def f(z):
x, y = z
fval = np.array([200 * x * (y - x**2) + 1 - x, 100 * (x**2 - y)])
fjac = np.array([[200 * (y - x**2) - 400 * x**2 - 1, 200 * x],
[200 * x, -100]])
return fval, fjac
def f(z):
x, y = z
fval = np.array([200 * x * (y - x ** 2) + 1 - x,
100 * (x ** 2 - y)])
fjac = np.array([[200 * (y - x ** 2) - 400 * x ** 2 - 1, 200 * x],
[200 * x, -100]])
return fval, fjac
def f(z):
x, y = z
fval = [x - x ** 2 - y ** 3,
y - x * y + 0.5]
fjac = [[1 - 2 * x, -3 * y **2],
[-y, 1 - x]]
return np.array(fval), np.array(fjac)
# delta discount factor
# Preliminary tasks
from compecon import BasisSpline, DPmodel
from compecon.quad import qnwnorm
from demos.setup import np, plt, demo
from warnings import simplefilter
simplefilter('ignore')
## FORMULATION
# 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
__author__ = 'Randall' from demos.setup import np from compecon import DDPmodel ## DEMDDP08 Job search model # Model Parameters u = 50 # weekly unemp. benefit v = 60 # weekly value of leisure pfind = 0.90 # prob. of finding job pfire = 0.10 # prob. of being fired delta = 0.99 # discount factor # State Space S = np.array([1, 2]) # vector of states n = S.size # number of states # Action Space (idle=1, active=2) X = ['idle', 'active'] # vector of actions m = len(X) # number of actions # Reward Function f = np.zeros((m, n)) f[0] = v # gets leisure f[1, 0] = u # gets benefit # 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
__author__ = 'Randall' from demos.setup import np from compecon import DDPmodel ## DEMDDP08 Job search model # Model Parameters u = 50 # weekly unemp. benefit v = 60 # weekly value of leisure pfind = 0.90 # prob. of finding job pfire = 0.10 # prob. of being fired delta = 0.99 # discount factor # State Space S = np.array([1, 2]) # vector of states n = S.size # number of states # Action Space (idle=1, active=2) X = ['idle', 'active'] # vector of actions m = len(X) # number of actions # Reward Function f = np.zeros((m, n)) f[0] = v # gets leisure f[1, 0] = u # gets benefit # State Transition Probability Matrix P = np.zeros((m, n, n)) P[0, :, 0] = 1 # remains unemployed
error = f2fit(x) - f2(x)
error.shape = nplot
x1.shape = nplot
x2.shape = nplot
fig = plt.figure(figsize=[15, 6])
ax = fig.gca(projection='3d',
title='Chebyshev Approximation Error',
xlabel='$x_1$',
ylabel='$x_2$',
zlabel='Error')
ax.plot_surface(x1,
x2,
error,
rstride=1,
cstride=1,
cmap=cm.coolwarm,
linewidth=0,
antialiased=False)
plt.show()
# Compute partial Derivatives
x = np.array([[0.5], [0.5]])
order = [[1, 0, 2, 1, 0], [0, 1, 0, 1, 2]]
ff = f2fit(x, order)
print(
'x = [0.5, 0.5]\nf1 = {:7.4f}\nf2 = {:7.4f}\nf11 = {:7.4f}\nf12 = {:7.4f}\nf22 = {:7.4f}'
.format(*ff))
f2 = lambda x: np.cos(x[0]) / np.exp(x[1])
# Set degree and domain interpolation
n, a, b = 7, 0.0, 1.0
f2fit = BasisChebyshev([n, n], a, b, f=f2)
# Nice plot of function approximation error
nplot = [101, 101]
x = nodeunif(nplot, a, b)
x1, x2 = x
error = f2fit(x) - f2(x)
error.shape = nplot
x1.shape = nplot
x2.shape = nplot
fig = plt.figure(figsize=[15, 6])
ax = fig.gca(projection='3d', title='Chebyshev Approximation Error',
xlabel='$x_1$', ylabel='$x_2$', zlabel='Error')
ax.plot_surface(x1, x2, error, rstride=1, cstride=1, cmap=cm.coolwarm,
linewidth=0, antialiased=False)
plt.show()
# Compute partial Derivatives
x = np.array([[0.5], [0.5]])
order = [[1, 0, 2, 1, 0],
[0, 1, 0, 1, 2]]
ff = f2fit(x, order)
print('x = [0.5, 0.5]\nf1 = {:7.4f}\nf2 = {:7.4f}\nf11 = {:7.4f}\nf12 = {:7.4f}\nf22 = {:7.4f}'.format(*ff))
__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
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
__author__ = 'Randall'
from demos.setup import np, plt, demo
from compecon import DDPmodel
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
# beta transition function state coefficients # gamma transition function action coefficients # omega central banker's preference weights # sbar equilibrium targets # delta discount factor # Preliminary tasks from compecon import BasisChebyshev, DPmodel from demos.setup import np, plt, demo from compecon.quad import qnwnorm ## FORMULATION # Model Parameters alpha = np.array([[0.9, -0.1]]).T # transition function constant coefficients beta = np.array([[-0.5, 0.2], [0.3, -0.4]]) # transition function state coefficients gamma = np.array([[-0.1, 0.0]]).T # transition function action coefficients omega = np.identity(2) # central banker's preference weights sbar = np.array([[1, 0]]).T # equilibrium targets sigma = 0.08 * np.identity(2), # shock covariance matrix delta = 0.9 # discount factor # Continuous State Shock Distribution m = [3, 3] # number of shocks mu = [0, 0] # means of shocks [e,w] = qnwnorm(m,mu,sigma) # shocks and probabilities # Approximation Structure n = 21 # number of collocation coordinates, per dimension
def f(x):
fval = [200 * x[0] * (x[1] - x[0] ** 2) + 1 - x[0], 100 * (x[0] ** 2 - x[1])]
fjac = [[200 * (x[1] - x[0] ** 2) - 400 * x[0] ** 2 - 1, 200 * x[0]],
[200 * x[0], -100]]
return np.array(fval), np.array(fjac)
def g(z):
x, y = z
return np.array([x **2 + y ** 3, x * y - 0.5])
# alpha transition function constant coefficients
# beta transition function state coefficients
# gamma transition function action coefficients
# omega central banker's preference weights
# sbar equilibrium targets
# delta discount factor
# Preliminary tasks
from compecon import BasisChebyshev, DPmodel
from demos.setup import np, plt, demo
from compecon.quad import qnwnorm
## FORMULATION
# Model Parameters
alpha = np.array([[0.9, -0.1]]).T # transition function constant coefficients
beta = np.array([[-0.5, 0.2],
[0.3, -0.4]]) # transition function state coefficients
gamma = np.array([[-0.1, 0.0]]).T # transition function action coefficients
omega = np.identity(2) # central banker's preference weights
sbar = np.array([[1, 0]]).T # equilibrium targets
sigma = 0.08 * np.identity(2), # shock covariance matrix
delta = 0.9 # discount factor
# Continuous State Shock Distribution
m = [3, 3] # number of shocks
mu = [0, 0] # means of shocks
[e, w] = qnwnorm(m, mu, sigma) # shocks and probabilities
# Approximation Structure
n = 21 # number of collocation coordinates, per dimension