# quant-econ Solutions: Search with Unknown Offer Distribution

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%matplotlib inline

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import numpy as np
import matplotlib.pyplot as plt

from quantecon import SearchProblem, compute_fixed_point

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## Exercise 1

This code solves the "Offer Distribution Unknown" model by iterating on a guess of the reservation wage function. You should find that the run time is much shorter than that of the value function approach in `odu_vfi.py`

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sp = SearchProblem(pi_grid_size=50)

phi_init = np.ones(len(sp.pi_grid))
w_bar = compute_fixed_point(sp.res_wage_operator, phi_init)

fig, ax = plt.subplots(figsize=(9, 7))
ax.plot(sp.pi_grid, w_bar, linewidth=2, color='black')
ax.set_ylim(0, 2)
ax.grid(axis='x', linewidth=0.25, linestyle='--', color='0.25')
ax.grid(axis='y', linewidth=0.25, linestyle='--', color='0.25')
ax.fill_between(sp.pi_grid, 0, w_bar, color='blue', alpha=0.15)
ax.fill_between(sp.pi_grid, w_bar, 2, color='green', alpha=0.15)
ax.text(0.42, 1.2, 'reject')
ax.text(0.7, 1.8, 'accept')

``````
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Computed iterate 1 with error 0.426161
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Computed iterate 3 with error 0.076090
Computed iterate 4 with error 0.046400
Computed iterate 5 with error 0.028295
Computed iterate 6 with error 0.018182
Computed iterate 7 with error 0.013566
Computed iterate 8 with error 0.009611
Computed iterate 9 with error 0.007113
Computed iterate 10 with error 0.005174
Computed iterate 11 with error 0.003732
Computed iterate 12 with error 0.002657
Computed iterate 13 with error 0.001876
Computed iterate 14 with error 0.001348
Computed iterate 15 with error 0.000965

Out:

<matplotlib.text.Text at 0x36f0290>

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The next piece of code is not one of the exercises from quant-econ, it's just a fun simulation to see what the effect of a change in the underlying distribution on the unemployment rate is.

At a point in the simulation, the distribution becomes significantly worse. It takes a while for agents to learn this, and in the meantime they are too optimistic, and turn down too many jobs. As a result, the unemployment rate spikes.

The code takes a few minutes to run.

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from scipy import interp
# Set up model and compute the function w_bar
sp = SearchProblem(pi_grid_size=50, F_a=1, F_b=1)
pi_grid, f, g, F, G = sp.pi_grid, sp.f, sp.g, sp.F, sp.G
phi_init = np.ones(len(sp.pi_grid))
w_bar_vals = compute_fixed_point(sp.res_wage_operator, phi_init)
w_bar = lambda x: interp(x, pi_grid, w_bar_vals)

class Agent:
"""
Holds the employment state and beliefs of an individual agent.
"""

def __init__(self, pi=1e-3):
self.pi = pi
self.employed = 1

def update(self, H):
"Update self by drawing wage offer from distribution H."
if self.employed == 0:
w = H.rvs()
if w >= w_bar(self.pi):
self.employed = 1
else:
self.pi = 1.0 / (1 + ((1 - self.pi) * g(w)) / (self.pi * f(w)))

num_agents = 5000
separation_rate = 0.025  # Fraction of jobs that end in each period
separation_num = int(num_agents * separation_rate)
agent_indices = range(num_agents)
agents = [Agent() for i in range(num_agents)]
sim_length = 600
change_date = 200  # Change to F after this many periods

unempl_rate = []
for i in range(sim_length):
print "date = ", i
if i == change_date:
H = F
# Randomly select separation_num agents and set employment status to 0
np.random.shuffle(agent_indices)
separation_list = agent_indices[:separation_num]
for agent_index in separation_list:
agents[agent_index].employed = 0
# Update agents
for agent in agents:
agent.update(H)
employed = [agent.employed for agent in agents]
unempl_rate.append(1 - np.mean(employed))

fig, ax = plt.subplots(figsize=(9, 7))
ax.plot(unempl_rate, lw=2, alpha=0.8, label='unemployment rate')
ax.axvline(change_date, color="red")
ax.legend()

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Computed iterate 1 with error 0.426161
Computed iterate 2 with error 0.127050
Computed iterate 3 with error 0.076090
Computed iterate 4 with error 0.046400
Computed iterate 5 with error 0.028295
Computed iterate 6 with error 0.018182
Computed iterate 7 with error 0.013566
Computed iterate 8 with error 0.009611
Computed iterate 9 with error 0.007113
Computed iterate 10 with error 0.005174
Computed iterate 11 with error 0.003732
Computed iterate 12 with error 0.002657
Computed iterate 13 with error 0.001876
Computed iterate 14 with error 0.001348
Computed iterate 15 with error 0.000965
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<matplotlib.legend.Legend at 0x3b3ce50>

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