)$',
'$\log_{10}$ Approximation Error', xlim, ylim)
plt.plot(c, d1, label='One-Sided')
plt.plot(c, d2, label='Two-Sided')
plt.vlines([e1, e2], *ylim, lcolor, linestyle=':')
plt.xticks(np.arange(-15, 5, 5))
plt.yticks(np.arange(-15, 10, 5))
demo.annotate(e1, 2, '$\sqrt{\epsilon}$', color=lcolor[0], ms=0)
demo.annotate(e2, 2, '$\sqrt[3]{\epsilon}$', color=lcolor[1], ms=0)
plt.legend(loc='lower left')
# In[7]:
demo.savefig([plt.gcf()])
# ### Plot Residuals
# In[13]:
S['resid2'] = 100*S.resid / S.value
fig2 = demo.figure('Bellman Equation Residual','','Percent Residual')
S['resid2'].plot(ax=plt.gca())
plt.hlines(0,0,smax,'k')
# ### Simulation
#
# The path followed by the biomass is computed by the ```simulate()``` method. Here we simulate 32 periods starting with a biomass level $s_0 = 0$.
# In[14]:
H = model.simulate(32, 0.0)
fig3 = demo.figure('Timber harvesting simulation','Period','Biomass')
H['biomass'].plot(ax=plt.gca())
# In[15]:
demo.savefig([fig1, fig2, fig3])