OOP in Action: The Samuelson Accelerator¶
Co-author: Natasha Watkins
Paul Samuelson’s (1939) multiplier-accelerator model¶
This lecture creates nonstochastic and stochastic versions of Paul Samuelson’s celebrated multiplier accelerator model, as described in [Sam39]
Samuelson used a second-order linear difference equation to represent a model of national output based on three components:
- a national output identity asserting that national outcome is the sum of output devoted to consumption plus investment plus government purchases
- a Keynesian consumption function asserting that consumption at time $ t $ is equal to a constant called the marginal propensity to consume times national output at time $ t-1 $
- an investment accelerator asserting that investment at time $ t $ equals a constant called the accelerator coefficient times the difference in output between period $ t-1 $ and $ t-2 $
- the idea that consumption plus investment plus government purchases constitute aggregate demand that automatically call forth an equal amount of aggregate supply
(To read about linear difference equations see here or chapter IX of [Sar87])
Samuelson used the model to analyze how particular values of the marginal propensity to consume and the accelerator coefficient might give rise to transient business cycles in national output with alternative dynamic properties:
- smooth convergence to a constant level of output
- damped business cycles that eventually converge to a constant level of output
- persistent business cycles that neither dampen nor explode
Later in the lecture we present an extension to Samuelson’s model that simply adds a random shock to the right side of the national income identity (the random shock represents random fluctuations in aggregate demand
This modification makes national output become governed by a second-order stochastic linear difference equation that, with appropriate parameter values, gives rise to recurrent irregular business cycles.
(To read about stochastic linear difference equations see chapter XI of [Sar87])
Structure of the model¶
In more detail, let’s assume that
- $ \{G_t\} $ is a sequence of levels of government expenditures. We’ll start by setting $ G_t = G \forall t $
- $ \{C_t\} $ is a sequence of levels of aggregate consumption expenditures, a key endogenous variable in the model
- $ \{I_t\} $ is a sequence of rates of investment, another key endogenous variable
- $ \{Y_t\} $ is a sequence of levels of national income, yet another endogenous variable
- $ a $ is the marginal propensity to consume in the Keynesian consumption function $ C_t = a Y_{t-1} + \gamma $
- $ b $ is the “accelerator coefficient” in the “investment accerator” $I_t = b (Y_{t-1} - Y_{t-2}) $
- $ \{\epsilon_{t}\} $ is an IID sequence standard normal random variables
- $ \sigma \geq 0 $ is a “volatility” parameter — setting $ \sigma = 0 $ recovers the nonstochastic case that we’ll start with
The model combines the consumption function
| $$
C_t = a Y_{t-1} + \gamma
$$
</td> | (1) | </tr></table>with the investment accelerator
| $$
I_t = b (Y_{t-1} - Y_{t-2})
$$
</td> | (2) | </tr></table>and the national income identity
| $$
Y_t = C_t + I_t + G_t
$$
</td> | (3) | </tr></table>- The parameter $ a $ is peoples’ marginal propensity to consume out of income - equation (1) asserts that people consume a fraction of math:a in (0,1) of each additional dollar of income
| $$
Y_t = (a+b) Y_{t-1} - b Y_{t-2} + (\gamma + G_t)
$$
</td> | </tr></table>or |
| $$
Y_t = \rho_1 Y_{t-1} + \rho_2 Y_{t-2} + (\gamma + G_t)
$$
</td> | (4) | </tr></table>where $ \rho_1 = (a+b) $ and $ \rho_2 = -b $
| $$
Y_{-1} = \bar Y_{-1}, \quad Y_{-2} = \bar Y_{-2}
$$
</td> | </tr></table>We’ll ordinarily set the parameters $ (a,b) $ so that starting from an arbitrary pair of initial conditions $ (\bar Y_{-1}, \bar Y_{-2}) $, national income $Y_t $ converges to a constant value as $ t $ becomes large |
| $$ Y_t = G_t + a (1-b) Y_{t-1} - a b Y_{t-2} + \sigma \epsilon_{t} $$ | (5) |
Mathematical analysis of the model¶
To get started, let’s set $ G_t \equiv 0 $, $ \sigma = 0 $, and $ \gamma = 0 $.
Then we can write equation (5) as
| $$
Y_t = \rho_1 Y_{t-1} + \rho_2 Y_{t-2}
$$
</td> | </tr></table>or |
| $$
Y_{t+2} - \rho_1 Y_{t+1} - \rho_2 Y_t = 0
$$
</td> | (6) | </tr></table>To discover the properties of the solution of (6), it is useful first to form the characteristic polynomial for (6):
| $$
z^2 - \rho_1 z - \rho_2
$$
</td> | (7) | </tr></table>where $ z $ is possibly a complex number
| $$
z^2 - \rho_1 z - \rho_2 = (z- \lambda_1 ) (z -\lambda_2) = 0
$$
</td> | (8) | </tr></table>Equation (8) is said to factor the characteristic polynomial
| $$
\lambda_1 = r e^{i \theta}, \ \lambda_2 = r e^{-i \theta}
$$
</td> | </tr></table>where $ r $ is the amplitude of the complex number and $ \theta $ is its angle. |
| $$
\lambda_1 = r (cos (\theta) + i \sin (\theta))
$$
</td> | </tr></table> |
| $$
\lambda_2 = r (cos (\theta) - i \sin(\theta))
$$
</td> | </tr></table>(To read about the polar form, see here) |
| $$
Y_t = \lambda_1^t c_1 + \lambda_2^t c_2
$$
</td> | </tr></table>where $ c_1 $ and $ c_2 $ are constants that depend on the two initial conditions and on $ \rho_1, \rho_2 $ |
| $$
Y_t = \tilde c_1 r^t \cos(\theta t + \tilde c_1)
$$
</td> | </tr></table>where $ \tilde c_1, \tilde c_2 $ is a pair of constants chosen to satisfy the given initial conditions for $ Y_{-1}, Y_{-2} $ |
| $$
| \lambda_j | < 1 \quad \quad \text{for } j = 1, 2
$$
</td> | </tr></table>Remark: When both eigenvalues $ \lambda_1, \lambda_2 $ have absolute values strictly less than one, the absolute value of the larger one governs the rate of convergence to the steady state of the non stochastic version of the model |
| $$ A = \begin{bmatrix} 1 & 0 & 0 \cr \gamma + G & \rho_1 & \rho_2 \cr 0 & 1 & 0 \end{bmatrix} $$ |
In [5]:
import numpy as np
import matplotlib.pyplot as plt
def param_plot():
"""this function creates the graph on page 189 of Sargent Macroeconomic Theory, second edition, 1987"""
fig, ax = plt.subplots(figsize=(12, 8))
ax.set_aspect('equal')
# Set axis
xmin, ymin = -3, -2
xmax, ymax = -xmin, -ymin
plt.axis([xmin, xmax, ymin, ymax])
# Set axis labels
ax.set(xticks=[], yticks=[])
ax.set_xlabel(r'$\rho_2$', fontsize=16)
ax.xaxis.set_label_position('top')
ax.set_ylabel(r'$\rho_1$', rotation=0, fontsize=16)
ax.yaxis.set_label_position('right')
# Draw (t1, t2) points
rho1 = np.linspace(-2, 2, 100)
ax.plot(rho1, -abs(rho1) + 1, c='black')
ax.plot(rho1, np.ones_like(rho1) * -1, c='black')
ax.plot(rho1, -(rho1**2 / 4), c='black')
# Turn normal axes off
for spine in ['left', 'bottom', 'top', 'right']:
ax.spines[spine].set_visible(False)
# Add arrows to represent axes
axes_arrows = {'arrowstyle': '<|-|>', 'lw': 1.3}
ax.annotate('', xy=(xmin, 0), xytext=(xmax, 0), arrowprops=axes_arrows)
ax.annotate('', xy=(0, ymin), xytext=(0, ymax), arrowprops=axes_arrows)
# Annotate the plot with equations
plot_arrowsl = {'arrowstyle': '-|>', 'connectionstyle': "arc3, rad=-0.2"}
plot_arrowsr = {'arrowstyle': '-|>', 'connectionstyle': "arc3, rad=0.2"}
ax.annotate(r'$\rho_1 + \rho_2 < 1$', xy=(0.5, 0.3), xytext=(0.8, 0.6),
arrowprops=plot_arrowsr, fontsize='12')
ax.annotate(r'$\rho_1 + \rho_2 = 1$', xy=(0.38, 0.6), xytext=(0.6, 0.8),
arrowprops=plot_arrowsr, fontsize='12')
ax.annotate(r'$\rho_2 < 1 + \rho_1$', xy=(-0.5, 0.3), xytext=(-1.3, 0.6),
arrowprops=plot_arrowsl, fontsize='12')
ax.annotate(r'$\rho_2 = 1 + \rho_1$', xy=(-0.38, 0.6), xytext=(-1, 0.8),
arrowprops=plot_arrowsl, fontsize='12')
ax.annotate(r'$\rho_2 = -1$', xy=(1.5, -1), xytext=(1.8, -1.3),
arrowprops=plot_arrowsl, fontsize='12')
ax.annotate(r'${\rho_1}^2 + 4\rho_2 = 0$', xy=(1.15, -0.35),
xytext=(1.5, -0.3), arrowprops=plot_arrowsr, fontsize='12')
ax.annotate(r'${\rho_1}^2 + 4\rho_2 < 0$', xy=(1.4, -0.7),
xytext=(1.8, -0.6), arrowprops=plot_arrowsr, fontsize='12')
# Label categories of solutions
ax.text(1.5, 1, 'Explosive\n growth', ha='center', fontsize=16)
ax.text(-1.5, 1, 'Explosive\n oscillations', ha='center', fontsize=16)
ax.text(0.05, -1.5, 'Explosive oscillations', ha='center', fontsize=16)
ax.text(0.09, -0.5, 'Damped oscillations', ha='center', fontsize=16)
# Add small marker to y-axis
ax.axhline(y=1.005, xmin=0.495, xmax=0.505, c='black')
ax.text(-0.12, -1.12, '-1', fontsize=10)
ax.text(-0.12, 0.98, '1', fontsize=10)
return fig
param_plot()
plt.show()
Explanation of the graph¶
The graph portrays regions in which the $ (\lambda_1, \lambda_2) $ root pairs implied by the $ (\rho_1 = (a+b), \rho_2 = - b) $ difference equation parameter pairs in the Samuelson model are such that:
- $ (\lambda_1, \lambda_2) $ are complex with modulus less than $ 1 $ - in this case, the $ \{Y_t\} $ sequence displays damped oscillations
- $ (\lambda_1, \lambda_2) $ are both real, but one is strictly greater than $ 1 $ - this leads to explosive growth
- $ (\lambda_1, \lambda_2) $ are both real, but one is strictly less than $ -1 $ - this leads to explosive oscillations
- $ (\lambda_1, \lambda_2) $ are both real and both are less than $ 1 $ in absolute value - in this case, there is smooth convergence to the steady state without damped cycles
Later we’ll present the graph with a red mark showing the particular point implied by the setting of $ (a,b) $
Function to describe implications of characteristic polynomial¶
In [6]:
def categorize_solution(rho1, rho2):
"""this function takes values of rho1 and rho2 and uses them to classify the type of solution"""
discriminant = rho1 ** 2 + 4 * rho2
if rho2 > 1 + rho1 or rho2 < -1:
print('Explosive oscillations')
elif rho1 + rho2 > 1:
print('Explosive growth')
elif discriminant < 0:
print('Roots are complex with modulus less than one; therefore damped oscillations')
else:
print('Roots are real and absolute values are less than zero; therfore get smooth convergence to a steady state')
In [7]:
### Test the categorize_solution function
categorize_solution(1.3, -.4)
none
Roots are real and absolute values are less than zero; therfore get smooth convergence to a steady stateFunction for plotting $ Y_t $ paths¶
A useful function for our work below
In [8]:
def plot_y(function=None):
"""function plots path of Y_t"""
plt.subplots(figsize=(12, 8))
plt.plot(function)
plt.xlabel('Time $t$')
plt.ylabel('$Y_t$', rotation=0)
plt.grid()
plt.show()
Manual or “by hand” root calculations¶
The following function calculates roots of the characteristic polynomial using high school algebra
(We’ll calculate the roots in other ways later)
The function also plots a $ Y_t $ starting from initial conditions that we set
In [9]:
from cmath import sqrt
##=== This is a 'manual' method ===#
def y_nonstochastic(y_0=100, y_1=80, alpha=.92, beta=.5, gamma=10, n=80):
"""Takes values of parameters and computes roots of characteristic polynomial.
It tells whether they are real or complex and whether they are less than unity in absolute value.
It also computes a simulation of length n starting from the two given initial conditions for national income"""
roots = []
rho1 = alpha + beta
rho2 = -beta
print('rho_1 is ', rho1)
print('rho_2 is ', rho2)
discriminant = rho1 ** 2 + 4 * rho2
if discriminant == 0:
roots.append(-rho1 / 2)
print('Single real root: ')
print(''.join(str(roots)))
elif discriminant > 0:
roots.append((-rho1 + sqrt(discriminant).real) / 2)
roots.append((-rho1 - sqrt(discriminant).real) / 2)
print('Two real roots: ')
print(''.join(str(roots)))
else:
roots.append((-rho1 + sqrt(discriminant)) / 2)
roots.append((-rho1 - sqrt(discriminant)) / 2)
print('Two complex roots: ')
print(''.join(str(roots)))
if all(abs(root) < 1 for root in roots):
print('Absolute values of roots are less than one')
else:
print('Absolute values of roots are not less than one')
def transition(x, t): return rho1 * x[t - 1] + rho2 * x[t - 2] + gamma
y_t = [y_0, y_1]
for t in range(2, n):
y_t.append(transition(y_t, t))
return y_t
plot_y(y_nonstochastic())
none
rho_1 is 1.42
rho_2 is -0.5
Two real roots:
[-0.6459687576256715, -0.7740312423743284]
Absolute values of roots are less than one
Reverse engineering parameters to generate damped cycles¶
The next cell writes code that takes as inputs the modulus $ r $ and phase $ \phi $ of a conjugate pair of complex numbers in polar form
| $$
\lambda_1 = r \exp(i \phi), \quad \lambda_2 = r \exp(- i \phi)
$$
</td> | </tr></table>- The code assumes that these two complex numbers are the roots of the characteristic polynomial |
| $$ \left [ \frac{\rho_{1}}{2} - \frac{1}{2} \sqrt{\rho_{1}^{2} + 4 \rho_{2}}, \quad \frac{\rho_{1}}{2} + \frac{1}{2} \sqrt{\rho_{1}^{2} + 4 \rho_{2}}\right ] $$ |
In [18]:
a = Symbol("alpha")
b = Symbol("beta")
r1 = a + b
r2 = -b
sympy.solve(z**2 - r1*z - r2, z)
Out[18]:
| $$ \left [ \frac{\alpha}{2} + \frac{\beta}{2} - \frac{1}{2} \sqrt{\alpha^{2} + 2 \alpha \beta + \beta^{2} - 4 \beta}, \quad \frac{\alpha}{2} + \frac{\beta}{2} + \frac{1}{2} \sqrt{\alpha^{2} + 2 \alpha \beta + \beta^{2} - 4 \beta}\right ] $$ |
Stochastic shocks¶
Now we’ll construct some code to simulate the stochastic version of the model that emerges when we add a random shock process to aggregate demand
In [19]:
def y_stochastic(y_0=0, y_1=0, alpha=0.8, beta=0.2, gamma=10, n=100, sigma=5):
"""This function takes parameters of a stochastic version of the model and proceeds to analyze
the roots of the characteristic polynomial and also generate a simulation"""
# Useful constants
rho1 = alpha + beta
rho2 = -beta
# Categorize solution
categorize_solution(rho1, rho2)
# Find roots of polynomial
roots = np.roots([1, -rho1, -rho2])
print(roots)
# Check if real or complex
if all(isinstance(root, complex) for root in roots):
print('Roots are complex')
else:
print('Roots are real')
# Check if roots are less than one
if all(abs(root) < 1 for root in roots):
print('Roots are less than one')
else:
print('Roots are not less than one')
# Generate shocks
epsilon = np.random.normal(0, 1, n)
# Define transition equation
def transition(x, t): return rho1 * \
x[t - 1] + rho2 * x[t - 2] + gamma + sigma * epsilon[t]
# Set initial conditions
y_t = [y_0, y_1]
# Generate y_t series
for t in range(2, n):
y_t.append(transition(y_t, t))
return y_t
plot_y(y_stochastic())
none
Roots are real and absolute values are less than zero; therfore get smooth convergence to a steady state
[ 0.7236068 0.2763932]
Roots are real
Roots are less than one
Let’s do a simulation in which there are shocks and the characteristic polynomial has complex roots
In [20]:
r = .97
period = 10 # length of cycle in units of time
phi = 2 * math.pi/period
### apply the reverse engineering function f
rho1, rho2, a, b = f(r, phi)
a = a.real # drop the imaginary part so that it is a valid input into y_nonstochastic
b = b.real
print("a, b = ", a, b)
plot_y(y_stochastic(y_0=40, y_1 = 42, alpha=a, beta=b, sigma=2, n=100))
none
a, b = 0.6285929690873979 0.9409000000000001
Roots are complex with modulus less than one; therefore damped oscillations
[ 0.78474648+0.57015169j 0.78474648-0.57015169j]
Roots are complex
Roots are less than one
Government spending¶
This function computes a response to either a permanent or one-off increase in government expenditures
In [21]:
def y_stochastic_g(y_0=20,
y_1=20,
alpha=0.8,
beta=0.2,
gamma=10,
n=100,
sigma=2,
g=0,
g_t=0,
duration='permanent'):
"""This program computes a response to a permanent increase in government expenditures that occurs
at time 20"""
# Useful constants
rho1 = alpha + beta
rho2 = -beta
# Categorize solution
categorize_solution(rho1, rho2)
# Find roots of polynomial
roots = np.roots([1, -rho1, -rho2])
print(roots)
# Check if real or complex
if all(isinstance(root, complex) for root in roots):
print('Roots are complex')
else:
print('Roots are real')
# Check if roots are less than one
if all(abs(root) < 1 for root in roots):
print('Roots are less than one')
else:
print('Roots are not less than one')
# Generate shocks
epsilon = np.random.normal(0, 1, n)
def transition(x, t, g):
# Non-stochastic - separated to avoid generating random series when not needed
if sigma == 0:
return rho1 * x[t - 1] + rho2 * x[t - 2] + gamma + g
# Stochastic
else:
epsilon = np.random.normal(0, 1, n)
return rho1 * x[t - 1] + rho2 * x[t - 2] + gamma + g + sigma * epsilon[t]
# Create list and set initial conditions
y_t = [y_0, y_1]
# Generate y_t series
for t in range(2, n):
# No government spending
if g == 0:
y_t.append(transition(y_t, t))
# Government spending (no shock)
elif g != 0 and duration == None:
y_t.append(transition(y_t, t))
# Permanent government spending shock
elif duration == 'permanent':
if t < g_t:
y_t.append(transition(y_t, t, g=0))
else:
y_t.append(transition(y_t, t, g=g))
# One-off government spending shock
elif duration == 'one-off':
if t == g_t:
y_t.append(transition(y_t, t, g=g))
else:
y_t.append(transition(y_t, t, g=0))
return y_t
A permanent government spending shock can be simulated as follows
In [22]:
plot_y(y_stochastic_g(g=10, g_t=20, duration='permanent'))
none
Roots are real and absolute values are less than zero; therfore get smooth convergence to a steady state
[ 0.7236068 0.2763932]
Roots are real
Roots are less than one
We can also see the response to a one time jump in government expenditures
In [23]:
plot_y(y_stochastic_g(g=500, g_t=50, duration='one-off'))
none
Roots are real and absolute values are less than zero; therfore get smooth convergence to a steady state
[ 0.7236068 0.2763932]
Roots are real
Roots are less than one
Wrapping everything into a class¶
Up to now we have written functions to do the work
Now we’ll roll up our sleeves and write a Python class called Samuelson for the Samuleson model
In [24]:
class Samuelson():
r"""This class represents the Samuelson model, otherwise known as the
multiple-accelerator model. The model combines the Keynesian multipler
with the accelerator theory of investment.
The path of output is governed by a linear second-order difference equation
.. math::
Y_t = + \alpha (1 + \beta) Y_{t-1} - \alpha \beta Y_{t-2}
Parameters
----------
y_0 : scalar
Initial condition for Y_0
y_1 : scalar
Initial condition for Y_1
alpha : scalar
Marginal propensity to consume
beta : scalar
Accelerator coefficient
n : int
Number of iterations
sigma : scalar
Volatility parameter. Must be greater than or equal to 0. Set
equal to 0 for non-stochastic model.
g : scalar
Government spending shock
g_t : int
Time at which government spending shock occurs. Must be specified
when duration != None.
duration : {None, 'permanent', 'one-off'}
Specifies type of government spending shock. If none, government
spending equal to g for all t.
"""
def __init__(self,
y_0=100,
y_1=50,
alpha=1.3,
beta=0.2,
gamma=10,
n=100,
sigma=0,
g=0,
g_t=0,
duration=None):
self.y_0, self.y_1, self.alpha, self.beta = y_0, y_1, alpha, beta
self.n, self.g, self.g_t, self.duration = n, g, g_t, duration
self.gamma, self.sigma = gamma, sigma
self.rho1 = alpha + beta
self.rho2 = -beta
self.roots = np.roots([1, -self.rho1, -self.rho2])
def root_type(self):
if all(isinstance(root, complex) for root in self.roots):
return 'Complex conjugate'
elif len(self.roots) > 1:
return 'Double real'
else:
return 'Single real'
def root_less_than_one(self):
if all(abs(root) < 1 for root in self.roots):
return True
def solution_type(self):
rho1, rho2 = self.rho1, self.rho2
discriminant = rho1 ** 2 + 4 * rho2
if rho2 >= 1 + rho1 or rho2 <= -1:
return 'Explosive oscillations'
elif rho1 + rho2 >= 1:
return 'Explosive growth'
elif discriminant < 0:
return 'Damped oscillations'
else:
return 'Steady state'
def _transition(self, x, t, g):
# Non-stochastic - separated to avoid generating random series when not needed
if self.sigma == 0:
return self.rho1 * x[t - 1] + self.rho2 * x[t - 2] + self.gamma + g
# Stochastic
else:
epsilon = np.random.normal(0, 1, self.n)
return self.rho1 * x[t - 1] + self.rho2 * x[t - 2] + self.gamma + g + self.sigma * epsilon[t]
def generate_series(self):
# Create list and set initial conditions
y_t = [self.y_0, self.y_1]
# Generate y_t series
for t in range(2, self.n):
# No government spending
if self.g == 0:
y_t.append(self._transition(y_t, t))
# Government spending (no shock)
elif self.g != 0 and self.duration == None:
y_t.append(self._transition(y_t, t))
# Permanent government spending shock
elif self.duration == 'permanent':
if t < self.g_t:
y_t.append(self._transition(y_t, t, g=0))
else:
y_t.append(self._transition(y_t, t, g=self.g))
# One-off government spending shock
elif self.duration == 'one-off':
if t == self.g_t:
y_t.append(self._transition(y_t, t, g=self.g))
else:
y_t.append(self._transition(y_t, t, g=0))
return y_t
def summary(self):
print('Summary\n' + '-' * 50)
print('Root type: ' + self.root_type())
print('Solution type: ' + self.solution_type())
print('Roots: ' + str(self.roots))
if self.root_less_than_one() == True:
print('Absolute value of roots is less than one')
else:
print('Absolute value of roots is not less than one')
if self.sigma > 0:
print('Stochastic series with sigma = ' + str(self.sigma))
else:
print('Non-stochastic series')
if self.g != 0:
print('Government spending equal to ' + str(self.g))
if self.duration != None:
print(self.duration.capitalize() +
' government spending shock at t = ' + str(self.g_t))
def plot(self):
fig, ax = plt.subplots(figsize=(12, 8))
ax.plot(self.generate_series())
ax.set(xlabel='Iteration', xlim=(0, self.n))
ax.set_ylabel('$Y_t$', rotation=0)
ax.grid()
# Add parameter values to plot
paramstr = '$\\alpha=%.2f$\n$\\beta=%.2f$\n$\\gamma=%.2f$\n$\\sigma=%.2f$\n$\\rho_1=%.2f$\n$\\rho_2=%.2f$'%(
self.alpha, self.beta, self.gamma, self.sigma, self.rho1, self.rho2)
props = dict(fc='white', pad=10, alpha=0.5)
ax.text(0.87, 0.05, paramstr, transform=ax.transAxes,
fontsize=12, bbox=props, va='bottom')
return fig
def param_plot(self):
# Uses the param_plot() function defined earlier (it is then able
# to be used standalone or as part of the model)
fig = param_plot()
ax = fig.gca()
# Add lambda values to legend
for i, root in enumerate(self.roots):
if isinstance(root, complex):
operator = ['+', ''] # Need to fill operator for positive as string is split apart
label = r'$\lambda_{0} = {1.real:.2f} {2} {1.imag:.2f}i$'.format(i+1, sam.roots[i], operator[i])
else:
label = r'$\lambda_{0} = {1.real:.2f}$'.format(i+1, sam.roots[i])
ax.scatter(0, 0, 0, label=label) # dummy to add to legend
# Add rho pair to plot
ax.scatter(self.rho1, self.rho2, 100, 'red', '+', label=r'$(\ \rho_1, \ \rho_2 \ )$', zorder=5)
plt.legend(fontsize=12, loc=3)
return fig
Illustration of Samuelson class¶
Now we’ll put our Samuelson class to work on an example
In [25]:
sam = Samuelson(alpha=0.8, beta=0.5, sigma=2, g=10, g_t=20, duration='permanent')
sam.summary()
none
Summary
---------------------------------------------------------
Root type: Complex conjugate
Solution type: Damped oscillations
Roots: [ 0.65+0.27838822j 0.65-0.27838822j]
Absolute value of roots is less than one
Stochastic series with sigma = 2
Government spending equal to 10
Permanent government spending shock at t = 20In [26]:
sam.plot()
plt.show()
Using the graph¶
We’ll use our graph to show where the roots lie and how their location is consistent with the behavior of the path just graphed
The red $+$ sign shows the location of the roots
In [27]:
sam.param_plot()
plt.show()
Using the LinearStateSpace class¶
It turns out that we can use the QuantEcon.py LinearStateSpace class to do much of the work that we have done from scratch above
Here is how we map the Samuelson model into an instance of a LinearStateSpace class
In [28]:
from quantecon import LinearStateSpace
""" This script maps the Samuelson model in the the LinearStateSpace class"""
alpha = 0.8
beta = 0.9
rho1 = alpha + beta
rho2 = -beta
gamma = 10
sigma = 1
g = 10
n = 100
A = [[1, 0, 0],
[gamma + g, rho1, rho2],
[0, 1, 0]]
G = [[gamma + g, rho1, rho2], # this is Y_{t+1}
[gamma, alpha, 0], # this is C_{t+1}
[0, beta, -beta]] # this is I_{t+1}
mu_0 = [1, 100, 100]
C = np.zeros((3,1))
C[1] = sigma # stochastic
sam_t = LinearStateSpace(A, C, G, mu_0=mu_0)
x, y = sam_t.simulate(ts_length=n)
fig, axes = plt.subplots(3, 1, sharex=True, figsize=(15, 8))
titles = ['Output ($Y_t$)', 'Consumption ($C_t$)', 'Investment ($I_t$)']
colors = ['darkblue', 'red', 'purple']
for ax, series, title, color in zip(axes, y, titles, colors):
ax.plot(series, color=color)
ax.set(title=title, xlim=(0, n))
ax.grid()
axes[-1].set_xlabel('Iteration')
plt.show()
Other methods in the LinearStateSpace class¶
Let’s plot impulse response functions for the instance of the Samuelson model using a method in the LinearStateSpace class
In [29]:
imres = sam_t.impulse_response()
imres = np.asarray(imres)
y1 = imres[:, :, 0]
y2 = imres[:, :, 1]
y1.shape
Out[29]:
| $$
\left ( 2, \quad 6, \quad 1\right )
$$
</td> | </tr></table>Now let’s compute the zeros of the characteristic polynomial by simply calculating the eigenvalues of $ A $ |
