From "Monetary Economics: An Integrated Approach to Credit, Money, Income, Production and Wealth, 2nd ed" by Wynne Godley and Marc Lavoie, 2012.
Assumptions
| 1.Households | 2.Production | 3.Government | Σ | |
|---|---|---|---|---|
| 1.Consumption | -C | +C | 0 | |
| 2.Govt expenditures | +G | -G | 0 | |
| 3.[Output] | [Y] | |||
| 4.Factor income (wages) | +WB | -WB | 0 | |
| 5.Taxes | -T | +T | 0 | |
| 6.Change in the stock of money | -ΔH | +ΔH | 0 | |
| Σ | 0 | 0 | 0 | 0 |
Definition of terms
C : Consumption goods demand by households
G : Government expenditures
Y : National income
WB : Wage bill
T : Taxes
ΔH : Change in cash money
In this model, people (as consumers and producers of income) have been separated.
| 1.Households | 2.Production | 3.Government | Σ | |
|---|---|---|---|---|
| 1.Consumption | -Cd | +Cs | 0 | |
| 2.Govt expenditures | +Gs | -Gd | 0 | |
| 3.[Output] | [Y] | |||
| 4.Factor income (wages) | +W•Ns | -W•Nd | 0 | |
| 5.Taxes | -Ts | +Td | 0 | |
| 6.Change in the stock of money | -ΔHh | +ΔHs | 0 | |
| Σ | 0 | 0 | 0 | 0 |
Differences from previous matrix:
Definition of terms
From here, I will be building the model in code. Because this is the first model, the Python code will be explained in more detail also.
The following piece of code is necessary to show the graphics inline for iPython notebooks. To view the graphs, matplotlib is required.
In [1]:
# This line configures matplotlib to show figures embedded in the notebook,
# instead of opening a new window for each figure. More about that later.
# If you are using an old version of IPython, try using '%pylab inline' instead.
%matplotlib inline
from pysolve.model import Model
from pysolve.utils import is_close,round_solution
import matplotlib.pyplot as plt
In order to build the model, we must first start off by importing several modules that will be used to build the model. pysolve is a Python module that I have developed to make it easier to specify and solve linear models.
The first line will import the main Model class. The second line imports several utility functions that will prove useful.
In [2]:
from pysolve.model import Model
from pysolve.utils import is_close,round_solution
The first step when developing a pysolve model is to create the model. This is just an empty model for now, but we will be adding the rest of the information to this.
In [3]:
model = Model()
The second step is to define the (endogenous) variables. These are the variables that we are allowed to manipulate within the model. This is pretty straigtforward.
As a useful step, I define the default value for all variables. This can be changed on an individual basis. This is the value that the variable will start off with if nothing is changed.
In [4]:
model.set_var_default(0)
Next, we create the variables used by the sim. Most of these have been explained above.
In [5]:
model.var('Cd', desc='Consumption goods demand by households')
model.var('Cs', desc='Consumption goods supply')
model.var('Gs', desc='Government goods, supply')
model.var('Hh', desc='Cash money held by households')
model.var('Hs', desc='Cash money supplied by the government')
model.var('Nd', desc='Demand for labor')
model.var('Ns', desc='Supply of labor')
model.var('Td', desc='Taxes, demand')
model.var('Ts', desc='Taxes, supply')
model.var('Y', desc='Income = GDP')
model.var('YD', desc='Disposable income of households');
As an aside, multiple variables can be created by the following code. But the above is more descriptive.
model.vars('Y', 'YD', 'Ts', 'Td', 'Hs', 'Hh', 'Gs', 'Cs',
'Cd', 'Ns', 'Nd')
The value of the variables may also be changed mid-iteration. They will then be used to seed the value of the next iteration.
For example
varx = model.var('x')
# ... later
varx.value = 22
# this will also work
model.variables['x'].value = 22
Aside: the semicolon ';' at the end of the last line of code is an iPython artifact, and is not needed by the python code. It is used to suppress output by the iPython interpreter.
The next step is to define the parameters. I do not differentiate between exogenous variables and parameters since both are set outside of the model. The solver will not be able to change these values. However, the user may change these values between calls to the solver.
Like the variables, there is a call that may be made to set a default value for all parameters, but I will be creating the parameters with their default values. The call would look like this
model.set_parameter_initial(1.0)
In addition the parameter values could be changed like this:
Gd = model.param('Gd', initial=10)
# ...
# at some later time
Gd.value = 20
# or this would work also
model.parameters['Gd'].value = 20
Some of the parameters (alpha1, alpha2 and theta) have not been explained yet, but will be explained when we add the equations that use them.
In [6]:
model.param('Gd', desc='Government goods, demand', default=20.)
model.param('W', desc='Wage rate', default=1.)
model.param('alpha1', desc='Propensity to consume out of income', default=0.6)
model.param('alpha2', desc='Propensity to consume o of wealth', default=0.4)
model.param('theta', desc='Tax rate', default=0.2);
Adding an equation is just adding the textual form of the equation. There are some restrictions. Linear systems only.
In [7]:
model.add('Cs = Cd')
model.add('Gs = Gd')
model.add('Ts = Td')
model.add('Ns = Nd');
These four equations imply that demand equals supply for this period, no supply constraints of any kind.
In [8]:
model.add('YD = (W*Ns) - Ts');
Disposable income (YD) is the wages earned by households minus taxes.
In [9]:
model.add('Td = theta * W * Ns');
Taxes are a fixed proportion (theta) of income. theta is decided by the government and is exogenous to the model.
In [10]:
model.add('Cd = alpha1*YD + alpha2*Hh(-1)');
This is a consumption function, the rates at which housholds consume. This is a combination of consumption of inherited wealth (Hh(-1)) and post-tax income (YD).
In [11]:
model.add('Hs - Hs(-1) = Gd - Td');
This comes from the transaction-flow matrix and represents the governments budget constraint. Government expenditures that are not paid for by taxes (Gd-Td), must be covered by differences in the money supply.
In [12]:
model.add('Hh - Hh(-1) = YD - Cd');
The difference in the cash that households carry is the difference between their income and their consumption.
In [13]:
model.add('Y = Cs + Gs');
The determination of national income.
In [14]:
model.add('Nd = Y/W');
The determination of employment.
We now have 11 equations and 11 unknowns. Each of the eleven unknowns has been set on the left-hand side of an equation (This implies that we can use the Gauss-Seidel algorithm to iterate to a solution, convergence is not guaranteed but we can try.)
We have set the default for all of the variables to 0, and that will be used as an initial solution.
In [15]:
model.solve(iterations=100, threshold=1e-4);
In [16]:
prev = round_solution(model.solutions[-2], decimals=1)
solution = round_solution(model.solutions[-1], decimals=1)
print "Y : " + str(solution['Y'])
print "T : " + str(solution['Ts'])
print "YD : " + str(solution['YD'])
print "C : " + str(solution['Cs'])
print "Hs-Hs(-1) : " + str(solution['Hs'] - prev['Hs'])
print "Hh-Hh(-1) : " + str(solution['Hh'] - prev['Hh'])
print "H : " + str(solution['Hh'])
To make the model easier to manipulate, I will encapsulate model creation into a single function.
In [17]:
def create_sim_model():
model = Model()
model.set_var_default(0)
model.var('Cd', desc='Consumption goods demand by households')
model.var('Cs', desc='Consumption goods supply')
model.var('Gs', desc='Government goods, supply')
model.var('Hh', desc='Cash money held by households')
model.var('Hs', desc='Cash money supplied by the government')
model.var('Nd', desc='Demand for labor')
model.var('Ns', desc='Supply of labor')
model.var('Td', desc='Taxes, demand')
model.var('Ts', desc='Taxes, supply')
model.var('Y', desc='Income = GDP')
model.var('YD', desc='Disposable income of households')
model.param('Gd', desc='Government goods, demand')
model.param('W', desc='Wage rate')
model.param('alpha1', desc='Propensity to consume out of income')
model.param('alpha2', desc='Propensity to consume out of wealth')
model.param('theta', desc='Tax rate')
model.add('Cs = Cd') # 3.1
model.add('Gs = Gd') # 3.2
model.add('Ts = Td') # 3.3
model.add('Ns = Nd') # 3.4
model.add('YD = (W*Ns) - Ts') # 3.5
model.add('Td = theta * W * Ns') # 3.6, theta < 1.0
model.add('Cd = alpha1*YD + alpha2*Hh(-1)') # 3.7, 0 < alpha2 < alpha1 < 1
model.add('Hs - Hs(-1) = Gd - Td') # 3.8
model.add('Hh - Hh(-1) = YD - Cd') # 3.9
model.add('Y = Cs + Gs') # 3.10
model.add('Nd = Y/W') # 3.11
return model
Now we can run the simulation using the model.
In [18]:
model = create_sim_model()
model.set_values({'alpha1': 0.6,
'alpha2': 0.4,
'theta': 0.2,
'Gd': 20,
'W': 1})
model.solve(iterations=100, threshold=1e-5)
prev = round_solution(model.solutions[-2], decimals=1)
solution = round_solution(model.solutions[-1], decimals=1)
print "Y : " + str(solution['Y'])
print "T : " + str(solution['Ts'])
print "YD : " + str(solution['YD'])
print "C : " + str(solution['Cs'])
print "Hs-Hs(-1) : " + str(solution['Hs'] - prev['Hs'])
print "Hh-Hh(-1) : " + str(solution['Hh'] - prev['Hh'])
print "H : " + str(solution['Hh'])
We now generate the steady-state solution by iterating until the solutions converge.
In [19]:
steady_state = create_sim_model()
steady_state.set_values({'alpha1': 0.6,
'alpha2': 0.4,
'theta': 0.2,
'Gd': 20,
'W': 1})
for _ in xrange(100):
steady_state.solve(iterations=100, threshold=1e-5)
prev_soln = steady_state.solutions[-2]
soln = steady_state.solutions[-1]
if is_close(prev_soln, soln, atol=1e-4):
break
prev = round_solution(steady_state.solutions[-2], decimals=1)
solution = round_solution(steady_state.solutions[-1], decimals=1)
print "Y : " + str(solution['Y'])
print "T : " + str(solution['Ts'])
print "YD : " + str(solution['YD'])
print "C : " + str(solution['Cs'])
print "Hs-Hs(-1) : " + str(solution['Hs'] - prev['Hs'])
print "Hh-Hh(-1) : " + str(solution['Hh'] - prev['Hh'])
print "H : " + str(solution['Hh'])
We can also generate table 3.4
In [20]:
from IPython.display import HTML
import numpy
from pysolve.utils import generate_html_table
data = list()
for var in [('Gd', 'G'), ('Y', 'Y'), ('Ts', 'T'), ('YD', 'YD'), ('Cs', 'C')]:
rowdata = list()
rowdata.append(var[1])
for i in [0, 1, 2, -1]:
rowdata.append(str(numpy.round(steady_state.solutions[i][var[0]], decimals=1)))
data.append(rowdata)
for var in [('Hs', 'ΔHs'), ('Hh', 'ΔHh')]:
rowdata = list()
rowdata.append(var[1])
rowdata.append(str(numpy.round(steady_state.solutions[0][var[0]], decimals=1)))
for i in [1, 2, -1]:
rowdata.append(str(numpy.round(steady_state.solutions[i][var[0]] -
steady_state.solutions[i-1][var[0]], decimals=1)))
data.append(rowdata)
for var in [('Hh', 'H')]:
rowdata = list()
rowdata.append(var[1])
for i in [0, 1, 2, -1]:
rowdata.append(str(numpy.round(steady_state.solutions[i][var[0]], decimals=1)))
data.append(rowdata)
s = generate_html_table(['Period', '1', '2', '3', '∞'], data)
HTML(s)
Out[20]:
In [21]:
step_model = create_sim_model()
step_model.set_values({'alpha1': 0.6,
'alpha2': 0.4,
'theta': 0.2,
'Gd': 20,
'W': 1})
# Use the steady state solution as a starting point
step_model.solutions = steady_state.solutions[-2:]
for i in xrange(45):
step_model.solve(iterations=100, threshold=1e-5)
if i == 2:
step_model.parameters['Gd'].value += 5
Calculate the solution but with an permanent increase in government expenditures (+5) and starting from the steady state solution.
In [22]:
caption = '''
Figure 3.1 Impact on national income Y and the steady state solution Y*,
following a permanent increase in government expenditures ($\\bigtriangleup$G = 5).'''
gdata = [s['Gd']/s['theta'] for s in step_model.solutions]
ydata = [s['Y'] for s in step_model.solutions]
# Now graph G/theta and Y
fig = plt.figure()
axes = fig.add_axes([0.1, 0.1, 1.1, 1.1])
axes.spines['top'].set_visible(False)
axes.spines['right'].set_visible(False)
axes.set_ylim(97, 129)
axes.plot(gdata, 'r') # plot G/theta
axes.plot(ydata, linestyle='--', color='g') # plot Y
# add labels
plt.text(10, 126, 'Steady-state solution Y*')
plt.text(15, 120, 'Income Y')
fig.text(.1, -.1, caption);
In [23]:
caption = '''
Figure 3.2 Disposable income and consumption starting from scratch (Table 3.4)'''
cdata = [s['Cd'] for s in steady_state.solutions]
yddata = [s['YD'] for s in steady_state.solutions]
fig = plt.figure()
axes = fig.add_axes([0.1, 0.1, 1.0, 1.0])
axes.tick_params(top='off', right='off')
axes.spines['top'].set_visible(False)
axes.spines['right'].set_visible(False)
axes.set_ylim(0, 85)
axes.set_xlim(-2, 50)
axes.plot(cdata, linestyle=':', color='r') # plot C
axes.plot(yddata, linestyle='--', color='g') # plot YD
plt.axhline(y=80, color='k')
# add labels
plt.text(2, 72, 'Disposable')
plt.text(2, 68, 'Income YD')
plt.text(10, 60, 'Consumption C')
fig.text(0.1, 0, caption);
In [24]:
caption = '''
Figure 3.3 Wealth change and wealth level starting from scratch (Table 3.4)'''
hdata = [s['Hh'] for s in steady_state.solutions]
deltahdata = [s['Hh'] for s in steady_state.solutions]
for i in xrange(1, len(steady_state.solutions)):
deltahdata[i] -= hdata[i-1]
fig = plt.figure()
axes = fig.add_axes([0.1, 0.1, 1.0, 1.0])
axes.tick_params(top='off')
axes.set_ylim(0, 13)
axes.set_xlim(-2, 50)
axes.plot(deltahdata, linestyle='--', color='g')
axes2 = axes.twinx()
axes2.set_ylim(0, 85)
axes2.set_xlim(-2, 50)
axes2.plot(hdata, 'r')
# add labels
plt.text(20, 16, 'Household saving')
plt.text(20, 12, '(the change in the money stock)')
plt.text(20, 70, 'Wealth level H (money stock)')
fig.text(0.1, -0.05, caption);
In [25]:
caption = '''
Figure 3.4 Evolution of wealth, target wealth, consumption and disposable income
following an increase in government expenditures ($\\bigtriangleup$G = 5) Model SIM '''
hdata = [s['Hh'] for s in step_model.solutions]
cdata = [s['Cs'] for s in step_model.solutions]
vtdata = [s['YD']*(1.-s['alpha1'])/s['alpha2'] for s in step_model.solutions]
fig = plt.figure()
axes = fig.add_axes([0.1, 0.1, 1.0, 1.0])
axes.tick_params(top='off', right='off')
axes.spines['top'].set_visible(False)
axes.spines['right'].set_visible(False)
axes.set_ylim(78, 102)
axes.set_xlim(-2, 50)
axes.plot(hdata, linestyle='-', color='g', label='Wealth')
axes.plot(cdata, linestyle=':', color='r', linewidth=2, label='Consumption')
axes.plot(vtdata, linestyle='--', color='b', label='Target wealth (and disposable income)')
plt.legend(loc=(0.35,0.2), frameon=False)
fig.text(0.1, -0.05, caption);
In [26]:
alpha_model = create_sim_model()
alpha_model.set_values({'alpha1': 0.6,
'alpha2': 0.4,
'theta': 0.2,
'Gd': 20,
'W': 1})
# Use the steady state solution as a starting point
alpha_model.solutions = steady_state.solutions[-2:]
for i in xrange(50):
alpha_model.solve(iterations=100, threshold=1e-4)
if i == 2:
alpha_model.parameters['alpha1'].value = 0.7
We will need to generate solutions that involve a change in alpha1 (from 0.6 to 0.7).
In [27]:
caption = '''
Figure 3.8 Evolution of consumption, disposable income and wealth following
an increase in the propensity to consume out of current income
($\\alpha_1$ moves from 0.6 to 0.7)'''
hdata = [s['Hh'] for s in alpha_model.solutions]
cdata = [s['Cs'] for s in alpha_model.solutions]
vtdata = [s['YD'] for s in alpha_model.solutions]
fig = plt.figure()
axes = fig.add_axes([0.1, 0.1, 1.0, 1.0])
axes.tick_params(top='off', right='off')
axes.spines['top'].set_visible(False)
axes.spines['right'].set_visible(False)
axes.set_ylim(58, 100)
axes.set_xlim(-2, 50)
axes.plot(hdata, linestyle='-', color='g')
axes.plot(cdata, linestyle=':', color='r', linewidth=2)
axes.plot(vtdata, linestyle='--', color='b')
plt.text(6, 97, 'Consumption')
plt.text(8, 79, 'Disposable income')
plt.text(20, 62, 'Wealth')
fig.text(0.1, -0.1, caption);
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