def compute_steady_state_unemployment(c, T):
"""
Compute the steady state unemployment rate given c and T using lambda, the
job finding rate, from the McCall model and then computing steady state
unemployment corresponding to alpha, lambda, b, d.
"""
lmda = compute_lambda(c, T)
lm = LakeModel(alpha=alpha, lmda=lmda, b=0, d=0)
x = lm.rate_steady_state()
e, u = x
return u
def compute_steady_state_quantities(c, tau):
"""
Compute the steady state unemployment rate given c and tau using optimal
quantities from the McCall model and computing corresponding steady state
quantities
"""
w_bar, lmda, V, U = compute_optimal_quantities(c, tau)
# Compute steady state employment and unemployment rates
lm = LakeModel(alpha=alpha_q, lmda=lmda, b=b, d=d)
x = lm.rate_steady_state()
e, u = x
# Compute steady state welfare
w = np.sum(V * p_vec * (w_vec - tau > w_bar)) / np.sum(p_vec * (w_vec - tau > w_bar))
welfare = e * w + u * U
return e, u, welfare
def compute_steady_state_quantities(c, tau):
"""
Compute the steady state unemployment rate given c and tau using optimal
quantities from the McCall model and computing corresponding steady state
quantities
"""
w_bar, lmda, V, U = compute_optimal_quantities(c, tau)
# Compute steady state employment and unemployment rates
lm = LakeModel(alpha=alpha_q, lmda=lmda, b=b, d=d)
x = lm.rate_steady_state()
e, u = x
# Compute steady state welfare
w = np.sum(V * p_vec *
(w_vec - tau > w_bar)) / np.sum(p_vec * (w_vec - tau > w_bar))
welfare = e * w + u * U
return e, u, welfare
import matplotlib.pyplot as plt
from lake_model import LakeModel
from quantecon import MarkovChain
import matplotlib
matplotlib.style.use('ggplot')
lm = LakeModel(d=0, b=0)
T = 5000 # Simulation length
alpha, lmda = lm.alpha, lm.lmda
P = [[1 - lmda, lmda], [alpha, 1 - alpha]]
mc = MarkovChain(P)
xbar = lm.rate_steady_state()
fig, axes = plt.subplots(2, 1, figsize=(10, 8))
s_path = mc.simulate(T, init=1)
s_bar_e = s_path.cumsum() / range(1, T + 1)
s_bar_u = 1 - s_bar_e
ax = axes[0]
ax.plot(s_bar_u, '-b', lw=2, alpha=0.5)
ax.hlines(xbar[1], 0, T, 'r', '--')
ax.set_title(r'Percent of time unemployed')
ax = axes[1]
ax.plot(s_bar_e, '-b', lw=2, alpha=0.5)
ax.hlines(xbar[0], 0, T, 'r', '--')
ax.set_title(r'Percent of time employed')
"""
Stock dynamics the a lake model.
"""
import numpy as np
import matplotlib.pyplot as plt
from lake_model import LakeModel
import matplotlib
matplotlib.style.use('ggplot')
lm = LakeModel()
e_0 = 0.92 # Initial employment rate
u_0 = 1 - e_0 # Initial unemployment rate
T = 50 # Simulation length
xbar = lm.rate_steady_state()
fig, axes = plt.subplots(2, 1, figsize=(10, 8))
x_0 = (e_0, u_0)
x_path = np.vstack(lm.simulate_rate_path(x_0, T))
ax = axes[0]
ax.plot(x_path[:,0], '-b', lw=2, alpha=0.5)
ax.hlines(xbar[0], 0, T, 'r', '--')
ax.set_title(r'Employment rate')
ax = axes[1]
ax.plot(x_path[:,1], '-b', lw=2, alpha=0.5)
ax.hlines(xbar[1], 0, T, 'r', '--')
ax.set_title(r'Unemployment rate')