def K_to_F(self, K):
"""
Compute agent 1's best value-maximizing response F, given K.
Parameters
----------
K : array_like(float, ndim=2)
A j x n array
Returns
-------
F : array_like(float, ndim=2)
The policy function for a given K
P : array_like(float, ndim=2)
The value function for a given K
"""
A1 = self.A + dot(self.C, K)
B1 = self.B
Q1 = self.Q
R1 = self.R - self.beta * self.theta * dot(K.T, K)
lq = LQ(Q1, R1, A1, B1, beta=self.beta)
P, F, d = lq.stationary_values()
return F, P
def F_to_K(self, F):
"""
Compute agent 2's best cost-minimizing response K, given F.
Parameters
----------
F : array_like(float, ndim=2)
A k x n array
Returns
-------
K : array_like(float, ndim=2)
Agent's best cost minimizing response for a given F
P : array_like(float, ndim=2)
The value function for a given F
"""
Q2 = self.beta * self.theta
R2 = - self.R - dot(F.T, dot(self.Q, F))
A2 = self.A - dot(self.B, F)
B2 = self.C
lq = LQ(Q2, R2, A2, B2, beta=self.beta)
P, neg_K, d = lq.stationary_values()
return - neg_K, P
def robust_rule(self):
"""
This method solves the robust control problem by tricking it
into a stacked LQ problem, as described in chapter 2 of Hansen-
Sargent's text "Robustness." The optimal control with observed
state is
.. math::
u_t = - F x_t
And the value function is -x'Px
Returns
-------
F : array_like(float, ndim=2)
The optimal control matrix from above
P : array_like(float, ndim=2)
The psoitive semi-definite matrix defining the value
function
K : array_like(float, ndim=2)
the worst-case shock matrix K, where
:math:`w_{t+1} = K x_t` is the worst case shock
"""
# == Simplify names == #
A, B, C, Q, R = self.A, self.B, self.C, self.Q, self.R
beta, theta = self.beta, self.theta
k, j = self.k, self.j
# == Set up LQ version == #
I = identity(j)
Z = np.zeros((k, j))
Ba = hstack([B, C])
Qa = vstack([hstack([Q, Z]), hstack([Z.T, -beta*I*theta])])
lq = LQ(Qa, R, A, Ba, beta=beta)
# == Solve and convert back to robust problem == #
P, f, d = lq.stationary_values()
F = f[:k, :]
K = -f[k:f.shape[0], :]
return F, K, P
def K_to_F(self, K):
"""
Compute agent 1's best value-maximizing response F, given K.
Parameters
==========
K : array_like
A self.j x self.n array
Returns
=======
F : array_like, dtype = float
P : array_like, dtype = float
"""
A1 = self.A + dot(self.C, K)
B1 = self.B
Q1 = self.Q
R1 = self.R - self.beta * self.theta * dot(K.T, K)
lq = LQ(Q1, R1, A1, B1, beta=self.beta)
P, F, d = lq.stationary_values()
return F, P
def F_to_K(self, F):
"""
Compute agent 2's best cost-minimizing response K, given F.
Parameters
==========
F : array_like
A self.k x self.n array
Returns
=======
K : array_like, dtype = float
P : array_like, dtype = float
"""
Q2 = self.beta * self.theta
R2 = - self.R - dot(F.T, dot(self.Q, F))
A2 = self.A - dot(self.B, F)
B2 = self.C
lq = LQ(Q2, R2, A2, B2, beta=self.beta)
P, neg_K, d = lq.stationary_values()
return - neg_K, P
def K_to_F(self, K):
"""
Compute agent 1's best value-maximizing response F, given K.
Parameters
==========
K : array_like
A self.j x self.n array
Returns
=======
F : array_like, dtype = float
P : array_like, dtype = float
"""
A1 = self.A + dot(self.C, K)
B1 = self.B
Q1 = self.Q
R1 = self.R - self.beta * self.theta * dot(K.T, K)
lq = LQ(Q1, R1, A1, B1, beta=self.beta)
P, F, d = lq.stationary_values()
return F, P
def F_to_K(self, F):
"""
Compute agent 2's best cost-minimizing response K, given F.
Parameters
==========
F : array_like
A self.k x self.n array
Returns
=======
K : array_like, dtype = float
P : array_like, dtype = float
"""
Q2 = self.beta * self.theta
R2 = -self.R - dot(F.T, dot(self.Q, F))
A2 = self.A - dot(self.B, F)
B2 = self.C
lq = LQ(Q2, R2, A2, B2, beta=self.beta)
P, neg_K, d = lq.stationary_values()
return -neg_K, P
def robust_rule(self):
"""
This method solves the robust control problem by tricking it into a
stacked LQ problem, as described in chapter 2 of Hansen-Sargent's
text "Robustness." The optimal control with observed state is
u_t = - F x_t
And the value function is -x'Px
Returns
=======
F : array_like, dtype = float
The optimal control matrix from above above
P : array_like, dtype = float
The psoitive semi-definite matrix defining the value function
K : array_like, dtype = float
the worst-case shock matrix K, where :math:`w_{t+1} = K x_t` is
the worst case shock
"""
# == Simplify names == #
A, B, C, Q, R = self.A, self.B, self.C, self.Q, self.R
beta, theta = self.beta, self.theta
k, j = self.k, self.j
# == Set up LQ version == #
I = identity(j)
Z = np.zeros((k, j))
Ba = hstack([B, C])
Qa = vstack([hstack([Q, Z]), hstack([Z.T, -beta * I * theta])])
lq = LQ(Qa, R, A, Ba, beta=beta)
# == Solve and convert back to robust problem == #
P, f, d = lq.stationary_values()
F = f[:k, :]
K = -f[k:f.shape[0], :]
return F, K, P
"""
Origin: QE by John Stachurski and Thomas J. Sargent
Filename: solution_ree_ex2.py
Authors: Chase Coleman, Spencer Lyon, Thomas Sargent, John Stachurski
Solves an exercise from the rational expectations module
"""
from __future__ import print_function
import numpy as np
from lqcontrol import LQ
from solution_ree_ex1 import beta, R, Q, B
candidates = ((94.0886298678, 0.923409232937), (93.2119845412, 0.984323478873),
(95.0818452486, 0.952459076301))
for kappa0, kappa1 in candidates:
# == Form the associated law of motion == #
A = np.array([[1, 0, 0], [0, kappa1, kappa0], [0, 0, 1]])
# == Solve the LQ problem for the firm == #
lq = LQ(Q, R, A, B, beta=beta)
P, F, d = lq.stationary_values()
F = F.flatten()
h0, h1, h2 = -F[2], 1 - F[0], -F[1]
# == Test the equilibrium condition == #
if np.allclose((kappa0, kappa1), (h0, h1 + h2)):
print('Equilibrium pair =', kappa0, kappa1)
print('(h0, h1, h2) = ', h0, h1, h2)
break
for theta in thetas:
df.ix[theta] = evaluate_policy(theta, F)
if df.ix[theta, 'entropy'] >= emax:
break
df = df.dropna(how='any')
return df
#-----------------------------------------------------------------------------#
# Main
#-----------------------------------------------------------------------------#
# == Compute the optimal rule == #
optimal_lq = LQ(Q, R, A, B, C, beta)
Po, Fo, do = optimal_lq.stationary_values()
# == Compute a robust rule given theta == #
baseline_robust = RBLQ(Q, R, A, B, C, beta, theta)
Fb, Kb, Pb = baseline_robust.robust_rule()
# == Check the positive definiteness of worst-case covariance matrix to == #
# == ensure that theta exceeds the breakdown point == #
test_matrix = np.identity(Pb.shape[0]) - np.dot(C.T, Pb.dot(C)) / theta
eigenvals, eigenvecs = eig(test_matrix)
assert (eigenvals >= 0).all(), 'theta below breakdown point.'
emax = 1.6e6
"""
from __future__ import print_function
import numpy as np
from lqcontrol import LQ
from solution_ree_ex1 import beta, R, Q, B
candidates = (
(94.0886298678, 0.923409232937),
(93.2119845412, 0.984323478873),
(95.0818452486, 0.952459076301)
)
for kappa0, kappa1 in candidates:
# == Form the associated law of motion == #
A = np.array([[1, 0, 0], [0, kappa1, kappa0], [0, 0, 1]])
# == Solve the LQ problem for the firm == #
lq = LQ(Q, R, A, B, beta=beta)
P, F, d = lq.stationary_values()
F = F.flatten()
h0, h1, h2 = -F[2], 1 - F[0], -F[1]
# == Test the equilibrium condition == #
if np.allclose((kappa0, kappa1), (h0, h1 + h2)):
print('Equilibrium pair =', kappa0, kappa1)
print('(h0, h1, h2) = ', h0, h1, h2)
break
theta_bar = 2.0
# == Useful constants == #
m0 = (a0 - c) / (2 * a1)
m1 = b / (2 * a1)
# == Formulate LQ problem == #
Q = gamma
R = [[a1, -a1, 0], [-a1, a1, 0], [0, 0, 0]]
A = [[0, 0, m0], [0, 1, 0], [0, 0, 1]]
B = [[0], [1], [0]]
C = [[m1], [0], [0]]
rlq = RBLQ(A, B, C, Q, R, beta, theta_bar)
lq = LQ(Q, R, A, B, beta=beta)
f, k, p = rlq.robust_rule()
print f
print rlq.K_to_F(k)
if 0:
F_opt, K_opt, P_opt = rlq.robust_rule_simple()
x0 = np.asarray((1, 0, 1)).reshape(3, 1)
num_thetas = 20
thetas = np.linspace(1, 5, num_thetas)
vals = np.empty((2, num_thetas))
ent = np.empty(num_thetas)
