__author__ = 'Randall'
from demos.setup import np, plt, demo
from compecon import DDPmodel
from compecon.tools import getindex
# DEMDDP01 Mine management model
# Model Parameters
price = 1 # price of ore
sbar = 100 # initial ore stock
delta = 0.9 # discount factor
# State Space
S = np.arange(sbar + 1) # vector of states
n = S.size # number of states
# Action Space
X = np.arange(sbar + 1) # vector of actions
m = X.size # number of actions
# Reward Function
f = np.full((m, n), -np.inf)
for c, s in enumerate(S):
for r, x in enumerate(X):
if x <= s:
f[r, c] = price * x - (x ** 2) / (1 + s)
# State Transition Function
g = np.empty_like(f)
from compecon import DDPmodel
from compecon.tools import gridmake, getindex
## DEMDDP06 Bioeconomic model
# Model Parameters
T = 10 # foraging periods
emax = 8 # energy capacity
e = [2, 4, 4] # energy from foraging
p = [1.0, 0.7, 0.8] # predation survival probabilities
q = [0.5, 0.8, 0.7] # foraging success probabilities
# State Space
S = np.arange(emax + 1) # energy levels
n = S.size # number of states
# Action Space
X = np.arange(3) + 1 # vector of actions
m = X.size # number of actions
# Reward Function
f = np.zeros((m, n))
# State Transition Probability Matrix
P = np.zeros((m, n, n))
for k in range(m):
P[k, 0, 0] = 1
i = range(1, n)
# does not survive predation
# In[4]:
nplot = 1001
x = np.linspace(a, b, nplot)
y = runge(x)
# Plot Runge's Function
# Initialize data matrices
# In[5]:
n = np.arange(3, 33, 2)
nn = n.size
errunif, errcheb = (np.zeros([nn, nplot]) for k in range(2))
nrmunif, nrmcheb, conunif, concheb = (np.zeros(nn) for k in range(4))
# Compute approximation errors on refined grid and interpolation matrix condition numbers
# In[6]:
for i in range(nn):
# Uniform-node monomial-basis approximant
xnodes = np.linspace(a, b, n[i])
c = np.polyfit(xnodes, runge(xnodes), n[i])
yfit = np.polyval(c, x)
nplot = 1001
x = np.linspace(a, b, nplot)
y = runge(x)
# Plot Runge's Function
fig1 = plt.figure(figsize=[6, 9])
ax1 = fig1.add_subplot(211,
title="Runge's Function",
xlabel='',
ylabel='y',
xticks=[])
ax1.plot(x, y)
ax1.text(-0.8, 0.8, r'$y = \frac{1}{1+25x^2}$', fontsize=18)
# Initialize data matrices
n = np.arange(3, 33, 2)
nn = n.size
errunif, errcheb = (np.zeros([nn, nplot]) for k in range(2))
nrmunif, nrmcheb, conunif, concheb = (np.zeros(nn) for k in range(4))
# Compute approximation errors on refined grid and interpolation matrix condition numbers
for i in range(nn):
# Uniform-node monomial-basis approximant
xnodes = np.linspace(a, b, n[i])
c = np.polyfit(xnodes, runge(xnodes), n[i])
yfit = np.polyval(c, x)
phi = xnodes.reshape(-1, 1)**np.arange(n[i])
errunif[i] = yfit - y
nrmunif[i] = np.log10(norm(yfit - y, np.inf))
conunif[i] = np.log10(cond(phi, 2))
g[k] = getindex(snext, S)
# Model Structure
model = DDPmodel(f, g, delta)
model.solve()
## Analysis
# Plot Optimal Policy
demo.figure('Optimal Harvest Policy', 'Stock', 'Harvest')
plt.plot(S, X[model.policy])
# Plot Value Function
demo.figure('Optimal Value Function', 'Stock', 'Value')
plt.plot(S, model.value)
# Simulate Model
nyrs = 20
t = np.arange(nyrs + 1)
spath, xpath = model.simulate(n - 1, nyrs)
# Plot State Path
demo.figure('Optimal State Path', 'Year', 'Stock')
plt.plot(t, S[spath])
# Plot Optimal Transition Function
demo.figure('Optimal State Transition Function', 'S(t)', 'S(t+1)')
ii, jj = np.where(model.transition)
plt.plot(S[ii], S[jj], S, S, '--')
plt.show()
__author__ = 'Randall' from demos.setup import np, plt, demo from compecon import DDPmodel from compecon.tools import gridmake, getindex # DEMDDP03 Asset replacement model with maintenance # Model Parameters maxage = 5 # maximum asset age repcost = 75 # replacement cost mancost = 10 # maintenance cost delta = 0.9 # discount factor # State Space s1 = np.arange(1, 1 + maxage) # asset age s2 = s1 - 1 # servicings S = gridmake(s1,s2) # combined state grid S1, S2 = S n = S1.size # total number of states # Action Space X = ['no action', 'service', 'replace'] # vector of actions m = len(X) # number of actions # Reward Function f = np.zeros((m, n)) q = 50 - 2.5 * S1 - 2.5 * S1 ** 2 f[0] = q * np.minimum(1, 1 - (S1 - S2) / maxage) f[1] = q * np.minimum(1, 1 - (S1 - S2 - 1) / maxage) - mancost f[2] = 50 - repcost
__author__ = 'Randall'
from demos.setup import np, plt, demo
from compecon import DDPmodel
## DEMDDP09 Deterministic cow replacement model
# Model Parameters
delta = 0.9 # discount factor
cost = 500 # replacement cost
price = 150 # milk price
# State Space
S = np.arange(1, 11) # lactation states
n = S.size # number of states
# Action Space (keep='K', replace='R')
X = ['Keep', 'Replace'] # keep or replace
m = len(X) # number of actions
# Reward Function
f = np.empty((m, n))
y = -0.2 * S**2 + 2 * S + 8 # yield per lactation
f[0] = price * y
f[1] = f[0] - cost
f[0, -1] = -np.inf # force replace at lactation 10
# State Transition Function
g = np.ones_like(f)
g[0] = np.minimum(np.arange(n) + 1,
n - 1) # Raise lactation number by 1, if keep
model = DDPmodel(f, g, delta)
model.solve()
## Analysis
# Plot Optimal Policy
demo.figure('Optimal Harvest Policy', 'Stock', 'Harvest')
plt.plot(S,X[model.policy])
# Plot Value Function
demo.figure('Optimal Value Function', 'Stock', 'Value')
plt.plot(S,model.value)
# Simulate Model
nyrs = 20
t = np.arange(nyrs + 1)
spath, xpath = model.simulate(n - 1, nyrs)
# Plot State Path
demo.figure('Optimal State Path', 'Year', 'Stock')
plt.plot(t, S[spath])
# Plot Optimal Transition Function
demo.figure('Optimal State Transition Function', 'S(t)', 'S(t+1)')
ii, jj = np.where(model.transition)
plt.plot(S[ii], S[jj], S, S, '--')
plt.show()
__author__ = 'Randall' from demos.setup import np, plt, demo from compecon import DDPmodel ## DEMDDP02 Asset replacement model # Model Parameters maxage = 5 # maximum machine age repcost = 75 # replacement cost delta = 0.9 # discount factor # State Space S = np.arange(1, 1 + maxage) # machine age n = S.size # number of states # Action Space (keep=1, replace=2) X = ['keep', 'replace'] # vector of actions m = len(X) # number of actions # Reward Function f = np.zeros((m, n)) f[0] = 50 - 2.5 * S - 2.5 * S**2 f[1] = 50 - repcost f[0, -1] = -np.inf # State Transition Function g = np.zeros_like(f) g[0] = np.arange(1, n + 1) g[0, -1] = n - 1 # adjust last state so it doesn't go out of bounds
from demos.setup import np, plt, tic, toc
from numpy.linalg import solve
from scipy.sparse.linalg import spsolve
from scipy.sparse import csc_matrix
""" Sparse linear equations"""
AA = np.random.rand(1000, 1000)
bb = np.random.rand(1000, 1)
for i in range(1000):
for j in range(1000):
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
__author__ = 'Randall' from demos.setup import np, plt, demo from compecon import DDPmodel ## DEMDDP09 Deterministic cow replacement model # Model Parameters delta = 0.9 # discount factor cost = 500 # replacement cost price = 150 # milk price # State Space S = np.arange(1, 11) # lactation states n = S.size # number of states # Action Space (keep='K', replace='R') X = ['Keep', 'Replace'] # keep or replace m = len(X) # number of actions # Reward Function f = np.empty((m, n)) y = -0.2 * S ** 2 + 2 * S + 8 # yield per lactation f[0] = price * y f[1] = f[0] - cost f[0, -1] = -np.inf # force replace at lactation 10 # State Transition Function g = np.ones_like(f)
## Analysis
# Plot Optimal Policy
demo.figure('Optimal Investment', 'Wealth', 'Investment')
plt.plot(S, X[model.policy] * S)
# Plot Optimal Policy
demo.figure('Optimal Consumption', 'Wealth', 'Consumption')
plt.plot(S, S - X[model.policy] * S)
# Plot Value Function
demo.figure('Optimal Value Function', 'Wealth', 'Value')
plt.plot(S, model.value)
# Simulate Model
nyrs = 20
t = np.arange(0, nyrs + 1)
st, xt = model.simulate(smin, nyrs)
# Plot State Path
demo.figure('Optimal State Path', 'Year', 'Wealth')
plt.plot(t, S[st])
# Compute Steady State Distribution and Mean
pi = model.markov()
avgstock = np.dot(S, pi)
print('Steady-state Wealth {:8.2f}, {:8.2f}'.format(*avgstock))
plt.show()
# Model Parameters
T = 0.5 # years to expiration
sigma = 0.2 # annual volatility
r = 0.05 # annual interest rate
strike = 2.1 # option strike price
p0 = 2.0 # current asset price
# Discretization Parameters
N = 100 # number of time intervals
tau = T / N # length of time intervals
delta = np.exp(-r * tau) # discount factor
u = np.exp(sigma * np.sqrt(tau)) # up jump factor
q = 0.5 + np.sqrt(tau) * (r - (sigma**2) / 2) / (2 * sigma) # up jump probability
# State Space
price = p0 * u ** np.arange(-N, N+1) # asset prices
n = price.size # number of states
# Action Space (hold=1, exercise=2)
X = ['hold', 'exercise'] # vector of actions
m = len(X) # number of actions
# Reward Function
f = np.zeros((m,n))
f[1] = strike - price
# State Transition Probability Matrix
P = np.zeros((m, n, n))
for i in range(n):
P[0, i, min(i + 1, n - 1)] = q
# State Transition Probability Matrix
P = np.zeros((m, n, n))
P[0, :, 0] = 1 # remains unemployed
P[1, 0, 0] = 1 - pfind # finds no job
P[1, 0, 1] = pfind # finds job
P[1, 1, 0] = pfire # gets fired
P[1, 1, 1] = 1 - pfire # keeps job
# Model Structure
model = DDPmodel(f, P, delta)
## Solution
# Solve Bellman Equation
wage = np.arange(55, 66)
xtable = np.zeros((wage.size, 2), dtype=int)
for i, w in enumerate(wage):
model.reward[1, 1] = w # vary wage
xtable[i] = model.solve().policy # solve via policy iteration
## Analysis
# Display Optimal Policy
print(' Optimal Job Search Strategy')
print(' (1=innactive, 2=active) ')
print(' Wage Unemployed Employed ')
print(*['{:4d} {:10s}{:10s}\n'.format(w, X[u], X[e]) for w, (u, e) in zip(wage, xtable)])
from compecon.tools import gridmake, getindex
# DEMDDP05 Water management model
# Model Parameters
alpha1 = 14 # producer benefit function parameter
beta1 = 0.8 # producer benefit function parameter
alpha2 = 10 # recreational user benefit function parameter
beta2 = 0.4 # recreational user benefit function parameter
maxcap = 30 # maximum dam capacity
r = np.array([0, 1, 2, 3, 4]) # rain levels
p = np.array([0.1, 0.2, 0.4, 0.2, 0.1]) # rain probabilities
delta = 0.9 # discount factor
# State Space
S = np.arange(1 + maxcap) # vector of states
n = S.size # number of states
# Action Space
X = np.arange(1 + maxcap) # vector of actions
m = X.size # number of actions
# Reward Function
f = np.full((m, n), -np.inf)
for k in range(m):
f[k, k:] = alpha1 * X[k]**beta1 + alpha2 * (S[k:] - X[k])**beta2
# State Transition Probability Matrix
P = np.zeros((m, n, n))
for k in range(m):
model = DDPmodel(f, g, delta).solve()
## Analysis
# Plot Optimal Policy
demo.figure('Optimal Investment', 'Wealth', 'Investment')
plt.plot(S, X[model.policy] * S)
# Plot Optimal Policy
demo.figure('Optimal Consumption', 'Wealth', 'Consumption')
plt.plot(S, S - X[model.policy] * S)
# Plot Value Function
demo.figure('Optimal Value Function', 'Wealth', 'Value')
plt.plot(S, model.value)
# Simulate Model
nyrs = 20
t = np.arange(0, nyrs + 1)
st, xt = model.simulate(smin, nyrs)
# Plot State Path
demo.figure('Optimal State Path', 'Year', 'Wealth')
plt.plot(t, S[st])
# Compute Steady State Distribution and Mean
pi = model.markov()
avgstock = np.dot(S, pi)
print('Steady-state Wealth {:8.2f}, {:8.2f}'.format(*avgstock))
plt.show()
__author__ = 'Randall' from demos.setup import np, plt, demo from compecon import DDPmodel from compecon.tools import gridmake ## DEMDDP10 Stochastic cow replacement model # Model Parameters delta = 0.9 # discount factor cost = 500 # replacement cost price = 150 # milk price # State Space s1 = np.arange(10) + 1 # lactation states s2 = np.array([0.8, 1.0, 1.2]) # productivity states n1 = s1.size # number of lactation states n2 = s2.size # number of productivity states S1, S2 = gridmake(s1,s2) # combined state grid n = n1 * n2 # total number of states # Action Space (keep='K', replace='R') X = np.array(['Keep','Replace']) # keep or replace m = X.size # number of actions # Reward Function f = np.empty((m, n)) y = (-0.2 * S1 ** 2 + 2 * S1 +8) * S2 # yield per lactation f[0] = price * y f[1] = f[0] - cost
# State Transition Probability Matrix
P = np.zeros((m, n, n))
P[0, :, 0] = 1 # remains unemployed
P[1, 0, 0] = 1 - pfind # finds no job
P[1, 0, 1] = pfind # finds job
P[1, 1, 0] = pfire # gets fired
P[1, 1, 1] = 1 - pfire # keeps job
# Model Structure
model = DDPmodel(f, P, delta)
## Solution
# Solve Bellman Equation
wage = np.arange(55, 66)
xtable = np.zeros((wage.size, 2), dtype=int)
for i, w in enumerate(wage):
model.reward[1, 1] = w # vary wage
xtable[i] = model.solve().policy # solve via policy iteration
## Analysis
# Display Optimal Policy
print(' Optimal Job Search Strategy')
print(' (1=innactive, 2=active) ')
print(' Wage Unemployed Employed ')
print(*[
'{:4d} {:10s}{:10s}\n'.format(w, X[u], X[e])
for w, (u, e) in zip(wage, xtable)
__author__ = 'Randall' from demos.setup import np, plt, demo from compecon import DDPmodel from compecon.tools import gridmake ## DEMDDP10 Stochastic cow replacement model # Model Parameters delta = 0.9 # discount factor cost = 500 # replacement cost price = 150 # milk price # State Space s1 = np.arange(10) + 1 # lactation states s2 = np.array([0.8, 1.0, 1.2]) # productivity states n1 = s1.size # number of lactation states n2 = s2.size # number of productivity states S1, S2 = gridmake(s1, s2) # combined state grid n = n1 * n2 # total number of states # Action Space (keep='K', replace='R') X = np.array(['Keep', 'Replace']) # keep or replace m = X.size # number of actions # Reward Function f = np.empty((m, n)) y = (-0.2 * S1**2 + 2 * S1 + 8) * S2 # yield per lactation f[0] = price * y f[1] = f[0] - cost f[0, S1 == 10] = -np.inf # force replace at lactation 10
# Model Parameters
A = 6 # maximum asset age
alpha = np.array([50, -2.5, -2.5]) # production function coefficients
kappa = 40 # net replacement cost
pbar = 1 # long-run mean unit profit
gamma = 0.5 # unit profit autoregression coefficient
sigma = 0.15 # standard deviation of unit profit shock
delta = 0.9 # discount factor
# Continuous State Shock Distribution
m = 5 # number of unit profit shocks
[e,w] = qnwnorm(m,0,sigma ** 2) # unit profit shocks and probabilities
# Deterministic Discrete State Transitions
h = np.zeros((2, A))
h[0, :-1] = np.arange(1, A)
# Approximation Structure
n = 200 # number of collocation nodes
pmin = 0 # minimum unit profit
pmax = 2 # maximum unit profit
basis = BasisSpline(n, pmin, pmax, labels=['unit profit']) # basis functions
# Model Structure
def profit(p, x, i, j):
a = i + 1
if j or a == A:
return p * 50 - kappa
else:
return p * (alpha[0] + alpha[1] * a + alpha[2] * a ** 2 )
__author__ = 'Randall' from demos.setup import np, plt, demo from compecon import DDPmodel ## DEMDDP02 Asset replacement model # Model Parameters maxage = 5 # maximum machine age repcost = 75 # replacement cost delta = 0.9 # discount factor # State Space S = np.arange(1, 1 + maxage) # machine age n = S.size # number of states # Action Space (keep=1, replace=2) X = ['keep', 'replace'] # vector of actions m = len(X) # number of actions # Reward Function f = np.zeros((m, n)) f[0] = 50 - 2.5 * S - 2.5 * S ** 2 f[1] = 50 - repcost f[0, -1] = -np.inf # State Transition Function g = np.zeros_like(f)
