def test_ddp_sorting():
beta = 0.95
# Sorted
s_indices = [0, 0, 1]
a_indices = [0, 1, 0]
a_indptr = [0, 2, 3]
R = [0, 1, 2]
Q = [(1, 0), (1 / 2, 1 / 2), (0, 1)]
Q_sparse = sparse.csr_matrix(Q)
# Shuffled
s_indices_shuffled = [0, 1, 0]
a_indices_shuffled = [0, 0, 1]
R_shuffled = [0, 2, 1]
Q_shuffled = [(1, 0), (0, 1), (1 / 2, 1 / 2)]
Q_shuffled_sparse = sparse.csr_matrix(Q_shuffled)
ddp0 = DiscreteDP(R, Q, beta, s_indices, a_indices)
ddp_sparse = DiscreteDP(R, Q_sparse, beta, s_indices, a_indices)
ddp_shuffled = DiscreteDP(R_shuffled, Q_shuffled, beta, s_indices_shuffled,
a_indices_shuffled)
ddp_shuffled_sparse = DiscreteDP(R_shuffled, Q_shuffled_sparse, beta,
s_indices_shuffled, a_indices_shuffled)
for ddp in [ddp0, ddp_sparse, ddp_shuffled, ddp_shuffled_sparse]:
assert_array_equal(ddp.s_indices, s_indices)
assert_array_equal(ddp.a_indices, a_indices)
assert_array_equal(ddp.a_indptr, a_indptr)
assert_array_equal(ddp.R, R)
if sparse.issparse(ddp.Q):
ddp_Q = ddp.Q.toarray()
else:
ddp_Q = ddp.Q
assert_array_equal(ddp_Q, Q)
def test_ddp_beta_0():
n, m = 3, 2
R = np.array([[0, 1], [1, 0], [0, 1]])
Q = np.empty((n, m, n))
Q[:] = 1 / n
beta = 0
sigma_star = [1, 0, 1]
v_star = [1, 1, 1]
v_init = [0, 0, 0]
ddp0 = DiscreteDP(R, Q, beta)
ddp1 = ddp0.to_sa_pair_form()
ddp2 = ddp0.to_sa_pair_form(sparse=False)
methods = ['vi', 'pi', 'mpi', 'lp']
for ddp in [ddp0, ddp1, ddp2]:
for method in methods:
if method == 'lp' and ddp._sparse:
assert_raises(NotImplementedError,
ddp.solve,
method=method,
v_init=v_init)
else:
res = ddp.solve(method=method, v_init=v_init)
assert_array_equal(res.sigma, sigma_star)
assert_array_equal(res.v, v_star)
def test_ddp_to_sa_and_to_product():
n, m = 3, 2
R = np.array([[0, 1], [1, 0], [-np.inf, 1]])
Q = np.empty((n, m, n))
Q[:] = 1/n
Q[0, 0, 0] = 0
Q[0, 0, 1] = 2/n
beta = 0.95
sparse_R = np.array([0, 1, 1, 0, 1])
_Q = np.full((5, 3), 1/3)
_Q[0, 0] = 0
_Q[0, 1] = 2/n
sparse_Q = sparse.coo_matrix(_Q)
ddp = DiscreteDP(R, Q, beta)
ddp_sa = ddp.to_sa_pair_form()
ddp_sa2 = ddp_sa.to_sa_pair_form()
ddp_sa3 = ddp.to_sa_pair_form(sparse=False)
ddp2 = ddp_sa.to_product_form()
ddp3 = ddp_sa2.to_product_form()
ddp4 = ddp.to_product_form()
# make sure conversion worked
for ddp_s in [ddp_sa, ddp_sa2, ddp_sa3]:
assert_allclose(ddp_s.R, sparse_R)
# allcose doesn't work on sparse
np.max(np.abs((sparse_Q - ddp_s.Q))) < 1e-15
assert_allclose(ddp_s.beta, beta)
# these two will have probability 0 in state 2, action 0 b/c
# of the infeasiability in R
funky_Q = np.empty((n, m, n))
funky_Q[:] = 1/n
funky_Q[0, 0, 0] = 0
funky_Q[0, 0, 1] = 2/n
funky_Q[2, 0, :] = 0
for ddp_f in [ddp2, ddp3]:
assert_allclose(ddp_f.R, ddp.R)
assert_allclose(ddp_f.Q, funky_Q)
assert_allclose(ddp_f.beta, ddp.beta)
# this one is just the original one.
assert_allclose(ddp4.R, ddp.R)
assert_allclose(ddp4.Q, ddp.Q)
assert_allclose(ddp4.beta, ddp.beta)
for method in ["pi", "vi", "mpi"]:
sol1 = ddp.solve(method=method)
for ddp_other in [ddp_sa, ddp_sa2, ddp_sa3, ddp2, ddp3, ddp4]:
sol2 = ddp_other.solve(method=method)
for k in ["v", "sigma", "num_iter"]:
assert_allclose(sol1[k], sol2[k])
def prices_to_capital_stock(am, r):
w = r_to_w(r)
am.set_prices(r, w)
aiyagari_ddp = DiscreteDP(am.R, am.Q, β)
# Compute the optimal policy
results = aiyagari_ddp.solve(method='policy_iteration')
# Compute the stationary distribution
stationary_probs = results.mc.stationary_distributions[0]
# Extract the marginal distribution for assets
asset_probs = asset_marginal(stationary_probs, am.a_size, am.z_size)
# Return K
return np.sum(asset_probs * am.a_vals)
def voter_dpp(self, etah, etal, tauy, pipar, prg, prb, Pm, pol_br, etam,
beta, delta):
R = self.rewardv(etam, beta)
Q = self.populate_Q(etah, etal, tauy, pipar, prg, prb, Pm, pol_br)
dpp = DiscreteDP(R=R,
Q=Q,
beta=delta,
s_indices=self.state_indices,
a_indices=self.action_indices)
results = dpp.solve(method='policy_iteration')
sol = DPPsol(results, R, Q)
return (sol)
def test_ddp_beta_1_not_implemented_error():
n, m = 3, 2
R = np.array([[0, 1], [1, 0], [0, 1]])
Q = np.empty((n, m, n))
Q[:] = 1/n
beta = 1
ddp0 = DiscreteDP(R, Q, beta)
ddp1 = ddp0.to_sa_pair_form()
ddp2 = ddp0.to_sa_pair_form(sparse=False)
solution_methods = \
['value_iteration', 'policy_iteration', 'modified_policy_iteration']
for ddp in [ddp0, ddp1, ddp2]:
assert_raises(NotImplementedError, ddp.evaluate_policy, np.zeros(n))
for method in solution_methods:
assert_raises(NotImplementedError, getattr(ddp, method))
def test_ddp_beta_1_not_implemented_error():
n, m = 3, 2
R = np.array([[0, 1], [1, 0], [0, 1]])
Q = np.empty((n, m, n))
Q[:] = 1 / n
beta = 1
ddp0 = DiscreteDP(R, Q, beta)
ddp1 = ddp0.to_sa_pair_form()
ddp2 = ddp0.to_sa_pair_form(sparse=False)
solution_methods = \
['value_iteration', 'policy_iteration', 'modified_policy_iteration']
for ddp in [ddp0, ddp1, ddp2]:
assert_raises(NotImplementedError, ddp.evaluate_policy, np.zeros(n))
for method in solution_methods:
assert_raises(NotImplementedError, getattr(ddp, method))
def test_ddp_beta_0():
n, m = 3, 2
R = np.array([[0, 1], [1, 0], [0, 1]])
Q = np.empty((n, m, n))
Q[:] = 1/n
beta = 0
sigma_star = [1, 0, 1]
v_star = [1, 1, 1]
v_init = [0, 0, 0]
ddp0 = DiscreteDP(R, Q, beta)
ddp1 = ddp0.to_sa_pair_form()
methods = ['vi', 'pi', 'mpi']
for ddp in [ddp0, ddp1]:
for method in methods:
res = ddp.solve(method=method, v_init=v_init)
assert_array_equal(res.sigma, sigma_star)
assert_array_equal(res.v, v_star)
def test_ddp_beta_0():
n, m = 3, 2
R = np.array([[0, 1], [1, 0], [0, 1]])
Q = np.empty((n, m, n))
Q[:] = 1 / n
beta = 0
sigma_star = [1, 0, 1]
v_star = [1, 1, 1]
v_init = [0, 0, 0]
ddp0 = DiscreteDP(R, Q, beta)
ddp1 = ddp0.to_sa_pair_form()
methods = ['vi', 'pi', 'mpi']
for ddp in [ddp0, ddp1]:
for method in methods:
res = ddp.solve(method=method, v_init=v_init)
assert_array_equal(res.sigma, sigma_star)
assert_array_equal(res.v, v_star)
def setUp(self):
# From Puterman 2005, Section 3.2, Section 4.6.1
# "single-product stochastic inventory control"
s_indices = [0, 0, 0, 0, 1, 1, 1, 2, 2, 3]
a_indices = [0, 1, 2, 3, 0, 1, 2, 0, 1, 0]
R = [0., -1., -2., -5., 5., 0., -3., 6., -1., 5.]
Q = [[1., 0., 0., 0.], [0.75, 0.25, 0., 0.], [0.25, 0.5, 0.25, 0.],
[0., 0.25, 0.5, 0.25], [0.75, 0.25, 0.,
0.], [0.25, 0.5, 0.25, 0.],
[0., 0.25, 0.5, 0.25], [0.25, 0.5, 0.25, 0.],
[0., 0.25, 0.5, 0.25], [0., 0.25, 0.5, 0.25]]
beta = 1
self.ddp = DiscreteDP(R, Q, beta, s_indices, a_indices)
def test_ddp_beta_1_not_implemented_error():
n, m = 3, 2
R = np.array([[0, 1], [1, 0], [0, 1]])
Q = np.empty((n, m, n))
Q[:] = 1 / n
beta = 1
ddp0 = DiscreteDP(R, Q, beta)
s_indices, a_indices = np.where(R > -np.inf)
R_sa = R[s_indices, a_indices]
Q_sa = Q[s_indices, a_indices]
ddp1 = DiscreteDP(R_sa, Q_sa, beta, s_indices, a_indices)
Q_sa_sp = sparse.csr_matrix(Q_sa)
ddp2 = DiscreteDP(R_sa, Q_sa_sp, beta, s_indices, a_indices)
solution_methods = \
['value_iteration', 'policy_iteration', 'modified_policy_iteration']
for ddp in [ddp0, ddp1, ddp2]:
assert_raises(NotImplementedError, ddp.evaluate_policy, np.zeros(n))
for method in solution_methods:
assert_raises(NotImplementedError, getattr(ddp, method))
def test_ddp_beta_0():
n, m = 3, 2
R = np.array([[0, 1], [1, 0], [0, 1]])
Q = np.empty((n, m, n))
Q[:] = 1 / n
beta = 0
sigma_star = [1, 0, 1]
v_star = [1, 1, 1]
v_init = [0, 0, 0]
ddp0 = DiscreteDP(R, Q, beta)
s_indices, a_indices = np.where(R > -np.inf)
R_sa = R[s_indices, a_indices]
Q_sa = Q[s_indices, a_indices]
ddp1 = DiscreteDP(R_sa, Q_sa, beta, s_indices, a_indices)
methods = ['vi', 'pi', 'mpi']
for ddp in [ddp0, ddp1]:
for method in methods:
res = ddp.solve(method=method, v_init=v_init)
assert_array_equal(res.sigma, sigma_star)
assert_array_equal(res.v, v_star)
def prices_to_capital_stock(am, r):
"""
Map prices to the induced level of capital stock.
Parameters:
----------
am : Household
An instance of an aiyagari_household.Household
r : float
The interest rate
"""
w = r_to_w(r)
am.set_prices(r, w)
aiyagari_ddp = DiscreteDP(am.R, am.Q, beta)
# Compute the optimal policy
results = aiyagari_ddp.solve(method='policy_iteration')
# Compute the stationary distribution
stationary_probs = results.mc.stationary_distributions[0]
# Extract the marginal distribution for assets
asset_probs = asset_marginal(stationary_probs, am.a_size, am.z_size)
# Return K
return np.sum(asset_probs * am.a_vals)
def prices_to_capital_stock(am, r):
"""
Map prices to the induced level of capital stock.
Parameters:
----------
am : Household
An instance of an aiyagari_household.Household
r : float
The interest rate
"""
w = r_to_w(r)
am.set_prices(r, w)
aiyagari_ddp = DiscreteDP(am.R, am.Q, beta)
# Compute the optimal policy
results = aiyagari_ddp.solve(method='policy_iteration')
# Compute the stationary distribution
stationary_probs = results.mc.stationary_distributions[0]
# Extract the marginal distribution for assets
asset_probs = asset_marginal(stationary_probs, am.a_size, am.z_size)
# Return K
return np.sum(asset_probs * am.a_vals)
def setUp(self):
# From Puterman 2005, Section 3.1
beta = 0.95
# Formulation with R: n x m, Q: n x m x n
n, m = 2, 2 # number of states, number of actions
R = [[5, 10], [-1, -np.inf]]
Q = np.empty((n, m, n))
Q[0, 0, :] = 0.5, 0.5
Q[0, 1, :] = 0, 1
Q[1, :, :] = 0, 1
ddp0 = DiscreteDP(R, Q, beta)
# Formulation with state-action pairs
L = 3 # number of state-action pairs
s_indices = [0, 0, 1]
a_indices = [0, 1, 0]
R_sa = [R[0][0], R[0][1], R[1][0]]
Q_sa = sparse.lil_matrix((L, n))
Q_sa[0, :] = Q[0, 0, :]
Q_sa[1, :] = Q[0, 1, :]
Q_sa[2, :] = Q[1, 0, :]
ddp_sa_sparse = DiscreteDP(R_sa, Q_sa, beta, s_indices, a_indices)
ddp_sa_dense = \
DiscreteDP(R_sa, Q_sa.toarray(), beta, s_indices, a_indices)
self.ddps = [ddp0, ddp_sa_sparse, ddp_sa_dense]
for ddp in self.ddps:
ddp.max_iter = 200
self.epsilon = 1e-2
# Analytical solution for beta > 10/11, Example 6.2.1
self.v_star = [(5 - 5.5 * beta) / ((1 - 0.5 * beta) * (1 - beta)),
-1 / (1 - beta)]
self.sigma_star = [0, 0]
def test_ddp_to_sa_and_to_product():
n, m = 3, 2
R = np.array([[0, 1], [1, 0], [-np.inf, 1]])
Q = np.empty((n, m, n))
Q[:] = 1 / n
Q[0, 0, 0] = 0
Q[0, 0, 1] = 2 / n
beta = 0.95
sparse_R = np.array([0, 1, 1, 0, 1])
_Q = np.full((5, 3), 1 / 3)
_Q[0, 0] = 0
_Q[0, 1] = 2 / n
sparse_Q = sparse.coo_matrix(_Q)
ddp = DiscreteDP(R, Q, beta)
ddp_sa = ddp.to_sa_pair_form()
ddp_sa2 = ddp_sa.to_sa_pair_form()
ddp_sa3 = ddp.to_sa_pair_form(sparse=False)
ddp2 = ddp_sa.to_product_form()
ddp3 = ddp_sa2.to_product_form()
ddp4 = ddp.to_product_form()
# make sure conversion worked
for ddp_s in [ddp_sa, ddp_sa2, ddp_sa3]:
assert_allclose(ddp_s.R, sparse_R)
# allcose doesn't work on sparse
np.max(np.abs((sparse_Q - ddp_s.Q))) < 1e-15
assert_allclose(ddp_s.beta, beta)
# these two will have probability 0 in state 2, action 0 b/c
# of the infeasiability in R
funky_Q = np.empty((n, m, n))
funky_Q[:] = 1 / n
funky_Q[0, 0, 0] = 0
funky_Q[0, 0, 1] = 2 / n
funky_Q[2, 0, :] = 0
for ddp_f in [ddp2, ddp3]:
assert_allclose(ddp_f.R, ddp.R)
assert_allclose(ddp_f.Q, funky_Q)
assert_allclose(ddp_f.beta, ddp.beta)
# this one is just the original one.
assert_allclose(ddp4.R, ddp.R)
assert_allclose(ddp4.Q, ddp.Q)
assert_allclose(ddp4.beta, ddp.beta)
for method in ["pi", "vi", "mpi"]:
sol1 = ddp.solve(method=method)
for ddp_other in [ddp_sa, ddp_sa2, ddp_sa3, ddp2, ddp3, ddp4]:
sol2 = ddp_other.solve(method=method)
for k in ["v", "sigma", "num_iter"]:
assert_allclose(sol1[k], sol2[k])
def prices_to_capital_stock(am, r):
w = r_to_w(r)
am.set_prices(r, w)
aiyagari_ddp = DiscreteDP(am.R, am.Q, β)
# Compute the optimal policy
results = aiyagari_ddp.solve(method='policy_iteration')
# Compute the stationary distribution
stationary_probs = results.mc.stationary_distributions[0]
# Extract the marginal distribution for assets
asset_probs = asset_marginal(stationary_probs, am.a_size, am.z_size)
# Return K
return np.sum(asset_probs * am.a_vals)
am = Household(a_max=30)
am_ddp = DiscreteDP(am.R, am.Q, am.β)
num_points = 20
r_vals = np.linspace(0.005, 0.04, num_points)
k_vals = np.empty(num_points)
for i, r in enumerate(r_vals):
k_vals[i] = prices_to_capital_stock(am, r)
fig, ax = plt.subplots()
ax.plot(k_vals, r_vals, lw=2, alpha=0.6, label='Supply')
ax.plot(k_vals, rd(k_vals), lw=2, alpha=0.6, label='Demand')
ax.set_xlabel('K')
ax.set_ylabel('r')
ax.legend()
aiyagari_ddp = DiscreteDP(am.R, am.Q, beta)
# Compute the optimal policy
results = aiyagari_ddp.solve(method='policy_iteration')
# Compute the stationary distribution
stationary_probs = results.mc.stationary_distributions[0]
# Extract the marginal distribution for assets
asset_probs = asset_marginal(stationary_probs, am.a_size, am.z_size)
# Return K
return np.sum(asset_probs * am.a_vals)
# Create an instance of Household
am = Household(a_max=20)
# Use the instance to build a discrete dynamic program
am_ddp = DiscreteDP(am.R, am.Q, am.beta)
# Create a grid of r values at which to compute demand and supply of capital
num_points = 20
r_vals = np.linspace(0.005, 0.04, num_points)
# Compute supply of capital
k_vals = np.empty(num_points)
for i, r in enumerate(r_vals):
k_vals[i] = prices_to_capital_stock(am, r)
# Plot against demand for capital by firms
fig, ax = plt.subplots(figsize=(11, 8))
ax.plot(k_vals, r_vals, lw=2, alpha=0.6, label='supply of capital')
ax.plot(k_vals, rd(k_vals), lw=2, alpha=0.6, label='demand for capital')
ax.grid()
aiyagari_ddp = DiscreteDP(am.R, am.Q, β)
# Compute the optimal policy
results = aiyagari_ddp.solve(method='policy_iteration')
# Compute the stationary distribution
stationary_probs = results.mc.stationary_distributions[0]
# Extract the marginal distribution for assets
asset_probs = asset_marginal(stationary_probs, am.a_size, am.z_size)
# Return K
return np.sum(asset_probs * am.a_vals)
# Create an instance of Household
am = Household(a_max=30)
# Use the instance to build a discrete dynamic program
am_ddp = DiscreteDP(am.R, am.Q, am.β)
# Create a grid of r values at which to compute demand and supply of capital
num_points = 30
r_vals = np.linspace(0.005, 0.04, num_points)
# Compute supply of capital
k_vals = np.empty(num_points)
for i, r in enumerate(r_vals):
k_vals[i] = prices_to_capital_stock(am, r)
# Plot against demand for capital by firms
fig, ax = plt.subplots(figsize=(11, 8))
ax.plot(k_vals, r_vals, lw=2, alpha=0.6, color='b', label='supply of capital')
ax.plot(k_vals,
rd(k_vals),
import numpy as np
import quantecon as qe
import matplotlib.pyplot as plt
from aiyagari_household import Household
from quantecon.markov import DiscreteDP
# Example prices
r = 0.03
w = 0.956
# Create an instance of Household
am = Household(a_max=20, r=r, w=w)
# Use the instance to build a discrete dynamic program
am_ddp = DiscreteDP(am.R, am.Q, am.beta)
# Solve using policy function iteration
results = am_ddp.solve(method='policy_iteration')
# Simplify names
z_size, a_size = am.z_size, am.a_size
z_vals, a_vals = am.z_vals, am.a_vals
n = a_size * z_size
# Get all optimal actions across the set of a indices with z fixed in each row
a_star = np.empty((z_size, a_size))
for s_i in range(n):
a_i = s_i // z_size
z_i = s_i % z_size
a_star[z_i, a_i] = a_vals[results.sigma[s_i]]
