# Lucas Asset Pricing with advanced Approximation Methods¶

## Introduction¶

This note describes why and how we modified the computer code of the original lucastree.py module. We briefly reformulate Lucas' asset pricing problem as found in the lecture notes . Denote by $y$ the fruit of the tree. The fruit’s growth rate follows the process $G(y,z') = y^\alpha z'$ with $z' \sim \log N(0,\sigma^2)$. The investor has CRRA preferences with curvature parameter $\gamma$ and discount factor $\beta$. Following Lucas (1978) , the pricing function, $p(y)$, solves the functional equation:

$$f(y) = h(y) + \beta \int_Z f(G(y,z')) Q(dz').$$ with \begin{align*} f(y) &= p(y)y^{-\gamma}, \\ h(y) &= \beta \int_Z \left( G(y,z') \right)^{1-\gamma} Q(dz') = \beta y^{ (1-\gamma)\alpha } \exp \left( (1-\gamma)^2 \sigma^2/2 \right). \end{align*}

We want the numeric solution $f$ to comply with theoretical predictions about its functional form. In the following, it is first documented under which circumstances $h$ transmits montoncity and concavity onto $f$. In particular, we prove that if $G$ is strictly increasing and concave 1, $h$ transmits the sign of its first and second derivatives onto $f$. Additionally, we show that if both $G$ and $h$ are strictly decreasing and convex, $f$ is strictly decreasing and convex as well. The solution to the functional equation is numerically obtained by iterating the contraction mapping $Tf(y) = h(y) + \beta \int_Z f(G(y,z')) Q(dz')$ until the distance between two successive iterations is smaller than a tolerance criteria. To compute the integral numerically, $f(G(y,z'))$ needs to be evaluated at arguments $y$ that are not on the grid. This is a chance to impose the properties of $h$ onto $f$ through an appropriate approximation routine. This note discusses how to implement such a routine at the end.

1. For the sake of brevity, when writing strictly increasing and concave we really mean strictly increasing and strictly concave. Also, strictly decreasing and convex refers to strictly decreasing and strictly convex, etc.

## Theoretical Predictions about the Functional Form of the Solution to Lucas' Asset Pricing Equation¶

This section documents under which circumstances $f$ inherits the sign of the first and second derivatives of $h$. In the following, suppose all necessary assumptions to guarantee a unique solution to Lucas' asset pricing problem are satisfied. One assumption is that the function $h$ is bounded in the supremum norm. Numercially, the assumption is satisfied because the lower end of the interval $Y$ is striclty positive and because $Y$ is bounded. Theoretically, one can prove that the $h$ needs only be bounded in a weighted supremum norm when the parameter $\alpha > 0$. Based on exercise 9.7 of the book by Stokey and Lucas with Prescott (1989), we prove the following proposition:

Proposition

1. Suppose $G$ is strictly increasing and concave in $y$. If $h$ is sctrictly increasing and convave, $f$ is strictly increasing and concave. If $h$ is sctrictly decreasing and convex, $f$ is strictly decreasing and convex.
2. Suppose $G$ is strictly decreasing and convex in $y$. If $h$ is strictly decreasing and convex, $f$ is strictly decreasing and convex.

Proof

1 Following the notation of the lecture notes, denote by $cb\mathbf{R}_+$ the set of continuous and bounded functions $f:\mathbf{R}_{+} \rightarrow \mathbf{R}_{+}$ . The set $cb'\mathbf{R}_{+} \subset cb\mathbf{R}_{+}$ is the set of continuous, bounded, nondecreasing and concave functions, and $cb''\mathbf{R}_{+} \subset cb'\mathbf{R}_{+}$ imposes additionally strict monotonicity and concavity. We want to show that the contraction operator $T$ maps any function $\tilde{f} \in cb'\mathbf{R}_{+}$ into the subset $cb''\mathbf{R}_{+}$. As the solution to the functional equation is characterized by $Tf = f$ and $cb'\mathbf{R}_{+}$ is a closed set, if the operator $T$ transforms any nondecreasing and concave function into a strictly increasing and concave function, then $f$ is strictly increasing and concave (Corollary 1 of the Contraction Mapping Theorem in Stokey and Lucas with Prescott (1989), p. 52).

To show the desired result, suppose first that $h$ is strictly increasing and concave and pick any $\tilde{f} \in cb'\mathbf{R}_{+}$. To begin, study whether $T\tilde{f}$ is strictly increasing. For any pair $\hat{y},y \in Y$ with $\hat{y} > y$, the function $T\tilde{f}$ satisfies:

\begin{align*} T\tilde{f}(\hat{y}) &= h(\hat{y}) + \beta \int_Z \tilde{f}( G(\hat{y},z')) Q(dz')\\ &> h(y) + \beta \int_Z \tilde{f}( G(y,z')) Q(dz')\\ &= T\tilde{f}(y). \end{align*}

The inequality holds because $G$ and $h$ are strictly increasing and $\tilde{f}$ is nondecreasing. Hence, $T\tilde{f}$ is strictly increasing.

To analyze concavity, define $y_{\omega} = \omega y + (1-\omega) y'$, for any $y,y' \in Y$, $y \neq y'$, and $0 < \omega < 1$. The strict concavity form of $h$ and $G$, together with $\tilde{f}$ being concave, ensure that:

\begin{align*} T\tilde{f}(y_\omega) &= h(y_\omega) + \beta \int_Z \tilde{f}( G(y_\omega,z')) Q(dz') \\ &> \omega \left[ h(y) + \beta \int_Z \tilde{f}( G(y,z')) Q(dz') \right] + (1 - \omega) \left[ h(y') + \beta \int_Z \tilde{f}( G(y',z')) Q(dz') \right] \\ &= \omega T\tilde{f}(y) + (1-\omega) T \tilde{f}(y'). \end{align*}

The function $T\tilde{f}$ is stricly concave. Taken together, we know that for any $\tilde{f} \in cb'\mathbf{R}_{+}$, $T\tilde{f} \in cb''\mathbf{R}_{+}$. Hence, $f$ must be an element of the set $cb''\mathbf{R}_+$, guaranteeing that $f$ has the same functional form as $h$.

Now, suppose $h$ is convex and decreasing. We could again define the operator $T$ as $Tf(y) = h(y) + \beta \int_Z f(G(y,z')) Q(dz')$ and study into which subset a candidate solution is mapped into. To facilitate analysis though, take a different route. Look at the modified operator $$Tf_{-} = h_{-} + \beta \int_Z f_{-} (G(y,z')) Q(z'),$$ with $h_{-} = -h$ and $f_{-} = -f$. Under the same assumptions guaranteeing a unique solution to the original contraction mapping, there exists a unique solution to the modified contraction mapping. As $h_{-}$ is strictly increasing and concave, the proof above applies to the modified contraction mapping. As $f_{-}$ is strictly increasing and concave, $f$ is strictly decreasing and convex and inherits the properties of $h$.

2 As both $G$ and $h$ are strictly decreasing and convex, one can proceed in a similar fashion as in case (1.) to show that $h$ transmits its functional form to $f$.

The different cases of the proposition can be rephrased in terms of the values of the parameters $\gamma,\alpha$. The functional form of $h$ is jointly determined by $\gamma,\alpha$ as $h(y) = y^{(1-\gamma)\alpha} \exp \left( (1-\gamma)^2 \sigma^2/2 \right)$. If $0 < \alpha < 1$, $G$ is strictly increasing and concave and case (1.) of the proposition applies. If $0 < \gamma < 1$, $f$ is strictly increasing and concave. If $\gamma > 1$, $f$ is strictly decreasing and convex. In contrast, suppose $-1 < \alpha < 0$. If $0 < \gamma < 1$ case (2.) of the proposition applies and $f$ is strictly decreasing and convex. If $\gamma > 1$, theory does not offer any help in determining the functional form of $f$. In this situation $G$ is decreasing and convex, while $h$ is increasing. Our proposition is deliberately more restrictive than the one in exercise 9.7 of Stokey and Lucas with Prescott (1989). Because we can calculate the functions $f$ analytically for the special cases of $\alpha \in \left\lbrace 0,1\right\rbrace$, numercial techniques are not needed.

## Imposing the functional form of $h$ onto $f$ through advanced approximation¶

This section describes how we impose the functional form of $h$ onto $f$. The solution to the functional equation is numerically obtained by iterating the contraction mapping $Tf(y) = h(y) + \beta \int_Z f(G(y,z')) Q(dz')$ until the distance between two successive iterations is smaller than a tolerance criteria. To compute the integral numerically, $f(G(y,z'))$ needs to be evaluated at arguments $x$ that are not on the grid through numerical approximation. This approximation is a chance to impose the properties of $h$ onto $f$. The grid points are a set $Y_{\text{Grid}} = \left\lbrace y_1,y_2,\ldots, y_{N-1},y_N \right\rbrace \subset Y$, with $y_l < y_m$ if $l < m$, $l,m \in \mathbf{N}$. Point $x \in Y$ is not on the gird. If $y_1 < x < y_N$ we interpolate the functional $f$ at $x$ by:

$$f(x) = f(y_L) + \dfrac{f(y_H) - f(y_L)}{h(y_H) - h(y_L)} \left( h(x) - h(y_L) \right).$$

with $y_L = \max \left\lbrace y_i \in Y_{\text{Grid}} : y_i < x \right\rbrace$ and $y_H = \min \left\lbrace y_i \in Y_{\text{Grid}}: y_i > x \right\rbrace$. For any point $x$ lower than $y_1$ or higher than $y_N$, we define the function value as:

\begin{align} f(x) = \begin{cases} f(y_1) + \dfrac{f(y_1) - f(y_2)}{h(y_1) - h(y_2)} \left(h(x) - h(y_1) \right) & \text{if } x < y_1,\\ f(y_N) + \dfrac{f(y_N) - f(y_{N-1})}{h(y_N) - h(y_{N-1})} \left( h(x) - h(y_N) \right) & \text{if } x > y_N. \end{cases} \end{align}

The approximation transmits the slope and shape of the function $h$ onto $f$ as $f'(x) \propto h'(x)$ and $f''(x) \propto h''(x)$ because the ratio in front of $h(x)$ is always positive. The function interpolationFunction of the modified lucastree.py module converts this idea into computer code. The entire module is contained in the next cell.

In [1]:
%%writefile ./lucastree.py
r"""
Filename: lucastree.py

Authors: Joao Brogueira and Fabian Schuetze

This file is a slight modification of the lucastree.py file
by Thomas Sargent, John Stachurski, Spencer Lyon under the
quant-econ project. We don't claim authorship of the entire file,
but full responsability for it and any existing mistakes.

Solves the price function for the Lucas tree in a continuous state
setting, using piecewise linear approximation for the sequence of
candidate price functions.  The consumption endownment follows the log
linear AR(1) process

.. math::

log y' = \alpha log y + \sigma \epsilon

where y' is a next period y and epsilon is an iid standard normal shock.
Hence

.. math::

y' = y^{\alpha} * \xi,

where

.. math::

\xi = e^(\sigma * \epsilon)

The distribution phi of xi is

.. math::

\phi = LN(0, \sigma^2),

where LN means lognormal.

"""
#from __future__ import division  # == Omit for Python 3.x == #
import numpy as np
from scipy.stats import lognorm
from quantecon.compute_fp import compute_fixed_point

class LucasTree(object):

"""
Class to solve for the price of a tree in the Lucas
asset pricing model

Parameters
----------
gamma : scalar(float)
The coefficient of risk aversion in the investor's CRRA utility
function
beta : scalar(float)
The investor's discount factor
alpha : scalar(float)
The correlation coefficient in the shock process
sigma : scalar(float)
The volatility of the shock process
grid : array_like(float), optional(default=None)
The grid points on which to evaluate the asset prices. Grid
points should be nonnegative. If None is passed, we will create
a reasonable one for you

Attributes
----------
gamma, beta, alpha, sigma, grid : see Parameters
grid_min, grid_max, grid_size : scalar(int)
Properties for grid upon which prices are evaluated
init_h : array_like(float)
The functional values h(y) with grid points being arguments
phi : scipy.stats.lognorm
The distribution for the shock process

Notes
-----
This file is a slight modification of the lucastree.py file
by Thomas Sargent, John Stachurski, Spencer Lyon, [SSL]_ under the
quant-econ project. We don't claim authorship of the entire file,
but full responsability for it and any existing mistakes.

References
----------
.. [SSL] Thomas Sargent, John Stachurski and Spencer Lyon, lucastree.py,
GitHub repository,
https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/models/lucastree.py

Examples
--------
>>> tree = LucasTree(gamma=2, beta=0.95, alpha=0.90, sigma=0.1)
>>> grid, price_vals = tree.grid, tree.compute_lt_price()

"""

def __init__(self, gamma, beta, alpha, sigma, grid=None):
self.gamma = gamma
self.beta = beta
self.alpha = alpha
self.sigma = sigma

# == set up grid == #
if grid is None:
(self.grid, self.grid_min,
self.grid_max, self.grid_size) = self._new_grid()
else:
self.grid = np.asarray(grid)
self.grid_min = min(grid)
self.grid_max = max(grid)
self.grid_size = len(grid)

# == set up distribution for shocks == #
self.phi = lognorm(sigma)

# == set up integration bounds. 4 Standard deviations. Make them
# private attributes b/c users don't need to see them, but we
# only want to compute them once. == #
self._int_min = np.exp(-5 * sigma)
self._int_max = np.exp(5 * sigma)

# == Set up h for the Lucas Operator == #
self.init_h = self.h(self.grid)

def h(self, x):
"""
Compute the function values of h in the Lucas operator.

Parameters
----------
x : array_like(float)
The arguments over which to computer the function values

Returns
-------
h : array_like(float)
The functional values

Notes
-----
Recall the functional form of h

.. math:: h(x) &= \beta * \int_Z u'(G(x,z)) phi(dz)
&= \beta x**((1-\gamma)*\alpha) * \exp((1-\gamma)**2 *\sigma /2)

"""
alpha, gamma, beta, sigma = self.alpha, self.gamma, self.beta, self.sigma
h = beta * x**((1 - gamma) * alpha) * \
np.exp((1 - gamma)**2 * sigma**2 / 2) * np.ones(x.size)

return h

def _new_grid(self):
"""
Construct the default grid for the problem

This is defined to be np.linspace(0, 10, 100) when alpha >= 1
and 100 evenly spaced points covering 4 standard deviations
when alpha < 1

"""
grid_size = 50
if abs(self.alpha) >= 1.0:
grid_min, grid_max = 0.1, 10
else:
# == Set the grid interval to contain most of the mass of the
# stationary distribution of the consumption endowment == #
ssd = self.sigma / np.sqrt(1 - self.alpha**2)
grid_min, grid_max = np.exp(-4 * ssd), np.exp(4 * ssd)

grid = np.linspace(grid_min, grid_max, grid_size)

return grid, grid_min, grid_max, grid_size

def integrate(self, g, int_min=None, int_max=None):
"""
Integrate the function g(z) * self.phi(z) from int_min to
int_max.

Parameters
----------
g : function
The function which to integrate

int_min, int_max : scalar(float), optional
The bounds of integration. If either of these parameters are
None (the default), they will be set to 4 standard
deviations above and below the mean.

Returns
-------
result : scalar(float)
The result of the integration

"""
# == Simplify notation == #
phi = self.phi
if int_min is None:
int_min = self._int_min
if int_max is None:
int_max = self._int_max

# == set up integrand and integrate == #
integrand = lambda z: g(z) * phi.pdf(z)
result, error = fixed_quad(integrand, int_min, int_max, n=20)
return result, error

def Approximation(self, x, grid, f):
r"""
Approximates the function f at given sample points x.

Parameters
----------
x: array_like(float)
Sample points over which the function f is evaluated

grid: array_like(float)
The grid values representing the domain of f

f: array_like(float)
The function values of f over the grid

Returns:
--------
fApprox: array_like(float)
The approximated function values at x

Notes
-----
Interpolation is done by the following function:

.. math:: f(x) = f(y_L) + \dfrac{f(y_H) - f(y_L)}{h(y_H) - h(y_L)} (h(x) - h(y_L) ).

Extrapolation is done as follows:

.. math:: f(x) =
\begin{cases}
f(y_1) + \dfrac{f(y_1) - f(y_2)}{h(y_1) - h(y_2)} \left(h(x) - h(y_1) \right) & \text{if } x < y_1,\\
f(y_N) + \dfrac{f(y_N) - f(y_{N-1})}{h(y_N) - h(y_{N-1})} \left( h(x) - h(y_N) \right) & \text{if } x > y_N.
\end{cases}

The approximation routine imposes the functional
form of the function :math:h onto the function math:f, as stated
in chapter 9.2 (in particular theorem 9.6 and 9.7 and exercise 9.7) of the
book by Stokey, Lucas and Prescott (1989).

"""
# == Initalize and create empty arrays to be filled in the == #
gamma, sigma, beta = self.gamma, self.sigma, self.beta
hX, hGrid = self.h(x), self.init_h
fL, fH, fApprox = np.empty_like(x), np.empty_like(x), np.empty_like(x)
hL, idxL, idxH, hH = np.empty_like(x), np.empty_like(
x), np.empty_like(x), np.empty_like(x)

# == Create Boolean array to determine which sample points are used for interpoltion
# and which are used for extrapolation == #
lower, middle, upper = (x < grid[0]), (x > grid[0]) & (
x < grid[-1]), (x > grid[-1])

# == Calcualte the indices of y_L, idxL[index], and y_H ,idxH[index], that are below and above a sample point, called value.
# In the notation of the interpolation routine, these indices are used to pick the function values
# f(y_L),f(y_H),h(y_L) and h(y_H) == #
for index, value in enumerate(x):
# Calculates the indices of y_L
idxL[index] = (np.append(grid[grid <= value], grid[0])).argmax()
idxH[index] = min(idxL[index] + 1, len(grid) - 1)
fL[index] = f[idxL[index]]
fH[index] = f[idxH[index]]
hL[index] = hGrid[idxL[index]]
hH[index] = hGrid[idxH[index]]

# == Interpolation == #
if self.alpha != 0:
ratio = (fH[middle] - fL[middle]) / (hH[middle] - hL[middle])
elif self.alpha == 0:
# If self.alpha ==0, ratio is zero, as hH == hL
ratio = (hH[middle] - hL[middle])
fApprox[middle] = fL[middle] + ratio * (hX[middle] - hL[middle])

# == Extrapolation == #
if self.alpha != 0:
fApprox[lower] = f[
0] + (f[0] - f[1]) / (hGrid[0] - hGrid[1]) * (hX[lower] - hGrid[0])
fApprox[upper] = f[-1] + \
(f[-1] - f[-2]) / (hGrid[-1] - hGrid[-2]) * \
(hX[upper] - hGrid[-1])
elif self.alpha == 0:
fApprox[lower] = f[0]
fApprox[upper] = f[-1]

return fApprox

def lucas_operator(self, f, Tf=None):
"""
The approximate Lucas operator, which computes and returns the
updated function Tf on the grid points.

Parameters
----------
f : array_like(float)
A candidate function on R_+ represented as points on a grid
and should be flat NumPy array with len(f) = len(grid)

Tf : array_like(float)
Optional storage array for Tf

Returns
-------
Tf : array_like(float)
The updated function Tf

Notes
-----
The argument Tf is optional, but recommended. If it is passed
into this function, then we do not have to allocate any memory
for the array here. As this function is often called many times
in an iterative algorithm, this can save significant computation
time.

"""
grid,  h = self.grid, self.init_h
alpha, beta = self.alpha, self.beta

# == set up storage if needed == #
if Tf is None:
Tf = np.empty_like(f)

# == Apply the T operator to f == #
Af = lambda x: self.Approximation(x, grid, f)

for i, y in enumerate(grid):
Tf[i] = h[i] + beta * self.integrate(lambda z: Af(y**alpha * z))[0]

return Tf

def compute_lt_price(self, error_tol=1e-7, max_iter=600, verbose=0):
"""
Compute the equilibrium price function associated with Lucas
tree lt

Parameters
----------
error_tol, max_iter, verbose
Arguments to be passed directly to
quantecon.compute_fixed_point. See that docstring for more
information

Returns
-------
price : array_like(float)
The prices at the grid points in the attribute grid of the
object

"""
# == simplify notation == #
grid, grid_size = self.grid, self.grid_size
lucas_operator, gamma = self.lucas_operator, self.gamma

# == Create storage array for compute_fixed_point. Reduces  memory
# allocation and speeds code up == #
Tf = np.empty(grid_size)

# == Initial guess, just a vector of ones == #
f_init = np.ones(grid_size)
f = compute_fixed_point(lucas_operator, f_init, error_tol,
max_iter, verbose, Tf=Tf)

price = f * grid**gamma

return price

Overwriting ./lucastree.py


The following two figures plot the functions $h,f$ and their first and second differences for parameters $(\gamma,\alpha) \in \left\lbrace (2,0.75),(0.5,0.75),(0.5,-0.75) \right\rbrace$. Note that the x-axis is in indices instead of grid values because the grid values change with different parameters. The graph illustrates that the sign of the slope and shape of $h$ is transmitted to $f$. We used $|\alpha| = 0.75$ because it generates a relatively strong visual slope of $h$. Our unit testing function also consider autoregressive parameters of $|\alpha| \in \left\lbrace 0.25, 0.5 \right\rbrace$.

In [2]:
%matplotlib inline
from lucastree import LucasTree
import numpy as np
import matplotlib.pyplot as plt

# first element gamma, second element alpha
vector = np.array([[2, 0.75], [0.5, 0.75], [0.5, -0.75]])
tree = LucasTree(gamma=2, beta=0.95, alpha=0.5, sigma=0.1)
h, hdiff, hdiff2 = np.empty((len(tree.grid), vector.shape[0])), np.empty(
(len(tree.grid) - 1, vector.shape[0])), np.empty((len(tree.grid) - 2, vector.shape[0]))
for idx, element in enumerate(vector):
tree = LucasTree(gamma=element[0], beta=0.95, alpha=element[1], sigma=0.1)
h[:, idx] = tree.h(tree.grid)
hdiff[:, idx] = np.ediff1d(h[:, idx])
hdiff2[:, idx] = np.ediff1d(hdiff[:, idx])
fig1, (ax1, ax2, ax3) = plt.subplots(3, 1, sharex='col')
annotation = ['(gamma,alpha): ' + str((i)) for i in vector]
ax1.plot(h)
ax2.plot(hdiff)
ax3.plot(hdiff2)
ax1.set_title('Plot of h')
ax2.set_title('Plot of the first difference of h')
ax3.set_title('Plot of the second difference of h')
ax3.legend(annotation, loc='upper center', bbox_to_anchor=(0.5, -0.07),
fig1.suptitle(
'Plot of the function h and its first and second difference', fontsize=15)
fig1.set_size_inches(15.5, 10.5)
fig1.show()

# first element gamma, second element alpha
vector = np.array([[2, 0.75], [0.5, 0.75], [0.5, -0.75]])
tree = LucasTree(gamma=2, beta=0.95, alpha=0.5, sigma=0.1)
f, fdiff, fdiff2 = np.empty((len(tree.grid), vector.shape[0])), np.empty(
(len(tree.grid) - 1, vector.shape[0])), np.empty((len(tree.grid) - 2, vector.shape[0]))
price = np.empty_like(f)
for idx, element in enumerate(vector):
tree = LucasTree(gamma=element[0], beta=0.95, alpha=element[1], sigma=0.1)
price[:, idx], grid = tree.compute_lt_price(), tree.grid
f[:, idx] = price[:, idx] * grid**(-element[0])
fdiff[:, idx] = np.ediff1d(f[:, idx])
fdiff2[:, idx] = np.ediff1d(fdiff[:, idx])
fig2, (ax1, ax2, ax3) = plt.subplots(3, 1, sharex='col')
annotation = ['(gamma,alpha): ' + str((i)) for i in vector]
ax1.plot(f)
ax2.plot(fdiff)
ax3.plot(fdiff2)
ax1.set_title('Plot of f')
ax2.set_title('Plot of the first difference of f')
ax3.set_title('Plot of the second difference of f')
ax3.legend(annotation, loc='upper center', bbox_to_anchor=(0.5, -0.05),
fig2.suptitle(
'Plot of the function f and its first and second difference', fontsize=15)
fig2.set_size_inches(15.5, 10.5)
fig2.show()


The two following figures graphs the solution $f$ if $\alpha \in \left\lbrace 0,1 \right\rbrace$. If dividends follow an i.i.d. process, ($\alpha = 0$) the function $f$ is constant. We reproduce the numerical results in the top panel of the following figure. The lower subplot graphs the price dividend ratio when dividend growth follows and i.i.d process ($\alpha =1$). As predicted by theory, the price dividend ratio is a constant.

In [3]:
%matplotlib inline
from lucastree import LucasTree
import numpy as np
import matplotlib.pyplot as plt

beta, gamma, sigma = 0.95, 2, 0.1

tree = LucasTree(gamma=gamma, beta=beta, alpha=1, sigma=sigma)
priceLinear, grid = tree.compute_lt_price(), tree.grid
fig1, (ax1, ax2) = plt.subplots(2, 1)
theoreticalPDRatio = np.ones(len(grid)) * beta * np.exp((1 - gamma) **
2 * sigma**2 / 2) / (1 - beta * np.exp((1 - gamma)**2 * sigma**2 / 2))
ax1.plot(grid, priceLinear / grid, grid, theoreticalPDRatio, 'g^')
annotation = ['Numerical Solution', 'Analytical Solution']
ax1.legend(annotation)
ax1.set_title('price dividend ratio for alpha = 1')
ax1.set_ylim([min(priceLinear / grid) - 1, max(priceLinear / grid) + 1])
tree = LucasTree(gamma=gamma, beta=beta, alpha=0, sigma=sigma)
priceFalling, grid = tree.compute_lt_price(), tree.grid
theoreticalF = np.ones(len(grid)) * beta * \
np.exp((1 - gamma)**2 * sigma**2 / 2) / (1 - beta)
f = priceFalling * grid**(-2)
ax2.plot(grid, f, grid, theoreticalF, 'g^')
ax2.set_ylim([min(f) - 1, max(f) + 1])
annotation = ['Numerical Solution', 'Analytical Solution']
ax2.legend(annotation)
ax2.set_title('function f for alpha = 0')
fig1.set_size_inches(15.5, 10.5)
fig1.suptitle(
'Plot of the function f  and the price dividend ratio for alpha=0 and alpha=1 respecitvely', fontsize=15)
fig1.show()


Finally, we report the unit testing function accompanying our lucastree.py module. This file tests if the functional form of $f$ adheres to the theoretical predicitons as outlined by the Proposition above. The file can be run from the Shell.

In [4]:
%%writefile ./test_lucastree.py
"""
filename: test_lucastree.py

Authors: Joao Brogueira and Fabian Schuetze

This file contains for different test for the
lucastree.py file

Functions
---------
compute_lt_price()      [Status: Tested in test_ConstantPDRatio, test_ConstantF,
test_slope_f, test_shape_f]

"""

import unittest
from lucastree import LucasTree  # This relative importing doesn't work!
import numpy as np

class Testlucastree(unittest.TestCase):

"""
Test Suite for lucastree.py based on the outout of the
LucasTree.compute_lt_price() function.

"""
# == Parameter values applicable to all test cases == #
beta = 0.95
sigma = 0.1

# == Paramter values for different tests == #
ConstantPD = np.array([2, 1])
ConstantF = np.array([2, 0])
FunctionalForm = np.array([[2, 0.75], [2, 0.5], [2, 0.25], [0.5, 0.75], [
0.5, 0.5], [0.5, 0.25], [0.5, -0.75], [0.5, -0.5], [0.5, -0.25]])

# == Tolerance Criteria == #
Tol = 1e-2

def setUp(self):
self.storage = lambda parameter0, parameter1: LucasTree(gamma=parameter0, beta=self.beta, alpha=parameter1,
sigma=self.sigma)

def test_ConstantPDRatio(self):
"""
Test whether the numerically computed price dividend ratio is
identical to its theoretical counterpart when dividend
growth follows an idd process

"""
gamma, alpha = self.ConstantPD
tree = self.storage(gamma, alpha)
grid = tree.grid
theoreticalPDRatio = np.ones(len(grid)) * self.beta * np.exp(
(1 - gamma)**2 * self.sigma**2 / 2) / (1 - self.beta * np.exp((1 - gamma)**2 * self.sigma**2 / 2))
self.assertTrue(
np.allclose(theoreticalPDRatio, tree.compute_lt_price() / grid, atol=self.Tol))

def test_ConstantF(self):
"""
Tests whether the numericlaly obtained solution, math:f
to the functional equation :math:f(y) = h(y) + \beta \int_Z f(G(y,z')) Q(z')
is identical to its theoretical counterpart, when divideds follow an
iid process

"""
gamma, alpha = self.ConstantF
tree = self.storage(gamma, alpha)
grid = tree.grid
theoreticalF = np.ones(len(
grid)) * self.beta * np.exp((1 - gamma)**2 * self.sigma**2 / 2) / (1 - self.beta)
self.assertTrue(np.allclose(
theoreticalF, tree.compute_lt_price() * grid**(-gamma), atol=self.Tol))

def test_slope_f(self):
"""
Tests whether the first difference of the numerically obtained function
:math:f is has the same sign as the first difference of the function
:math:h.

Notes
-----
This test is motivated by Theorem 9.7 ans exercise 9.7c) of the
book by Stokey, Lucas and Prescott (1989)

"""
for parameters in self.FunctionalForm:
gamma, alpha = parameters
tree = self.storage(gamma, alpha)
f = tree.compute_lt_price() * tree.grid ** (-gamma)
h = tree.h(tree.grid)
fdiff, hdiff = np.ediff1d(f), np.ediff1d(h)
if all(hdiff > 0):
self.assertTrue(all(fdiff > 0))
elif all(hdiff < 0):
self.assertTrue(all(fdiff < 0))

def test_shape_f(self):
"""
Tests whether the second difference of the numerically obtained function
:math:f is has the same sign as the second difference of the function
:math:h.

Notes
-----
This test is motivated by Theorem 9.8 ans exercise 9.7d) of the
book by Stokey, Lucas and Prescott (1989)

"""
for parameters in self.FunctionalForm:
gamma, alpha = parameters
tree = self.storage(gamma, alpha)
f = tree.compute_lt_price() * tree.grid ** (-gamma)
h = tree.h(tree.grid)
fdiff, hdiff = np.ediff1d(f), np.ediff1d(h)
fdiff2, hdiff2 = np.ediff1d(fdiff), np.ediff1d(hdiff)
if all(hdiff2 > 0):
self.assertTrue(all(fdiff2 > 0))
elif all(hdiff2 < 0):
self.assertTrue(all(fdiff2 < 0))

def tearDown(self):
pass

Overwriting ./test_lucastree.py