# Compute and Plot Critical Unit Profit Contributions
pcrit = [NLP(lambda s: model.Value_j(s)[i].dot([1,-1])).broyden(0.0)[0] for i in range(A)]
vcrit = [model.Value(s)[i] for i, s in enumerate(pcrit)]
# In[13]:
fig1 = demo.figure('Action-Contingent Value Functions', 'Net Unit Profit','Value', figsize=[10,5])
cc = np.linspace(0.3,0.9,model.dims.ni)
for a, i in enumerate(dstates):
plt.plot(S.loc[i,'value[keep]'] ,color=plt.cm.Blues(cc[a]), label='Keep ' + i)
if pmin < pcrit[a] < pmax:
demo.annotate(pcrit[a], vcrit[a], f'$p^*_{a+1}$', 'wo',(0, 0), fs=11, ms=18)
print(f'Age {a+1:d} Profit {pcrit[a]:5.2f}')
plt.plot(S.loc['a=1','value[replace]'], color=plt.cm.Oranges(0.5),label='Replace')
plt.legend()
# ### Plot Residual
# In[14]:
S['resid2'] = 100*S['resid'] / S['value']
fig2 = demo.figure('Bellman Equation Residual','Net Unit Profit','Percent Residual')
demo.qplot('unit profit','resid2','i',S)
print(f'Critical Search Wage = {wcrit0:5.1f}')
wcrit1, vcrit1 = critical(S.loc['employed'])
print(f'Critical Quit Wage = {wcrit1:5.1f}')
# ### Plot Action-Contingent Value Function
# In[10]:
vv = ['value[idle]', 'value[active]']
fig1 = plt.figure(figsize=[12, 4])
# UNEMPLOYED
demo.subplot(1, 2, 1, 'Action-Contingent Value, Unemployed', 'Wage', 'Value')
plt.plot(S.loc['unemployed', vv])
demo.annotate(wcrit0, vcrit0, f'$w^*_0 = {wcrit0:.1f}$', 'wo', (5, -5), fs=12)
plt.legend(['Do Not Search', 'Search'], loc='upper left')
# EMPLOYED
demo.subplot(1, 2, 2, 'Action-Contingent Value, Employed', 'Wage', 'Value')
plt.plot(S.loc['employed', vv])
demo.annotate(wcrit1, vcrit1, f'$w^*_0 = {wcrit1:.1f}$', 'wo', (5, -5), fs=12)
plt.legend(['Quit', 'Work'], loc='upper left')
# ### Plot Residual
# In[11]:
S['resid2'] = 100 * (S['resid'] / S['value'])
fig2 = demo.figure('Bellman Equation Residual', 'Wage', 'Percent Residual')
plt.plot(S.loc['unemployed', 'resid2'])
offs = [(4, -8), (-8, 5)]
msgs = ['Profit Entry', 'Profit Exit']
for i,iname in enumerate(dstates):
plt.axes(axs[i])
subdata = S.loc[iname,['value[close]', 'value[open]']]
plt.plot(subdata)
plt.title('Value of a currently ' + iname + ' firm')
plt.xlabel('Potential Profit')
if not i: plt.ylabel('Value of Firm')
plt.legend(dactions)
subdata['v.diff'] = subdata['value[open]'] - subdata['value[close]']
pcrit = np.interp(0, subdata['v.diff'], subdata.index)
vcrit = np.interp(pcrit, subdata.index, subdata['value[close]'])
demo.annotate(pcrit, vcrit, f'$p^*_{i:d} = {pcrit:.2f}$', 'wo', offs[i], fs=11, ms=12,)
print(f'{msgs[i]:12s} = {pcrit:5.2f}')
# ### Plot Residual
#
# We normalize the residuals as percentage of the value function. Notice the spikes at the "Profit entry" and "Profit exit" points.
# In[10]:
S['resid2'] = 100 * (S.resid / S.value)
fig2 = demo.figure('Bellman Equation Residual','Potential Profit','Percent Residual')
demo.qplot('profit','resid2','i',S)
# ### Plot Conditional Value Functions
# In[10]:
fig1 =demo.figure('Conditional Value Functions','Biomass','Value of Stand')
plt.plot(ss,vhat0(cc,ss),label='Grow')
plt.plot(ss,vhat1(cc,ss),label='Clear-Cut')
plt.legend()
vcrit = vhat(cc,scrit)
ymin = plt.ylim()[0]
plt.vlines(scrit, ymin,vcrit,'grey',linestyles='--')
demo.annotate(scrit,ymin,'$s^*$',ms=10)
demo.bullet(scrit,vcrit)
print(f'Optimal Biomass Harvest Level = {scrit:.4f}')
# ### Plot Value Function Residual
# In[ ]:
fig2 = demo.figure('Bellman Equation Residual', 'Biomass', 'Percent Residual')
plt.plot(ss, 100*resid(cc,ss) / vhat(cc,ss))
plt.hlines(0,0,smax,linestyles='--')
plt.plot(snodes,resid(cc),'ro')
ss = np.linspace(0, smax, 1000)
# ### Plot Conditional Value Functions
# In[10]:
fig1 = demo.figure('Conditional Value Functions', 'Biomass', 'Value of Stand')
plt.plot(ss, vhat0(cc, ss), label='Grow')
plt.plot(ss, vhat1(cc, ss), label='Clear-Cut')
plt.legend()
vcrit = vhat(cc, scrit)
ymin = plt.ylim()[0]
plt.vlines(scrit, ymin, vcrit, 'grey', linestyles='--')
demo.annotate(scrit, ymin, '$s^*$', ms=10)
demo.bullet(scrit, vcrit)
print(f'Optimal Biomass Harvest Level = {scrit:.4f}')
# ### Plot Value Function Residual
# In[ ]:
fig2 = demo.figure('Bellman Equation Residual', 'Biomass', 'Percent Residual')
plt.plot(ss, 100 * resid(cc, ss) / vhat(cc, ss))
plt.hlines(0, 0, smax, linestyles='--')
plt.plot(snodes, resid(cc), 'ro')
# ### Compute ergodic mean annual harvest
# In[ ]:
plt.axes(axs[i])
subdata = S.loc[iname, ['value[close]', 'value[open]']]
plt.plot(subdata)
plt.title('Value of a currently ' + iname + ' firm')
plt.xlabel('Potential Profit')
if not i: plt.ylabel('Value of Firm')
plt.legend(dactions)
subdata['v.diff'] = subdata['value[open]'] - subdata['value[close]']
pcrit = np.interp(0, subdata['v.diff'], subdata.index)
vcrit = np.interp(pcrit, subdata.index, subdata['value[close]'])
demo.annotate(
pcrit,
vcrit,
f'$p^*_{i:d} = {pcrit:.2f}$',
'wo',
offs[i],
fs=11,
ms=12,
)
print(f'{msgs[i]:12s} = {pcrit:5.2f}')
# ### Plot Residual
#
# We normalize the residuals as percentage of the value function. Notice the spikes at the "Profit entry" and "Profit exit" points.
# In[10]:
S['resid2'] = 100 * (S.resid / S.value)
fig2 = demo.figure('Bellman Equation Residual', 'Potential Profit',
'Percent Residual')
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()])
fig1 = demo.figure('Action-Contingent Value Functions',
'Net Unit Profit',
'Value',
figsize=[10, 5])
cc = np.linspace(0.3, 0.9, model.dims.ni)
for a, i in enumerate(dstates):
plt.plot(S.loc[i, 'value[keep]'],
color=plt.cm.Blues(cc[a]),
label='Keep ' + i)
if pmin < pcrit[a] < pmax:
demo.annotate(pcrit[a],
vcrit[a],
f'$p^*_{a+1}$',
'wo', (0, 0),
fs=11,
ms=18)
print(f'Age {a+1:d} Profit {pcrit[a]:5.2f}')
plt.plot(S.loc['a=1', 'value[replace]'],
color=plt.cm.Oranges(0.5),
label='Replace')
plt.legend()
# ### Plot Residual
# In[14]:
S['resid2'] = 100 * S['resid'] / S['value']
# ## 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()])
plt.plot(prices, resid(S.c))
# ### Plot demand and effective supply for m=1, 3, 5, 10, 15, 20 firms
# In[10]:
fig3 = demo.figure('Industry Supply and Demand Functions',
'Quantity', 'Price', [0, 12], figsize=[9,4])
lcolor = [z['color'] for z in plt.rcParams['axes.prop_cycle']]
for i, m in enumerate([1, 3, 5, 10, 15, 20]):
plt.plot(m*S(prices), prices) # supply
demo.annotate(m*S(1.2)-0.25,1.4-i/12,f'm={m:d}',color=lcolor[i],ms=0,fs=12)
plt.plot(D(prices), prices, linewidth=4, color='grey') # demand
demo.annotate(10,0.5,'demand',color='grey', ms=0, fs=12)
# ### Plot equilibrium price as a function of number of firms m
# In[14]:
pp = (b + a) / 2
dp = (b - a) / 2
m = np.arange(1, 26)
for i in range(50):
dp /= 2
n, a, z = 11, 0, 1
def f(x):
return np.sqrt(1/(2*np.pi))*np.exp(-0.5*x**2)
# In[3]:
x, w = qnwsimp(n, a, z)
prob = 0.5 + w.dot(f(x))
# In[4]:
a, b, n = -4, 4, 500
x = np.linspace(a, b, n)
xz = np.linspace(a, z, n)
plt.figure(figsize=[8,4])
plt.fill_between(xz,f(xz), color='yellow')
plt.hlines(0, a, b,'k','solid')
plt.vlines(z, 0, f(z),'r',linewidth=2)
plt.plot(x,f(x), linewidth=3)
demo.annotate(-1, 0.08,r'$\Pr\left(\tilde Z\leq z\right)$',fs=18,ms=0)
plt.yticks([])
plt.xticks([z],['$z$'],size=20)
demo.savefig([plt.gcf()])
wcrit1, vcrit1 = critical(S.loc['employed'])
print(f'Critical Quit Wage = {wcrit1:5.1f}')
# ### Plot Action-Contingent Value Function
# In[10]:
vv = ['value[idle]','value[active]']
fig1 = plt.figure(figsize=[12,4])
# UNEMPLOYED
demo.subplot(1,2,1,'Action-Contingent Value, Unemployed', 'Wage', 'Value')
plt.plot(S.loc['unemployed',vv])
demo.annotate(wcrit0, vcrit0, f'$w^*_0 = {wcrit0:.1f}$', 'wo', (5, -5), fs=12)
plt.legend(['Do Not Search', 'Search'], loc='upper left')
# EMPLOYED
demo.subplot(1,2,2,'Action-Contingent Value, Employed', 'Wage', 'Value')
plt.plot(S.loc['employed',vv])
demo.annotate(wcrit1, vcrit1, f'$w^*_0 = {wcrit1:.1f}$', 'wo',(5, -5), fs=12)
plt.legend(['Quit', 'Work'], loc='upper left')
# ### Plot Residual
# In[11]:
return 50 - cos(pi*x)*(2*pi*x - pi + 0.5)**2 # In[3]: xmin, xmax = 0, 1 ymin, ymax = 25, 65 a, b, n = 0.25, 0.75, 401 # In[4]: z = linspace(a,b,n) x = linspace(xmin, xmax, n) plt.figure(figsize=[8,4]) plt.fill_between(z,ymin,f(z), color='yellow') plt.hlines(ymin, xmin, xmax, 'k',linewidth=2) plt.vlines(a, ymin, f(a), 'r',linestyle='--',linewidth=2) plt.vlines(b, ymin, f(b), 'r',linestyle='--',linewidth=2) plt.plot(x,f(x), linewidth=3) plt.xlim([xmin, xmax]) plt.ylim([ymin-5, ymax]) plt.yticks([ymin], ['0'], size=20) plt.xticks([a, b],['$a$', '$b$'],size=20) demo.annotate(xmax-0.1, f(xmax)-9, r'$f(x)$',fs=18,ms=0) demo.annotate((a+b)/2, ymin+10 ,r'$A = \int_a^bf(x)dx$',fs=18,ms=0) demo.savefig([plt.gcf()])