10. Geometric Series for Elementary Economics#

10.2.1. Infinite geometric series#

The first type of geometric that interests us is the infinite series

Where $\cdots$ means that the series continues without end.

The key formula is

To prove key formula (10.1), multiply both sides by $(1-c)$ and verify that if $c \in (-1,1)$, then the outcome is the equation $1 = 1$.

10.2.2. Finite geometric series#

The second series that interests us is the finite geometric series

where $T$ is a positive integer.

The key formula here is

We now move on to describe some famous economic applications of geometric series.

10.3.1. A simple model#

There is a set of banks named $i = 0, 1, 2, \ldots$.

Bank $i$’s loans $L_i$, deposits $D_i$, and reserves $R_i$ must satisfy the balance sheet equation (because balance sheets balance):

The left side of the above equation is the sum of the bank’s assets, namely, the loans $L_i$ it has outstanding plus its reserves of cash $R_i$.

The right side records bank $i$’s liabilities, namely, the deposits $D_i$ held by its depositors; these are IOU’s from the bank to its depositors in the form of either checking accounts or savings accounts (or before 1914, bank notes issued by a bank stating promises to redeem notes for gold or silver on demand).

Each bank $i$ sets its reserves to satisfy the equation

where $r \in (0,1)$ is its reserve-deposit ratio or reserve ratio for short

• the reserve ratio is either set by a government or chosen by banks for precautionary reasons

Next we add a theory stating that bank $i+1$’s deposits depend entirely on loans made by bank $i$, namely

Thus, we can think of the banks as being arranged along a line with loans from bank $i$ being immediately deposited in $i+1$

• 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 $0$’s deposits

We can think of $D_0$ as being the amount of cash that a first depositor put into the first bank in the system, bank number $i=0$.

Now we do a little algebra.

Combining equations (10.2) and (10.3) tells us that

This states that bank $i$ loans a fraction $(1-r)$ of its deposits and keeps a fraction $r$ as cash reserves.

Combining equation (10.5) with equation (10.4) tells us that

which implies that

Equation (10.6) expresses $D_i$ as the $i$ th term in the product of $D_0$ and the geometric series

Therefore, the sum of all deposits in our banking system $i=0, 1, 2, \ldots$ is

10.3.2. Money multiplier#

The money multiplier is a number that tells the multiplicative factor by which an exogenous injection of cash into bank $0$ leads to an increase in the total deposits in the banking system.

Equation (10.7) asserts that the money multiplier is $\frac{1}{r}$

• 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 $D_0 = \sum_{i=0}^\infty R_i$.

10.4.1. Static version#

An elementary Keynesian model of national income determination consists of three equations that describe aggregate demand for $y$ and its components.

The first equation is a national income identity asserting that consumption $c$ plus investment $i$ equals national income $y$:

The second equation is a Keynesian consumption function asserting that people consume a fraction $b \in (0,1)$ of their income:

The fraction $b \in (0,1)$ is called the marginal propensity to consume.

The fraction $1-b \in (0,1)$ is called the marginal propensity to save.

The third equation simply states that investment is exogenous at level $i$.

• exogenous means determined outside this model.

Substituting the second equation into the first gives $(1-b) y = i$.

Solving this equation for $y$ gives

The quantity $\frac{1}{1-b}$ is called the investment multiplier or simply the multiplier.

Applying the formula for the sum of an infinite geometric series, we can write the above equation as

where $t$ is a nonnegative integer.

So we arrive at the following equivalent expressions for the multiplier:

The expression $\sum_{t=0}^\infty b^t$ motivates an interpretation of the multiplier as the outcome of a dynamic process that we describe next.

10.4.2. Dynamic version#

We arrive at a dynamic version by interpreting the nonnegative integer $t$ as indexing time and changing our specification of the consumption function to take time into account

• we add a one-period lag in how income affects consumption

We let $c_t$ be consumption at time $t$ and $i_t$ be investment at time $t$.

We modify our consumption function to assume the form

so that $b$ is the marginal propensity to consume (now) out of last period’s income.

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 $t \rightarrow + \infty$,

Remark 1: The above formula is often applied to assert that an exogenous increase in investment of $\Delta i$ at time $0$ ignites a dynamic process of increases in national income by successive amounts

at times $0, 1, 2, \ldots$.

Remark 2 Let $g_t$ be an exogenous sequence of government expenditures.

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 $\frac{1}{1-b}$, so that a permanent increase in government expenditures ultimately leads to an increase in national income equal to the multiplier times the increase in government expenditures.

10.5.1. Accumulation#

Geometric sequence (10.8) 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 $x$ dollars at time $0$ and reinvest the proceeds, then the sequence

tells how our account accumulates at dates $t=0, 1, 2, \ldots$.

10.5.2. Discounting#

Geometric sequence (10.9) tells us how much future dollars are worth in terms of today’s dollars.

Remember that the units of $R$ are dollars at $t+1$ per dollar at $t$.

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 $x$ dollars at time $t+j$, it is worth $x R^{-j}$ dollars at time $t$ (e.g., today).

10.5.3. Application to asset pricing#

A lease requires a payments stream of $x_t$ dollars at times $t = 0, 1, 2, \ldots$ where

where $G = (1+g)$ and $g \in (0,1)$.

Thus, lease payments increase at $g$ percent per period.

For a reason soon to be revealed, we assume that $G < R$.

The present value of the lease is

where the last line uses the formula for an infinite geometric series.

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

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

and the fact that $r$ is small to approximate $\frac{1}{1+r} \approx 1 - r$.

Use this approximation to write $p_0$ as

where the last step uses the approximation $r g \approx 0$.

The approximation

is known as the Gordon formula for the present value or current price of an infinite payment stream $x_0 G^t$ when the nominal one-period interest rate is $r$ and when $r > g$.

We can also extend the asset pricing formula so that it applies to finite leases.

Let the payment stream on the lease now be $x_t$ for $t= 1,2, \dots,T$, where again

The present value of this lease is:

Applying the Taylor series to $R^{-(T+1)}$ about $r=0$ we get:

Similarly, applying the Taylor series to $G^{T+1}$ about $g=0$:

Thus, we get the following approximation:

Expanding:

We could have also approximated by removing the second term $rgx_0(T+1)$ when $T$ is relatively small compared to $1/(rg)$ to get $x_0(T+1)$ as in the finite stream approximation.

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

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

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 $T$.

However, holding $g$ and r fixed, our approximations deteriorate as $T$ increases.

Next we compare the infinite and finite duration lease present values over different lease lengths $T$.

# 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 $T \rightarrow +\infty$, the value of a lease of duration $T$ approaches the value of a perpetual lease.

Now we consider two different views of what happens as $r$ and $g$ covary

# 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 $r > g$ is necessary if a lease of length $T = +\infty$ is to have finite value.

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!

# 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 $p_0$ of a lease varies with $r$ and $g$.

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

Our formula is:

print('dp0 / dg is:')
dp_dg = sym.diff(p0, g)
dp_dg

dp0 / dg is:

print('dp0 / dr is:')
dp_dr = sym.diff(p0, r)
dp_dr

dp0 / dr is:

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

Similarly, $\frac{\partial p_0}{\partial g}>0$ as long as $r>g$, $r>0$ and $g>0$ and $x_0$ is positive, so $\frac{\partial p_0}{\partial g}$ will always be positive.