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Edits in geom_series.rst #821

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213eab6
Edits in geom_series.rst - code repetition reduced and typos fixed
mtiley Dec 11, 2019
1b89842
bracket fix in geom_series
mtiley Dec 12, 2019
86cb919
neaten code in geom_series
mtiley Dec 12, 2019
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145 changes: 68 additions & 77 deletions source/rst/geom_series.rst
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Original file line number Diff line number Diff line change
Expand Up @@ -142,7 +142,7 @@ Economists and financiers often define the **supply of money** as an
economy-wide sum of **cash** plus **deposits**.

In a **fractional reserve banking system** (one in which the reserve
ratio :math:`r` satisfying :math:`0 < r < 1`), **banks create money** by issuing deposits *backed* by fractional reserves plus loans that they make to their customers.
ratio :math:`r` satisfies :math:`0 < r < 1`), **banks create money** by issuing deposits *backed* by fractional reserves plus loans that they make to their customers.

A geometric series is a key tool for understanding how banks create
money (i.e., deposits) in a fractional reserve system.
Expand Down Expand Up @@ -177,7 +177,7 @@ bank stating promises to redeem note for gold or silver on demand).
.. Dongchen: is there a way to add a little balance sheet here?
.. with assets on the left side and liabilities on the right side?

Ecah bank :math:`i` sets its reserves to satisfy the equation
Each bank :math:`i` sets its reserves to satisfy the equation

.. math::
:label: reserves
Expand Down Expand Up @@ -554,15 +554,15 @@ The **present value** of the lease is
where the last line uses the formula for an infinite geometric series.

Recall that :math:`R = 1+r` and :math:`G = 1+g` and that :math:`R > G`
and :math:`r > g` and that :math:`r` and\ :math:`g` are typically small
and :math:`r > g` and that :math:`r` and :math:`g` are typically small
numbers, e.g., .05 or .03.

Use the Taylor series of :math:`\frac{1}{1+r}` about :math:`r=0`,
namely,

.. math:: \frac{1}{1+r} = 1 - r + r^2 - r^3 + \cdots

and the fact that :math:`r` is small to aproximate
and the fact that :math:`r` is small to approximate
:math:`\frac{1}{1+r} \approx 1 - r`.

Use this approximation to write :math:`p_0` as
Expand Down Expand Up @@ -619,26 +619,26 @@ We could have also approximated by removing the second term
approximation.

We will plot the true finite stream present-value and the two
approximations, under different values of :math:`T`, and :math:`g` and :math:`r` in python.
approximations, under different values of :math:`T`, and :math:`g` and :math:`r` in Python.

First we plot the true finite stream present-value after computing it
below

.. code-block:: python3

# True present value of a finite lease
def finite_lease_pv(T, g, r, x_0):
def finite_lease_pv_true(T, g, r, x_0):
G = (1 + g)
R = (1 + r)
return (x_0 * (1 - G**(T + 1) * R**(-T - 1))) / (1 - G * R**(-1))
# First approximation for our finite lease

def finite_lease_pv_approx_f(T, g, r, x_0):
def finite_lease_pv_approx_1(T, g, r, x_0):
p = x_0 * (T + 1) + x_0 * r * g * (T + 1) / (r - g)
return p

# Second approximation for our finite lease
def finite_lease_pv_approx_s(T, g, r, x_0):
def finite_lease_pv_approx_2(T, g, r, x_0):
return (x_0 * (T + 1))

# Infinite lease
Expand All @@ -648,25 +648,32 @@ below
return x_0 / (1 - G * R**(-1))


Now that we have test run our functions, we can plot some outcomes.
Now that we have defined our functions, we can plot some outcomes.

First we study the quality of our approximations

.. code-block:: python3

def plot_function(axes, x_vals, func, args):
axes.plot(x_vals, func(*args), label=func.__name__)

T_max = 50

T = np.arange(0, T_max+1)
g = 0.02
r = 0.03
x_0 = 1
T_max = 50
T = np.arange(0, T_max+1)

our_args = (T, g, r, x_0)
funcs = [finite_lease_pv_true,
finite_lease_pv_approx_1,
finite_lease_pv_approx_2]
## the three functions we want to compare

fig, ax = plt.subplots()
ax.set_title('Finite Lease Present Value $T$ Periods Ahead')
y_1 = finite_lease_pv(T, g, r, x_0)
y_2 = finite_lease_pv_approx_f(T, g, r, x_0)
y_3 = finite_lease_pv_approx_s(T, g, r, x_0)
ax.plot(T, y_1, label='True T-period Lease PV')
ax.plot(T, y_2, label='T-period Lease First-order Approx.')
ax.plot(T, y_3, label='T-period Lease First-order Approx. adj.')
for f in funcs:
plot_function(ax, T, f, our_args)
ax.legend()
ax.set_xlabel('$T$ Periods Ahead')
ax.set_ylabel('Present Value, $p_0$')
Expand All @@ -686,18 +693,18 @@ over different lease lengths :math:`T`.
T = np.arange(0, T_max+1)
fig, ax = plt.subplots()
ax.set_title('Infinite and Finite Lease Present Value $T$ Periods Ahead')
y_1 = finite_lease_pv(T, g, r, x_0)
y_2 = np.ones(T_max+1)*infinite_lease(g, r, x_0)
ax.plot(T, y_1, label='T-period lease PV')
ax.plot(T, y_2, '--', label='Infinite lease PV')
f_1 = finite_lease_pv_true(T, g, r, x_0)
f_2 = np.ones(T_max+1)*infinite_lease(g, r, x_0)
ax.plot(T, f_1, label='T-period lease PV')
ax.plot(T, f_2, '--', label='Infinite lease PV')
ax.set_xlabel('$T$ Periods Ahead')
ax.set_ylabel('Present Value, $p_0$')
ax.legend()
plt.show()

The above graphs shows how as duration :math:`T \rightarrow +\infty`,
The graph above shows how as duration :math:`T \rightarrow +\infty`,
the value of a lease of duration :math:`T` approaches the value of a
perpetural lease.
perpetual lease.

Now we consider two different views of what happens as :math:`r` and
:math:`g` covary
Expand All @@ -712,30 +719,18 @@ Now we consider two different views of what happens as :math:`r` and
ax.set_xlabel('$T$ periods ahead')
T_max = 10
T=np.arange(0, T_max+1)
# r >> g, much bigger than g
r = 0.9
g = 0.4
ax.plot(finite_lease_pv(T, g, r, x_0), label='$r\gg g$')
# r > g
r = 0.5
g = 0.4
ax.plot(finite_lease_pv(T, g, r, x_0), label='$r>g$', color='green')

# r ~ g, not defined when r = g, but approximately goes to straight
# line with slope 1
r = 0.4001
g = 0.4
ax.plot(finite_lease_pv(T, g, r, x_0), label=r'$r \approx g$', color='orange')

# r < g
r = 0.4
g = 0.5
ax.plot(finite_lease_pv(T, g, r, x_0), label='$r<g$', color='red')

rs, gs = (0.9, 0.5, 0.4001, 0.4), (0.4, 0.4, 0.4, 0.5),
comparisons = ('$\gg$', '$>$', r'$\approx$', '$<$')
for r, g, comp in zip(rs, gs, comparisons):
ax.plot(finite_lease_pv_true(T, g, r, x_0), label=f'r(={r}) {comp} g(={g})')

ax.legend()
plt.show()


The above graphs gives a big hint for why the condition :math:`r > g` is

This graph gives a big hint for why the condition :math:`r > g` is
necessary if a lease of length :math:`T = +\infty` is to have finite
value.

Expand All @@ -755,7 +750,7 @@ visualization!
g = np.arange(0.011, 0.991, 0.005)

rr, gg = np.meshgrid(r, g)
z = finite_lease_pv(T, gg, rr, x_0)
z = finite_lease_pv_true(T, gg, rr, x_0)

# Removes points where undefined
same = (rr == gg)
Expand Down Expand Up @@ -809,10 +804,10 @@ After that, we'll use SymPy to compute derivatives

We can see that for :math:`\frac{\partial p_0}{\partial r}<0` as long as
:math:`r>g`, :math:`r>0` and :math:`g>0` and :math:`x_0` is positive,
this equation will always be negative.
so :math:`\frac{\partial p_0}{\partial r}` will always be negative.

Similarly, :math:`\frac{\partial p_0}{\partial g}>0` as long as :math:`r>g`, :math:`r>0` and :math:`g>0` and :math:`x_0` is positive, this equation
will always be postive.
Similarly, :math:`\frac{\partial p_0}{\partial g}>0` as long as :math:`r>g`, :math:`r>0` and :math:`g>0` and :math:`x_0` is positive, so :math:`\frac{\partial p_0}{\partial g}`
will always be positive.



Expand Down Expand Up @@ -859,54 +854,50 @@ i.e., the fraction of income that is consumed

.. code-block:: python3

# Changing fraction of consumption
b_0 = 1/3
b_1 = 2/3
b_2 = 5/6
b_3 = 0.9
bs = (1/3, 2/3, 5/6, 0.9)

fig,ax = plt.subplots()
ax.set_title('Changing Consumption as a Fraction of Income')
ax.set_ylabel('$y_t$')
ax.set_xlabel('$t$')
x = np.arange(0, T+1)
for b in (b_0, b_1, b_2, b_3):
for b in bs:
y = calculate_y(i_0, b, g_0, T, y_init)
ax.plot(x, y, label=r'$b=$'+f"{b:.2f}")
ax.legend()
plt.show()

Increasing the marginal propensity to consumer :math:`b` increases the
path of output over time

Increasing the marginal propensity to consume :math:`b` increases the
path of output over time.

Now we will compare the effects on output of increases in investment and government spending.

.. code-block:: python3

x = np.arange(0, T+1)
y_0 = calculate_y(i_0, b, g_0, T, y_init)
fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(6, 10))
fig.subplots_adjust(hspace=0.3)

# Changing initial investment:
i_1 = 0.4
y_1 = calculate_y(i_1, b, g_0, T, y_init)
ax1.set_title('An Increase in Investment on Output')
ax1.plot(x, y_0, label=r'$i=0.3$', linestyle='--')
ax1.plot(x, y_1, label=r'$i=0.4$')
ax1.legend()
ax1.set_ylabel('$y_t$')
ax1.set_xlabel('$t$')

# Changing government spending
g_1 = 0.4
y_1 = calculate_y(i_0, b, g_1, T, y_init)
ax2.set_title('An Increase in Government Spending on Output')
ax2.plot(x, y_0, label=r'$g=0.3$', linestyle='--')
ax2.plot(x, y_1, label=r'$g=0.4$')
ax2.legend()
ax2.set_ylabel('$y_t$')
ax2.set_xlabel('$t$')

x = np.arange(0, T+1)
values = [0.3, 0.4]

for i in values:
y = calculate_y(i, b, g_0, T, y_init)
ax1.plot(x, y, label=f"i={i}")
for g in values:
y = calculate_y(i_0, b, g, T, y_init)
ax2.plot(x, y, label=f"g={g}")

axes = ax1, ax2
param_labels = "Investment", "Government Spending"
for ax, param in zip(axes, param_labels):
ax.set_title(f'An Increase in {param} on Output')
ax.legend(loc ="lower right")
ax.set_ylabel('$y_t$')
ax.set_xlabel('$t$')
plt.show()


Notice here, whether government spending increases from 0.3 to 0.4 or
investment increases from 0.3 to 0.4, the shifts in the graphs are
identical.