<jupyter_start><jupyter_text>Part 3.2 Python OOP 2 with Keynesian Samuelson Accelerator Example This lecture creates nonstochastic and stochastic versions of <NAME>’s celebrated multiplier accelerator model [Sam39]In doing so, we extend the example of the Solow model class in our second OOP lectureOur objectives are to* provide a more detailed example of OOP and classes* review a famous model* review linear difference equations, both deterministic and stochastic  Samuelson’s ModelSamuelson 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 consumption plus investment plus government purchases* a Keynesian consumption function asserting that consumption at time $t$ is equal to a constant 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, which automatically calls forth an equal amount of aggregate supply* *(To read about linear difference equations see [Linear Diff Eq's on Wikipedia](https://en.wikipedia.org/wiki/Linear_difference_equation) or chapter IX of [Sar87](https://lectures.quantecon.org/py/zreferences.htmlsargent1987))*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 outputPossible dynamic properties include:* 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 explodeLater we present an extension that adds a random shock to the right side of the national income identity representing random fluctuations in aggregate demandThis 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](https://lectures.quantecon.org/py/zreferences.htmlsargent1987))  DetailsLet’s assume that* ${G_t}$ is a sequence of levels of government expenditures. We’ll start by setting **$G_t=G$** *for all $t$** ${C_t}$ 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 accelerator”** $I_t = b (Y_{t-1} - Y_{t-2})$* ${ε_t}$ is an IID sequence standard normal random variables.* $σ≥0$ is a “volatility” parameter — setting $σ=0$ recovers the nonstochastic case that we’ll start with.The model combines the consumption function $C_t = a Y_{t-1} + \gamma$ $(1)$ with the investment accelerator $I_t = b (Y_{t-1} - Y_{t-2})$ $(2)$ and the national income identity $Y_t=C_t+I_t+G_t$ $(3)$* 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* The parameter $b>0$ is the investment accelerator coefficient - equation $(2)$ asserts that people invest in physical capital when income is increasing and disinvest when it is decreasingEquations $(1), (2), and (3)$ imply the following second-order linear difference equation for national income: * $Y_t = (a+b) Y_{t-1} - b Y_{t-2} + (\gamma + G_t)$or* $Y_t = \rho_1 Y_{t-1} + \rho_2 Y_{t-2} + (\gamma + G_t)$    $(4)$where $ρ_1=(a+b)$ and $ρ_2=−b$To complete the model, we require two initial conditionsIf the model is to generate time series for $t=0, \ldots, T$, we require initial values,* $Y_{-1} = \bar Y_{-1}, \quad  Y_{-2} = \bar Y_{-2}$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.We are interested in studying* the transient fluctuations in $Y_t$ as it converges to its **steady state** level* the **rate** at which it converges to a steady state levelThe deterministic version of the model described so far — meaning that no random shocks hit aggregate demand — has only transient fluctuations.We can convert the model to one that has persistent irregular fluctuations by adding a random shock to aggregate demand Stochastic version of the modelWe create a **random** or **stochastic** version of the model by adding a random process of **shocks** or **disturbances** $\{\sigma \epsilon_t \}$ to the right side of equation $(4)$, leading to the second-order scalar linear stochastic difference equation:* $Y_t = G_t + a (1-b) Y_{t-1} - a b Y_{t-2} + \sigma \epsilon_{t}$    $(5)$ Mathematical analysis of the modelTo get started, let’s set $G_t≡0$, $σ=0$, and $γ=0$.Then we can write equation $(5)$ as* $Y_t = \rho_1 Y_{t-1} + \rho_2 Y_{t-2}$or* $Y_{t+2} - \rho_1 Y_{t+1} - \rho_2 Y_t  = 0$ $(6)$To discover the properties of the solution of $(6)$, it is useful first to form the characteristic polynomial for $(6)$:* $z^2−ρ_1z−ρ_2$    $(7)$where $z$ is possibly a complex numberWe want to find the two zeros (a.k.a. roots) – namely $λ_1,λ_2$ – of the characteristic polynomial.These are two special values of $z$, say $z=λ_1$ and $z=λ_2$, such that if we set $z$ equal to one of these values in expression $(7)$, the characteristic polynomial $(7)$ equals zero:* $z^2−ρ_1z−ρ_2=(z−λ_1)(z−λ_2)=0$    $(8)$Equation $(8)$ is said to **factor** the characteristic polynomialWhen the roots are complex, they will occur as a *complex conjugate pair*When the roots are complex, it is convenient to represent them in the polar form* $λ_1=re^{iω}, λ2=re^{−iω}$where $r$ is the amplitude of the complex number and $ω$ is its angle or phaseThese can also be represented as* $λ_1=r(cos(ω)+isin(ω))\\ λ_2=r(cos(ω)−isin(ω))$Given initial conditions $Y_{−1},Y_{−2}$, we want to generate a solution of the difference equation $(6)$It can be represented as* $Y_t = λ_1^t c_1 + λ_2^t c_2$where $c1$ and $c2$ are constants that depend on the two initial conditions and on $ρ_1,ρ_2$When the roots are complex, it is useful to pursue the following calculationsNotice that:* $$\begin{eqnarray}Y_t & =  & c_1 (r e^{i ω})^t + c_2 (r e^{-i ω})^t  \\& = & c_1 r^t e^{iω t} + c_2 r^t e^{-i ω t} \\& = &  c_1 r^t [\cos(ω t) + i \sin(ω t) ] + c_2 r^t [\cos(ω t) - i \sin(ω t) ] \\& = & (c_1 + c_2) r^t \cos(ω t) + i (c_1 - c_2) r^t \sin(ω t)\end{eqnarray}$$The only way that $Y_t$can be a real number for each t is if $c_1+c_2$ is a real number and $c_1−c_2$ is an imaginary number This happens only when $c_1$ and $c_2$ are complex conjugates, in which case they can be written in the polar forms $c_1=ve^{iθ},  c_2=ve^{−iθ}$So we can write* $$\begin{eqnarray}Y_t & = & v e^{i \theta} r^t e^{i \omega t} + v e ^{- i \theta} r^t e^{-i \omega t} \\& = & v r^t [ e^{i(\omega t + \theta)} + e^{-i (\omega t +\theta)}]  \\& = & 2 v r^t  \cos (\omega t + \theta)\end{eqnarray}$$where $v$ and $θ$ are constants that must be chosen to satisfy initial conditions for $Y_1,Y_2$where $~c_1, ~c_2$ is a pair of constants chosen to satisfy the given initial conditions for $Y_1,Y_2$This formula shows that when the roots are complex, $Y_t$ displays oscillations with period $\check p =\frac{2π}{ω}$ and damping factor $r$ We say that $\check p$ is the period because in that amount of time the cosine wave $cos(ωt+θ)$goes through exactly one complete cycles (Draw a cosine funtion to convince yourself of this please)Remark: Following [Sam39], we want to choose the parameters $a,b$of the model so that the absolute values (of the possibly complex) roots $λ_1,λ_2$ of the characteristic polynomial are both strictly less than one:* $|λ_j|<1 \text{ for } j=1,2$Remark: When both roots $λ_1,λ_2$ of the characteristic polynomial 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 Things this lecture doesWe write a function to generate simulations of a ${Y_t}$ sequence as a function of timeThe function requires that we put in initial conditions for $Y_1,Y_2$The function checks that $a,b$ are set so that $λ_1,λ_2$ are less than unity in absolute value (also called “modulus”)The function also tells us whether the roots are complex, and, if they are complex, returns both their real and complex partsIf the roots are both real, the function returns their values We use our function written to simulate paths that are stochastic (when $σ>0$)We have written the function in a way that allows us to input ${G_t}$ paths of a few simple forms, e.g.,* one time jumps in G at some time* a permanent jump in G that occurs at some timeWe proceed to use the Samuelson multiplier-accererator model as a laboratory to make a simple OOP exampleThe “state” that determines next period’s $Y_{t+1}$ is now not just the current value $Y_t$ but also the once lagged value $Y_{t−1}$This involves a little more bookkeeping than is required in the Solow model class definition**We use the Samuelson multiplier-accelerator model as a vehicle for teaching how we can gradually add more features to the class.**We want to have a method in the class that automatically generates a simulation, either nonstochastic $(σ=0)$ or stochastic $(σ>0)$We also show how to map the Samuelson model into a simple instance of the `LinearStateSpace` class described hereWe can use a `LinearStateSpace` instance to do various things that we did above with our homemade function and classAmong other things, we show by example that the eigenvalues of the matrix $A$that we use to form the instance of the LinearStateSpace class for the Samuelson model equal the roots of the characteristic polynomial $(7)$ for the Samuelson multiplier accelerator modelHere is the formula for the matrix $A$in the linear state space system in the case that government expenditures are a constant $G$* $A = \begin{bmatrix} 1 & 0 & 0 \cr \gamma + G & \rho_1 & \rho_2 \cr 0 & 1 & 0 \end{bmatrix}$  ImplementationWe’ll start by drawing an informative graph from page 189 of [Sar87]<jupyter_code>import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

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
    ρ1 = np.linspace(-2, 2, 100)
    ax.plot(ρ1, -abs(ρ1) + 1, c='black')
    ax.plot(ρ1, np.ones_like(ρ1) * -1, c='black')
    ax.plot(ρ1, -(ρ1**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()<jupyter_output><empty_output><jupyter_text>The graph portrays regions in which the $(λ_1,λ_2)$ root pairs implied by the $(ρ_1=(a+b),ρ_2=−b)$ difference equation parameter pairs in the Samuelson model are such that:* $(λ_1,λ_2)$ are complex with modulus less than 1 - in this case, the ${Y_t}$ sequence displays damped oscillations* $(λ_1,λ_2)$ are both real, but one is strictly greater than 1 - this leads to explosive growth* $(λ_1,λ_2)$ are both real, but one is strictly less than −1 - this leads to explosive oscillations* $(λ_1,λ_2)$ 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 cyclesLater 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**<jupyter_code>def categorize_solution(ρ1, ρ2):
    """this function takes values of ρ1 and ρ2 and uses them to classify the type of solution"""

    discriminant = ρ1 ** 2 + 4 * ρ2
    if ρ2 > 1 + ρ1 or ρ2 < -1:
        print('Explosive oscillations')
    elif ρ1 + ρ2 > 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; therefore get smooth convergence to a steady state')
### Test the categorize_solution function

categorize_solution(1.3, -.4)<jupyter_output>Roots are real and absolute values are less than zero; therefore get smooth convergence to a steady state<jupyter_text>**Function for plotting $Y_t$ paths*** A useful function for our work below<jupyter_code>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()<jupyter_output><empty_output><jupyter_text>**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<jupyter_code>from cmath import sqrt

##=== This is a 'manual' method ===#

def y_nonstochastic(y_0=100, y_1=80, α=.92, β=.5, γ=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 = []

    ρ1 = α + β
    ρ2 = -β

    print(f'ρ_1 is {ρ1}')
    print(f'ρ_2 is {ρ2}')

    discriminant = ρ1 ** 2 + 4 * ρ2

    if discriminant == 0:
        roots.append(-ρ1 / 2)
        print('Single real root: ')
        print(''.join(str(roots)))
    elif discriminant > 0:
        roots.append((-ρ1 + sqrt(discriminant).real) / 2)
        roots.append((-ρ1 - sqrt(discriminant).real) / 2)
        print('Two real roots: ')
        print(''.join(str(roots)))
    else:
        roots.append((-ρ1 + sqrt(discriminant)) / 2)
        roots.append((-ρ1 - 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 ρ1 * x[t - 1] + ρ2 * x[t - 2] + γ

    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())<jupyter_output>ρ_1 is 1.42
ρ_2 is -0.5
Two real roots: 
[-0.6459687576256715, -0.7740312423743284]
Absolute values of roots are less than one<jupyter_text>**Reverse engineering parameters to generate damped cycles**The next cell writes code that takes as inputs the modulus $r$ and phase $ϕ$ of a conjugate pair of complex numbers in polar form$λ_1=r \space exp(iϕ),\space λ_2=r \space exp(−iϕ)$* The code assumes that these two complex numbers are the roots of the characteristic polynomial* It then reverse engineers $(a,b)$ and $(ρ_1,ρ_2)$, pairs that would generate those roots<jupyter_code>### code to reverse engineer a  cycle
### y_t = r^t (c_1 cos(ϕ t) + c2 sin(ϕ t))
###

import cmath
import math

def f(r, ϕ):
    """
    Takes modulus r and angle ϕ of complex number r exp(j ϕ)
    and creates ρ1 and ρ2 of characteristic polynomial for which
    r exp(j ϕ) and r exp(- j ϕ) are complex roots.

    Returns the multiplier coefficient a and the accelerator coefficient b
    that verifies those roots.
    """
    g1 = cmath.rect(r, ϕ)  # Generate two complex roots
    g2 = cmath.rect(r, -ϕ)
    ρ1 = g1 + g2           # Implied ρ1, ρ2
    ρ2 = -g1 * g2
    b = -ρ2                # Reverse engineer a and b that validate these
    a = ρ1 - b
    return ρ1, ρ2, a, b

## Now let's use the function in an example
## Here are the example paramters

r = .95
period = 10                # Length of cycle in units of time
ϕ = 2 * math.pi/period

## Apply the function

ρ1, ρ2, a, b = f(r, ϕ)

print(f"a, b = {a}, {b}")
print(f"ρ1, ρ2 = {ρ1}, {ρ2}")


## Print the real components of ρ1 and ρ2

ρ1 = ρ1.real
ρ2 = ρ2.real

ρ1, ρ2<jupyter_output><empty_output><jupyter_text>**Root finding using numpy**Here we’ll use numpy to compute the roots of the characteristic polynomial<jupyter_code>r1, r2 = np.roots([1, -ρ1, -ρ2])

p1 = cmath.polar(r1)
p2 = cmath.polar(r2)

print(f"r, ϕ = {r}, {ϕ}")
print(f"p1, p2 = {p1}, {p2}")
# print(f"g1, g2 = {g1}, {g2}")

print(f"a, b = {a}, {b}")
print(f"ρ1, ρ2 = {ρ1}, {ρ2}")
##=== This method uses numpy to calculate roots ===#


def y_nonstochastic(y_0=100, y_1=80, α=.9, β=.8, γ=10, n=80):

    """ Rather than computing the roots of the characteristic polynomial by hand as we did earlier, this function
    enlists numpy to do the work for us """

    # Useful constants
    ρ1 = α + β
    ρ2 = -β

    categorize_solution(ρ1, ρ2)

    # Find roots of polynomial
    roots = np.roots([1, -ρ1, -ρ2])
    print(f'Roots are {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')

    # Define transition equation
    def transition(x, t): return ρ1 * x[t - 1] + ρ2 * x[t - 2] + γ

    # 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_nonstochastic())<jupyter_output>Roots are complex with modulus less than one; therefore damped oscillations
Roots are [0.85+0.27838822j 0.85-0.27838822j]
Roots are complex
Roots are less than one<jupyter_text>**Reverse engineered complex roots: example**The next cell studies the implications of reverse engineered complex rootsWe’ll generate an **undamped** cycle of period **10**<jupyter_code>r = 1   # generates undamped, nonexplosive cycles

period = 10   #  length of cycle in units of time
ϕ = 2 * math.pi/period

## Apply the reverse engineering function f

ρ1, ρ2, a, b = f(r, ϕ)

a = a.real  # drop the imaginary part so that it is a valid input into y_nonstochastic
b = b.real

print(f"a, b = {a}, {b}")

ytemp = y_nonstochastic(α=a, β=b, y_0=20, y_1=30)
plot_y(ytemp)<jupyter_output>a, b = 0.6180339887498949, 1.0
Roots are complex with modulus less than one; therefore damped oscillations
Roots are [0.80901699+0.58778525j 0.80901699-0.58778525j]
Roots are complex
Roots are less than one<jupyter_text>**Digression**: *using sympy to find roots*We can also use sympy to compute analytic formulas for the roots<jupyter_code>import sympy
from sympy import Symbol, init_printing
init_printing()

r1 = Symbol("ρ_1")
r2 = Symbol("ρ_2")
z = Symbol("z")

sympy.solve(z**2 - r1*z - r2, z)
a = Symbol("α")
b = Symbol("β")
r1 = a + b
r2 = -b

sympy.solve(z**2 - r1*z - r2, z)<jupyter_output><empty_output><jupyter_text>**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<jupyter_code>def y_stochastic(y_0=0, y_1=0, α=0.8, β=0.2, γ=10, n=100, σ=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
    ρ1 = α + β
    ρ2 = -β

    # Categorize solution
    categorize_solution(ρ1, ρ2)

    # Find roots of polynomial
    roots = np.roots([1, -ρ1, -ρ2])
    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
    ϵ = np.random.normal(0, 1, n)

    # Define transition equation
    def transition(x, t): return ρ1 * \
        x[t - 1] + ρ2 * x[t - 2] + γ + σ * ϵ[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())<jupyter_output>Roots are real and absolute values are less than zero; therefore get smooth convergence to a steady state
[0.7236068 0.2763932]
Roots are real
Roots are less than one<jupyter_text>Let’s do a simulation in which there are shocks and the characteristic polynomial has complex roots<jupyter_code>r = .97

period = 10   #  length of cycle in units of time
ϕ = 2 * math.pi/period

### apply the  reverse engineering function f

ρ1, ρ2, a, b = f(r, ϕ)

a = a.real  # drop the imaginary part so that it is a valid input into y_nonstochastic
b = b.real

print(f"a, b = {a}, {b}")
plot_y(y_stochastic(y_0=40, y_1 = 42, α=a, β=b, σ=2, n=100))<jupyter_output>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<jupyter_text>**Government spending**This function computes a response to either a permanent or one-off increase in government expenditures<jupyter_code>def y_stochastic_g(y_0=20,
                   y_1=20,
                   α=0.8,
                   β=0.2,
                   γ=10,
                   n=100,
                   σ=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
    ρ1 = α + β
    ρ2 = -β

    # Categorize solution
    categorize_solution(ρ1, ρ2)

    # Find roots of polynomial
    roots = np.roots([1, -ρ1, -ρ2])
    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
    ϵ = np.random.normal(0, 1, n)

    def transition(x, t, g):

        # Non-stochastic - separated to avoid generating random series when not needed
        if σ == 0:
            return ρ1 * x[t - 1] + ρ2 * x[t - 2] + γ + g

        # Stochastic
        else:
            ϵ = np.random.normal(0, 1, n)
            return ρ1 * x[t - 1] + ρ2 * x[t - 2] + γ + g + σ * ϵ[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<jupyter_output><empty_output><jupyter_text>A permanent government spending shock can be simulated as follows<jupyter_code>plot_y(y_stochastic_g(g=10, g_t=20, duration='permanent'))<jupyter_output>Roots are real and absolute values are less than zero; therefore get smooth convergence to a steady state
[0.7236068 0.2763932]
Roots are real
Roots are less than one<jupyter_text>We can also see the response to a one time jump in government expenditures<jupyter_code>plot_y(y_stochastic_g(g=500, g_t=50, duration='one-off'))<jupyter_output>Roots are real and absolute values are less than zero; therefore get smooth convergence to a steady state
[0.7236068 0.2763932]
Roots are real
Roots are less than one<jupyter_text>**Wrapping everything into a class**Up to now we have written functions to do the workNow we’ll roll up our sleeves and write a Python class called `Samuelson` for the Samuleson model<jupyter_code>class Samuelson():

    r"""This class represents the Samuelson model, otherwise known as the
    multiple-accelerator model. The model combines the Keynesian multiplier
    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
    α : scalar
        Marginal propensity to consume
    β : scalar
        Accelerator coefficient
    n : int
        Number of iterations
    σ : 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,
                 α=1.3,
                 β=0.2,
                 γ=10,
                 n=100,
                 σ=0,
                 g=0,
                 g_t=0,
                 duration=None):

        self.y_0, self.y_1, self.α, self.β = y_0, y_1, α, β
        self.n, self.g, self.g_t, self.duration = n, g, g_t, duration
        self.γ, self.σ = γ, σ
        self.ρ1 = α + β
        self.ρ2 = -β
        self.roots = np.roots([1, -self.ρ1, -self.ρ2])

    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):
        ρ1, ρ2 = self.ρ1, self.ρ2
        discriminant = ρ1 ** 2 + 4 * ρ2
        if ρ2 >= 1 + ρ1 or ρ2 <= -1:
            return 'Explosive oscillations'
        elif ρ1 + ρ2 >= 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.σ == 0:
            return self.ρ1 * x[t - 1] + self.ρ2 * x[t - 2] + self.γ + g

        # Stochastic
        else:
            ϵ = np.random.normal(0, 1, self.n)
            return self.ρ1 * x[t - 1] + self.ρ2 * x[t - 2] + self.γ + g + self.σ * ϵ[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(f'Root type: {self.root_type()}')
        print(f'Solution type: {self.solution_type()}')
        print(f'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.σ > 0:
            print('Stochastic series with σ = ' + str(self.σ))
        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 = f'$\\alpha={self.α:.2f}$ \n $\\beta={self.β:.2f}$ \n $\\gamma={self.γ:.2f}$ \n \
$\\sigma={self.σ:.2f}$ \n $\\rho_1={self.ρ1:.2f}$ \n $\\rho_2={self.ρ2:.2f}$'
        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 λ 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 = rf'$\lambda_{i+1} = {sam.roots[i].real:.2f} {operator[i]} {sam.roots[i].imag:.2f}i$'
            else:
                label = rf'$\lambda_{i+1} = {sam.roots[i].real:.2f}$'
            ax.scatter(0, 0, 0, label=label) # dummy to add to legend

        # Add ρ pair to plot
        ax.scatter(self.ρ1, self.ρ2, 100, 'red', '+', label=r'$(\ \rho_1, \ \rho_2 \ )$', zorder=5)

        plt.legend(fontsize=12, loc=3)

        return fig<jupyter_output><empty_output><jupyter_text>**Illustration of Samuelson class**Now we’ll put our Samuelson class to work on an example<jupyter_code>sam = Samuelson(α=0.8, β=0.5, σ=2, g=10, g_t=20, duration='permanent')
sam.summary()
sam.plot()
plt.show()<jupyter_output><empty_output><jupyter_text>Using the graphWe’ll use our graph to show where the roots lie and how their location is consistent with the behavior of the path just graphedThe red $+$ sign shows the location of the roots<jupyter_code>sam.param_plot()
plt.show()<jupyter_output><empty_output><jupyter_text>**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 aboveHere is how we map the Samuelson model into an instance of a `LinearStateSpace` class<jupyter_code>from quantecon import LinearStateSpace

""" This script maps the Samuelson model in the the LinearStateSpace class"""
α = 0.8
β = 0.9
ρ1 = α + β
ρ2 = -β
γ = 10
σ = 1
g = 10
n = 100

A = [[1,        0,      0],
     [γ + g,   ρ1,     ρ2],
     [0,        1,      0]]

G = [[γ + g, ρ1,   ρ2],         # this is Y_{t+1}
     [γ,      α,    0],         # this is C_{t+1}
     [0,      β,   -β]]         # this is I_{t+1}

μ_0 = [1, 100, 100]
C = np.zeros((3,1))
C[1] = σ # stochastic

sam_t = LinearStateSpace(A, C, G, mu_0=μ_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()<jupyter_output><empty_output><jupyter_text>**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<jupyter_code>imres = sam_t.impulse_response()
imres = np.asarray(imres)
y1 = imres[:, :, 0]
y2 = imres[:, :, 1]
y1.shape<jupyter_output><empty_output><jupyter_text>Now let’s compute the zeros of the characteristic polynomial by simply calculating the eigenvalues of $A$<jupyter_code>A = np.asarray(A)
w, v = np.linalg.eig(A)
print(w)<jupyter_output>[0.85+0.42130749j 0.85-0.42130749j 1.  +0.j        ]<jupyter_text>**Inheriting methods from** `LinearStateSpace`We could also create a subclass of `LinearStateSpace` (inheriting all its methods and attributes) to add more functions to use<jupyter_code>class SamuelsonLSS(LinearStateSpace):

    """
    this subclass creates a Samuelson multiplier-accelerator model
    as a linear state space system
    """
    def __init__(self,
                 y_0=100,
                 y_1=100,
                 α=0.8,
                 β=0.9,
                 γ=10,
                 σ=1,
                 g=10):

        self.α, self.β = α, β
        self.y_0, self.y_1, self.g = y_0, y_1, g
        self.γ, self.σ = γ, σ

        # Define intial conditions
        self.μ_0 = [1, y_0, y_1]

        self.ρ1 = α + β
        self.ρ2 = -β

        # Define transition matrix
        self.A = [[1,                 0,         0],
                  [γ + g,       self.ρ1,   self.ρ2],
                  [0,                 1,         0]]

        # Define output matrix
        self.G = [[γ + g, self.ρ1, self.ρ2],         # this is Y_{t+1}
                  [γ,           α,       0],         # this is C_{t+1}
                  [0,           β,      -β]]         # this is I_{t+1}

        self.C = np.zeros((3, 1))
        self.C[1] = σ  # stochastic

        # Initialize LSS with parameters from Samuleson model
        LinearStateSpace.__init__(self, self.A, self.C, self.G, mu_0=self.μ_0)

    def plot_simulation(self, ts_length=100, stationary=True):

        # Temporarily store original parameters
        temp_μ = self.μ_0
        temp_Σ = self.Sigma_0

        # Set distribution parameters equal to their stationary values for simulation
        if stationary == True:
            try:
                self.μ_x, self.μ_y, self.σ_x, self.σ_y = self.stationary_distributions()
                self.μ_0 = self.μ_y
                self.Σ_0 = self.σ_y
            # Exception where no convergence achieved when calculating stationary distributions
            except ValueError:
                print('Stationary distribution does not exist')

        x, y = self.simulate(ts_length)

        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')

        # Reset distribution parameters to their initial values
        self.μ_0 = temp_μ
        self.Sigma_0 = temp_Σ

        return fig

    def plot_irf(self, j=5):

        x, y = self.impulse_response(j)

        # Reshape into 3 x j matrix for plotting purposes
        yimf = np.array(y).flatten().reshape(j+1, 3).T

        fig, axes = plt.subplots(3, 1, sharex=True, figsize=(15, 8))
        labels = ['$Y_t$', '$C_t$', '$I_t$']
        colors = ['darkblue', 'red', 'purple']
        for ax, series, label, color in zip(axes, yimf, labels, colors):
            ax.plot(series, color=color)
            ax.set(xlim=(0, j))
            ax.set_ylabel(label, rotation=0, fontsize=14, labelpad=10)
            ax.grid()

        axes[0].set_title('Impulse Response Functions')
        axes[-1].set_xlabel('Iteration')

        return fig

    def multipliers(self, j=5):
        x, y = self.impulse_response(j)
        return np.sum(np.array(y).flatten().reshape(j+1, 3), axis=0)<jupyter_output><empty_output><jupyter_text>**Illustrations**Let’s show how we can use the `SamuelsonLSS`<jupyter_code>samlss = SamuelsonLSS()
samlss.plot_simulation(100, stationary=False)
plt.show()
samlss.plot_simulation(100, stationary=True)
plt.show()
samlss.plot_irf(100)
plt.show()
samlss.multipliers()<jupyter_output><empty_output><jupyter_text>**Pure multiplier model**Let’s shut down the accelerator by setting $b=0$ to get a pure multiplier model* the absence of cycles gives an idea about why Samuelson included the accelerator<jupyter_code>pure_multiplier = SamuelsonLSS(α=0.95, β=0)
pure_multiplier.plot_simulation()
pure_multiplier = SamuelsonLSS(α=0.8, β=0)
pure_multiplier.plot_simulation()
pure_multiplier.plot_irf(100)<jupyter_output><empty_output>
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