/
code.py
187 lines (155 loc) · 5.67 KB
/
code.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
# Python 3.4 code
from matplotlib import pyplot as plt
from numpy import log
from scipy.optimize import brentq as findroot
try:
import seaborn as sns
sns.set_style("white") # optional seabird settings
sns.set_palette('Greys_d') # optional seabird settings
except ImportError:
pass
class SimpleModel:
def u(self, c, n):
"Utility function"
return ((((c ** (1 - self.sigma)) / (1 - self.sigma) if self.sigma != 1
else log(c))) - self.psi * (n ** (1 + self.gamma)) /
(1 + self.gamma))
def u_prime_c(self, c):
"Derivative of utility function"
return c ** (-self.sigma)
def u_inverse(self, x):
return x ** (- 1 / self.sigma)
def u_prime_n(self, n):
"Inverse of derivative of utility function"
return - self.psi * n ** self.gamma
def u_prime_of_n_inverse(self, x):
return (x / (- self.psi)) ** (1 / self.gamma)
def __init__(self, parameters):
try:
self.beta = parameters['be']
self.r = parameters['r']
self.sigma = parameters['sigma']
self.gamma = parameters['gamma']
self.psi = parameters['psi']
self.g = parameters['g']
self.R = self.r + 1
self.bR = self.R * self.beta
except KeyError as k:
print("Parameter error. Expecting " + str(k))
raise
def multiplier(eta):
x = findroot(lambda l: m.u(m.u_inverse((l + eta *
(1 - m.sigma)) ** (-1)),
m.u_prime_of_n_inverse(
- (l + eta * (1 + m.gamma)) **
(-1))) - udev,
0.001, 10)
return x
def c_at_w(eta):
return m.u_inverse((multiplier(eta) + eta * (1 - m.sigma)) ** (-1))
def n_at_w(eta):
return m.u_prime_of_n_inverse(
- (multiplier(eta) + eta * (1 + m.gamma)) ** (-1))
def c_before_w(mult, eta):
return m.u_inverse((mult + eta * (1 - m.sigma)) ** (-1))
def n_before_w(mult, eta):
return m.u_prime_of_n_inverse(-(mult + eta * (1 + m.gamma)) ** (-1))
def tax(c, n):
return - 1 - m.u_prime_c(c) / m.u_prime_n(n)
# PARAMETERS
ssdebt = .5 # steady state debt
r = 0.05
g = 0.2
par = {'r': r,
'be': .94,
'sigma': 1,
'gamma': .5,
'g': g,
'psi': 1 / (1 - g - r * ssdebt)}
pTminus = 10
pTplus = 10
m = SimpleModel(par)
udev = m.u(1 - m.g - m.r * ssdebt, 1)
bigA_ss = m.R * (c_at_w(0) + m.g - n_at_w(0)) / m.r
a_ss = (c_at_w(0) - n_at_w(0)) / (1 - m.beta)
eta0 = eta1 = .18
c_list = []
n_list = []
cap_t = 500
for t in range(cap_t):
c_list.append(c_at_w(eta1))
n_list.append(n_at_w(eta1))
eta1 = eta1 * m.bR
print(t)
c_before = []
n_before = []
l1 = multiplier(eta0)
eta1 = eta0
for t in range(cap_t):
l1 = l1 / m.bR
eta1 = eta1 / m.bR
c_before.append(c_before_w(l1, eta1))
n_before.append(n_before_w(l1, eta1))
c_before.reverse()
n_before.reverse()
c_list = c_before + c_list
n_list = n_before + n_list
big_A = [bigA_ss]
a = [a_ss]
for j in range(len(c_list)):
c_item, n_item = c_list[-j-1], n_list[-j-1]
cprime = c_list[-j] if j > 0 else c_list[-1]
big_A.append(c_item + m.g - n_item + big_A[-1] / m.R)
a.append(c_item + m.u_prime_n(n_item) / m.u_prime_c(c_item) *
n_item + m.beta * m.u_prime_c(cprime) / m.u_prime_c(c_item) *
a[-1])
a.reverse()
big_A.reverse()
plt.rc('text', usetex=True)
# plt.rc('font', **{'family': 'sans-serif',
# 'sans-serif': ['Computer Modern Roman']})
fig, axes = plt.subplots(nrows=2, ncols=2, figsize=(10, 7))
axes[0, 0].plot(c_list[cap_t-pTminus:cap_t+pTplus], 'k-',
lw=2, label='$c$')
axes[0, 0].plot(n_list[cap_t-pTminus:cap_t+pTplus], 'k--',
lw=2, label='$n$')
axes[0, 0].legend(('$c$', '$n$'), loc=2, fontsize=14)
axes[0, 0].set_title('(a) Consumption and labor')
axes[0, 0].set_xlabel('')
axes[0, 0].set_ylim([0.55, 0.85])
axes[0, 0].set_xticks([0, 5, 10, 15, 20])
axes[0, 0].set_xticklabels([0, 5, 'T', 15, 20])
axes[1, 0].plot([tax(c, n) for (c, n) in
zip(c_list[cap_t-pTminus:cap_t+pTplus],
n_list[cap_t-pTminus:cap_t+pTplus])],
'k-', lw=2, label='$\\tau_n$')
axes[1, 0].plot([1 - 1 / (m.R - 1) * (c2 / (m.beta * c1) - 1)
for (c1, c2) in
zip(c_list[cap_t-pTminus-1:cap_t+pTplus-1],
c_list[cap_t-pTminus:cap_t+pTplus])],
'k--', lw=2, label='$\\phi_k$')
axes[1, 0].legend(('$\\tau_n$', '$\\phi_k$'), loc=3, fontsize=14)
axes[1, 0].set_title('(c) Labor and capital income tax')
axes[1, 0].set_xticks([0, 5, 10, 15, 20])
axes[1, 0].set_xticklabels([0, 5, 'T', 15, 20])
axes[1, 0].set_xlabel('time')
axes[1, 1].plot(a[cap_t-pTminus:cap_t+pTplus], 'k-', lw=2, label='$a$')
axes[1, 1].plot([at - At for (At, at) in
zip(big_A[cap_t-pTminus:cap_t+pTplus],
a[cap_t-pTminus:])],
'k--', lw=2, label='$b$')
axes[1, 1].plot(big_A[cap_t-pTminus:cap_t+pTplus], 'k-.', lw=2,
label='$A$')
axes[1, 1].legend(('$a$', '$b$', '$A$'), loc=1, fontsize=14)
axes[1, 1].set_title('(d) Assets and liabilities')
axes[1, 1].set_xticks([0, 5, 10, 15, 20])
axes[1, 1].set_xticklabels([0, 5, 'T', 15, 20])
axes[1, 1].set_xlabel('time')
axes[0, 1].plot([m.u(c, n) - udev for c, n in zip(
c_list[cap_t-pTminus:cap_t+pTplus],
n_list[cap_t-pTminus:cap_t+pTplus])], 'k-', lw=2)
axes[0, 1].set_title('(b) Utility relative to deviation utility')
axes[0, 1].set_xlabel('')
axes[0, 1].set_xticks([0, 5, 10, 15, 20])
axes[0, 1].set_xticklabels([0, 5, 'T', 15, 20])
plt.show()