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The lecture describes important ideas in economics that use the mathematics of geometric series.
Among these are
- the Keynesian multiplier
- the money multiplier that prevails in fractional reserve banking systems
- interest rates and present values of streams of payouts from assets
(As we shall see below, the term multiplier comes down to meaning sum of a convergent geometric series)
These and other applications prove the truth of the wise crack that
"In economics, a little knowledge of geometric series goes a long way."
Below we'll use the following imports:
import matplotlib.pyplot as plt
plt.rcParams["figure.figsize"] = (11, 5) #set default figure size
import numpy as np
import sympy as sym
from sympy import init_printing
from matplotlib import cm
To start, let
- We often write this as
$c \in (-1,1)$ . - Here
$(-1,1)$ denotes the collection of all real numbers that are strictly less than$1$ and strictly greater than$-1$ . - The symbol
$\in$ means in or belongs to the set after the symbol.
We want to evaluate geometric series of two types -- infinite and finite.
The first type of geometric that interests us is the infinite series
Where
The key formula is
:label: infinite
1 + c + c^2 + c^3 + \cdots = \frac{1}{1 -c }
To prove key formula {eq}infinite, multiply both sides by
The second series that interests us is the finite geometric series
where
The key formula here is
:label: geom_formula
The above formula works for any value of the scalar
$c$. We don't have to restrict $c$ to be in the
set $(-1,1)$.
We now move on to describe some famous economic applications of geometric series.
In a fractional reserve banking system, banks hold only a fraction
- In recent times
- cash consists of pieces of paper issued by the government and
called dollars or pounds or
$\ldots$ - a deposit is a balance in a checking or savings account that entitles the owner to ask the bank for immediate payment in cash
- cash consists of pieces of paper issued by the government and
called dollars or pounds or
- When the UK and France and the US were on either a gold or silver
standard (before 1914, for example)
- cash was a gold or silver coin
- a deposit receipt was a bank note that the bank promised to convert into gold or silver on demand; (sometimes it was also a checking or savings account balance)
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
A geometric series is a key tool for understanding how banks create money (i.e., deposits) in a fractional reserve system.
The geometric series formula {eq}infinite is at the heart of the classic model of the money creation process -- one that leads us to the celebrated
money multiplier.
There is a set of banks named
Bank
:label: balance
L_i + R_i = D_i
The left side of the above equation is the sum of the bank's assets,
namely, the loans
The right side records bank
Each bank
:label: reserves
R_i = r D_i
where
- the reserve ratio is either set by a government or chosen by banks for precautionary reasons
Next we add a theory stating that bank
:label: deposits
D_{i+1} = L_i
Thus, we can think of the banks as being arranged along a line with
loans from bank
- in this way, the debtors to bank
$i$ become creditors of bank$i+1$
Finally, we add an initial condition about an exogenous level of bank
We can think of
Now we do a little algebra.
Combining equations {eq}balance and {eq}reserves tells us that
:label: fraction
L_i = (1-r) D_i
This states that bank
Combining equation {eq}fraction with equation {eq}deposits tells us that
which implies that
:label: geomseries
D_i = (1 - r)^i D_0 \ \text{ for } i \geq 0
Equation {eq}geomseries expresses
Therefore, the sum of all deposits in our banking system
:label: sumdeposits
\sum_{i=0}^\infty (1-r)^i D_0 = \frac{D_0}{1 - (1-r)} = \frac{D_0}{r}
The money multiplier is a number that tells the multiplicative
factor by which an exogenous injection of cash into bank
Equation {eq}sumdeposits asserts that the money multiplier is
- An initial deposit of cash of
$D_0$ in bank$0$ leads the banking system to create total deposits of$\frac{D_0}{r}$ . - The initial deposit
$D_0$ is held as reserves, distributed throughout the banking system according to$D_0 = \sum_{i=0}^\infty R_i$ .
The famous economist John Maynard Keynes and his followers created a
simple model intended to determine national income
- there are substantial unemployed resources, in particular excess supply of labor and capital
- prices and interest rates fail to adjust to make aggregate supply equal demand (e.g., prices and interest rates are frozen)
- national income is entirely determined by aggregate demand
An elementary Keynesian model of national income determination consists
of three equations that describe aggregate demand for
The first equation is a national income identity asserting that
consumption
The second equation is a Keynesian consumption function asserting that
people consume a fraction
The fraction
The fraction
The third equation simply states that investment is exogenous at level
- exogenous means determined outside this model.
Substituting the second equation into the first gives
Solving this equation for
The quantity
Applying the formula for the sum of an infinite geometric series, we can write the above equation as
where
So we arrive at the following equivalent expressions for the multiplier:
The expression
We arrive at a dynamic version by interpreting the nonnegative integer
- we add a one-period lag in how income affects consumption
We let
We modify our consumption function to assume the form
so that
We begin with an initial condition stating that
We also assume that
so that investment is constant over time.
It follows that
and
and
and more generally
or
Evidently, as
Remark 1: The above formula is often applied to assert that an
exogenous increase in investment of
at times
Remark 2 Let
If we generalize the model so that the national income identity becomes
then a version of the preceding argument shows that the government
expenditures multiplier is also
We can apply our formula for geometric series to study how interest rates affect values of streams of dollar payments that extend over time.
We work in discrete time and assume that
We let
- if the nominal interest rate is
$5$ percent, then$r= .05$
A one-period gross nominal interest rate
- if
$r=.05$ , then$R = 1.05$
Remark: The gross nominal interest rate
When people borrow and lend, they trade dollars now for dollars later or dollars later for dollars now.
The price at which these exchanges occur is the gross nominal interest rate.
- If I sell
$x$ dollars to you today, you pay me$R x$ dollars tomorrow. - This means that you borrowed
$x$ dollars for me at a gross interest rate$R$ and a net interest rate$r$ .
We assume that the net nominal interest rate
Two important geometric sequences are
:label: geom1
1, R, R^2, \cdots
and
:label: geom2
1, R^{-1}, R^{-2}, \cdots
Sequence {eq}geom1 tells us how dollar values of an investment accumulate
through time.
Sequence {eq}geom2 tells us how to discount future dollars to get their
values in terms of today's dollars.
Geometric sequence {eq}geom1 tells us how one dollar invested and re-invested
in a project with gross one period nominal rate of return accumulates
- here we assume that net interest payments are reinvested in the project
- thus,
$1$ dollar invested at time$0$ pays interest$r$ dollars after one period, so we have$r+1 = R$ dollars at time$1$ - at time
$1$ we reinvest$1+r =R$ dollars and receive interest of$r R$ dollars at time$2$ plus the principal$R$ dollars, so we receive$r R + R = (1+r)R = R^2$ dollars at the end of period$2$ - and so on
Evidently, if we invest
tells how our account accumulates at dates
Geometric sequence {eq}geom2 tells us how much future dollars are worth in terms of today's dollars.
Remember that the units of
It follows that
- the units of
$R^{-1}$ are dollars at$t$ per dollar at$t+1$ - the units of
$R^{-2}$ are dollars at$t$ per dollar at$t+2$ - and so on; the units of
$R^{-j}$ are dollars at$t$ per dollar at$t+j$
So if someone has a claim on
A lease requires a payments stream of
where
Thus, lease payments increase at
For a reason soon to be revealed, we assume that
The present value of the lease is
where the last line uses the formula for an infinite geometric series.
Recall that
Use the Taylor series of
and the fact that
Use this approximation to write
where the last step uses the approximation
The approximation
is known as the Gordon formula for the present value or current
price of an infinite payment stream
We can also extend the asset pricing formula so that it applies to finite leases.
Let the payment stream on the lease now be
The present value of this lease is:
Applying the Taylor series to
Similarly, applying the Taylor series to
Thus, we get the following approximation:
Expanding:
We could have also approximated by removing the second term
We will plot the true finite stream present-value and the two
approximations, under different values of
First we plot the true finite stream present-value after computing it below
# True present value of a finite lease
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_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_2(T, g, r, x_0):
return (x_0 * (T + 1))
# Infinite lease
def infinite_lease(g, r, x_0):
G = (1 + g)
R = (1 + r)
return x_0 / (1 - G * R**(-1))
Now that we have defined our functions, we can plot some outcomes.
First we study the quality of our approximations
---
mystnb:
figure:
caption: "Finite lease present value $T$ periods ahead"
name: finite_lease_present_value
---
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
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()
for f in funcs:
plot_function(ax, T, f, our_args)
ax.legend()
ax.set_xlabel('$T$ Periods Ahead')
ax.set_ylabel('Present Value, $p_0$')
plt.show()
Evidently our approximations perform well for small values of
However, holding
Next we compare the infinite and finite duration lease present values
over different lease lengths
---
mystnb:
figure:
caption: "Infinite and finite lease present value $T$ periods ahead"
name: infinite_and_finite_lease_present_value
---
# Convergence of infinite and finite
T_max = 1000
T = np.arange(0, T_max+1)
fig, ax = plt.subplots()
f_1 = finite_lease_pv_true(T, g, r, x_0)
f_2 = np.full(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 graph above shows how as duration
Now we consider two different views of what happens as
---
mystnb:
figure:
caption: "Value of lease of length $T$"
name: value_of_lease
---
# First view
# Changing r and g
fig, ax = plt.subplots()
ax.set_ylabel('Present Value, $p_0$')
ax.set_xlabel('$T$ periods ahead')
T_max = 10
T=np.arange(0, T_max+1)
rs, gs = (0.9, 0.5, 0.4001, 0.4), (0.4, 0.4, 0.4, 0.5),
comparisons = (r'$\gg$', '$>$', 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()
This graph gives a big hint for why the condition
For fans of 3-d graphs the same point comes through in the following graph.
If you aren't enamored of 3-d graphs, feel free to skip the next visualization!
---
mystnb:
figure:
caption: "Three period lease PV with varying $g$ and $r$"
name: three_period_lease_PV
---
# Second view
fig = plt.figure(figsize = [16, 5])
T = 3
ax = plt.subplot(projection='3d')
r = np.arange(0.01, 0.99, 0.005)
g = np.arange(0.011, 0.991, 0.005)
rr, gg = np.meshgrid(r, g)
z = finite_lease_pv_true(T, gg, rr, x_0)
# Removes points where undefined
same = (rr == gg)
z[same] = np.nan
surf = ax.plot_surface(rr, gg, z, cmap=cm.coolwarm,
antialiased=True, clim=(0, 15))
fig.colorbar(surf, shrink=0.5, aspect=5)
ax.set_xlabel('$r$')
ax.set_ylabel('$g$')
ax.set_zlabel('Present Value, $p_0$')
ax.view_init(20, 8)
plt.show()
We can use a little calculus to study how the present value
We will use a library called SymPy.
SymPy enables us to do symbolic math calculations including computing derivatives of algebraic equations.
We will illustrate how it works by creating a symbolic expression that represents our present value formula for an infinite lease.
After that, we'll use SymPy to compute derivatives
# Creates algebraic symbols that can be used in an algebraic expression
g, r, x0 = sym.symbols('g, r, x0')
G = (1 + g)
R = (1 + r)
p0 = x0 / (1 - G * R**(-1))
init_printing(use_latex='mathjax')
print('Our formula is:')
p0
print('dp0 / dg is:')
dp_dg = sym.diff(p0, g)
dp_dg
print('dp0 / dr is:')
dp_dr = sym.diff(p0, r)
dp_dr
We can see that for
Similarly,
We will now go back to the case of the Keynesian multiplier and plot the
time path of
---
mystnb:
figure:
caption: "Path of aggregate output tver time"
name: path_of_aggregate_output_over_time
---
# Function that calculates a path of y
def calculate_y(i, b, g, T, y_init):
y = np.zeros(T+1)
y[0] = i + b * y_init + g
for t in range(1, T+1):
y[t] = b * y[t-1] + i + g
return y
# Initial values
i_0 = 0.3
g_0 = 0.3
# 2/3 of income goes towards consumption
b = 2/3
y_init = 0
T = 100
fig, ax = plt.subplots()
ax.set_xlabel('$t$')
ax.set_ylabel('$y_t$')
ax.plot(np.arange(0, T+1), calculate_y(i_0, b, g_0, T, y_init))
# Output predicted by geometric series
ax.hlines(i_0 / (1 - b) + g_0 / (1 - b), xmin=-1, xmax=101, linestyles='--')
plt.show()
In this model, income grows over time, until it gradually converges to the infinite geometric series sum of income.
We now examine what will happen if we vary the so-called marginal propensity to consume, i.e., the fraction of income that is consumed
---
mystnb:
figure:
caption: "Changing consumption as a fraction of income"
name: changing_consumption_as_fraction_of_income
---
bs = (1/3, 2/3, 5/6, 0.9)
fig,ax = plt.subplots()
ax.set_ylabel('$y_t$')
ax.set_xlabel('$t$')
x = np.arange(0, T+1)
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 consume
Now we will compare the effects on output of increases in investment and government spending.
---
mystnb:
figure:
caption: "Different increase on output"
name: different_increase_on_output
---
fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(6, 10))
fig.subplots_adjust(hspace=0.3)
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.