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+.. _geom_series:
+
+.. include:: /_static/includes/header.raw
+
+.. highlight:: julia
+
+*****************************************
+Geometric Series for Elementary Economics
+*****************************************
+
+.. contents:: :depth: 2
+
+Overview
+========
+
+
+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
+
+.. epigraph::
+
+    "in economics, a little knowledge of geometric series goes a long way "
+
+Setup
+------------------
+
+Below we'll use the following imports:
+
+.. literalinclude:: /_static/includes/deps_generic.jl
+      :class: hide-output
+
+.. code-block:: julia
+
+  using Plots
+
+.. code-block:: julia 
+    :class: test 
+
+    using Test 
+
+Key Formulas
+============
+
+To start, let :math:`c` be a real number that lies strictly between
+:math:`-1` and :math:`1`.
+
+-  We often write this as :math:`c \in (-1,1)`.
+
+-  Here :math:`(-1,1)` denotes the collection of all real numbers that
+   are strictly less than :math:`1` and strictly greater than :math:`-1`.
+
+-  The symbol :math:`\in` means *in* or *belongs to the set after the symbol*.
+
+We want to evaluate geometric series of two types -- infinite and finite.
+
+Infinite Geometric Series
+-------------------------
+
+The first type of geometric that interests us is the infinite series
+
+.. math:: 1 + c + c^2 + c^3 + \cdots
+
+Where :math:`\cdots` means that the series continues without limit.
+
+The key formula is
+
+.. math::
+   :label: infinite
+
+   1 + c + c^2 + c^3 + \cdots = \frac{1}{1 -c }
+
+To prove key formula :eq:`infinite`, multiply both sides  by :math:`(1-c)` and verify
+that if :math:`c \in (-1,1)`, then the outcome is the
+equation :math:`1 = 1`.
+
+Finite Geometric Series
+-----------------------
+
+The second series that interests us is the finite geometric series
+
+.. math:: 1 + c + c^2 + c^3 + \cdots + c^T
+
+where :math:`T` is a positive integer.
+
+The key formula here is
+
+.. math:: 1 + c + c^2 + c^3 + \cdots + c^T  = \frac{1 - c^{T+1}}{1-c}
+
+**Remark:** The above formula works for any value of the scalar
+:math:`c`. We don't have to restrict :math:`c` to be in the
+set :math:`(-1,1)`.
+
+
+
+
+We now move on to describe some famous economic applications of
+geometric series.
+
+
+
+Example: The Money Multiplier in Fractional Reserve Banking
+===========================================================
+
+In a fractional reserve banking system, banks hold only a fraction
+:math:`r \in (0,1)` of cash behind each **deposit receipt** that they
+issue
+
+*  In recent times
+
+   -  cash consists of pieces of paper issued by the government and
+      called dollars or pounds or :math:`\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
+
+*  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 :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.
+
+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**.
+
+
+
+A Simple Model
+--------------
+
+There is a set of banks named :math:`i = 0, 1, 2, \ldots`.
+
+Bank :math:`i`'s loans :math:`L_i`, deposits :math:`D_i`, and
+reserves :math:`R_i` must satisfy the balance sheet equation (because
+**balance sheets balance**):
+
+.. math:: L_i + R_i = D_i
+
+The left side of the above equation is the sum of the bank's **assets**,
+namely, the loans :math:`L_i` it has outstanding plus its reserves of
+cash :math:`R_i`.
+
+The right side records bank :math:`i`'s liabilities,
+namely, the deposits :math:`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 note for gold or silver on demand).
+
+.. TO REMOVE:
+.. Dongchen: is there a way to add a little balance sheet here?
+.. with assets on the left side and liabilities on the right side?
+
+Each bank :math:`i` sets its reserves to satisfy the equation
+
+.. math::
+  :label: reserves
+
+  R_i = r D_i
+
+where :math:`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 :math:`i+1`'s deposits depend
+entirely on loans made by bank :math:`i`, namely
+
+.. math::
+  :label: deposits
+
+  D_{i+1} = L_i
+
+Thus, we can think of the banks as being arranged along a line with
+loans from bank :math:`i` being immediately deposited in :math:`i+1`
+
+-  in this way, the debtors to bank :math:`i` become creditors of
+   bank :math:`i+1`
+
+Finally, we add an *initial condition* about an exogenous level of bank
+:math:`0`'s deposits
+
+.. math:: D_0 \ \text{ is given exogenously}
+
+We can think of :math:`D_0` as being the amount of cash that a first
+depositor put into the first bank in the system, bank number :math:`i=0`.
+
+Now we do a little algebra.
+
+Combining equations :eq:`reserves` and :eq:`deposits` tells us that
+
+.. math::
+  :label: fraction
+
+  L_i = (1-r) D_i
+
+This states that bank :math:`i` loans a fraction :math:`(1-r)` of its
+deposits and keeps a fraction :math:`r` as cash reserves.
+
+Combining equation :eq:`fraction` with equation :eq:`deposits` tells us that
+
+.. math:: D_{i+1} = (1-r) D_i  \ \text{ for } i \geq 0
+
+which implies that
+
+.. math::
+  :label: geomseries
+
+  D_i = (1 - r)^i D_0  \ \text{ for } i \geq 0
+
+Equation :eq:`geomseries` expresses :math:`D_i` as the :math:`i` th term in the
+product of :math:`D_0` and the geometric series
+
+.. math::  1, (1-r), (1-r)^2, \cdots
+
+Therefore, the sum of all deposits in our banking system
+:math:`i=0, 1, 2, \ldots` is
+
+.. math::
+  :label: sumdeposits
+
+  \sum_{i=0}^\infty (1-r)^i D_0 =  \frac{D_0}{1 - (1-r)} = \frac{D_0}{r}
+
+
+Money Multiplier
+----------------
+
+The **money multiplier** is a number that tells the multiplicative
+factor by which an exogenous injection of cash into bank :math:`0` leads
+to an increase in the total deposits in the banking system.
+
+Equation :eq:`sumdeposits` asserts that the **money multiplier** is
+:math:`\frac{1}{r}`
+
+-  An initial deposit of cash of :math:`D_0` in bank :math:`0` leads
+   the banking system to create total deposits of :math:`\frac{D_0}{r}`.
+
+-  The initial deposit :math:`D_0` is held as reserves, distributed
+   throughout the banking system according to :math:`D_0 = \sum_{i=0}^\infty R_i`.
+
+.. Dongchen: can you think of some simple Python examples that
+.. illustrate how to create sequences and so on? Also, some simple
+.. experiments like lowering reserve requirements? Or others you may suggest?
+
+
+Example: The Keynesian Multiplier
+=================================
+
+The famous economist John Maynard Keynes and his followers created a
+simple model intended to determine national income :math:`y` in
+circumstances in which
+
+-  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
+
+
+Static Version
+--------------
+
+
+An elementary Keynesian model of national income determination consists
+of three equations that describe aggregate demand for :math:`y` and its
+components.
+
+The first equation is a national income identity asserting that
+consumption :math:`c` plus investment :math:`i` equals national income
+:math:`y`:
+
+.. math:: c+ i = y
+
+The second equation is a Keynesian consumption function asserting that
+people consume a fraction :math:`b \in (0,1)` of their income:
+
+.. math:: c = b y
+
+The fraction :math:`b \in (0,1)` is called the **marginal propensity to
+consume**.
+
+The fraction :math:`1-b \in (0,1)` is called the **marginal propensity
+to save**.
+
+The third equation simply states that investment is exogenous at level
+:math:`i`.
+
+- *exogenous* means *determined outside this model*.
+
+Substituting the second equation into the first gives :math:`(1-b) y = i`.
+
+Solving this equation for :math:`y` gives
+
+.. math:: y = \frac{1}{1-b} i
+
+The quantity :math:`\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
+
+.. math:: y = i \sum_{t=0}^\infty b^t
+
+where :math:`t` is a nonnegative integer.
+
+So we arrive at the following equivalent expressions for the multiplier:
+
+.. math:: \frac{1}{1-b} =   \sum_{t=0}^\infty b^t
+
+The expression :math:`\sum_{t=0}^\infty b^t` motivates an interpretation
+of the multiplier as the outcome of a dynamic process that we describe
+next.
+
+Dynamic Version
+---------------
+
+We arrive at a dynamic version by interpreting the nonnegative integer
+:math:`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 :math:`c_t` be consumption at time :math:`t` and :math:`i_t` be
+investment at time :math:`t`.
+
+We modify our consumption function to assume the form
+.. math:: c_t = b y_{t-1}
+
+so that :math:`b` is the marginal propensity to consume (now) out of
+last period's income.
+
+We begin with an initial condition stating that
+
+.. math:: y_{-1} = 0
+
+We also assume that
+
+.. math:: i_t = i \ \ \textrm {for all }  t \geq 0
+
+so that investment is constant over time.
+
+It follows that
+
+.. math:: y_0 = i + c_0 = i + b y_{-1} =  i
+
+and
+
+.. math:: y_1 = c_1 + i = b y_0 + i = (1 + b) i
+
+and
+
+.. math:: y_2 = c_2 + i = b y_1 + i = (1 + b + b^2) i
+
+and more generally
+
+.. math:: y_t = b y_{t-1} + i = (1+ b + b^2 + \cdots + b^t) i
+
+or
+
+.. math:: y_t = \frac{1-b^{t+1}}{1 -b } i
+
+Evidently, as :math:`t \rightarrow + \infty`,
+
+.. math:: y_t \rightarrow \frac{1}{1-b} i
+
+**Remark 1:** The above formula is often applied to assert that an
+exogenous increase in investment of :math:`\Delta i` at time :math:`0`
+ignites a dynamic process of increases in national income by amounts
+
+.. math:: \Delta i, (1 + b )\Delta i, (1+b + b^2) \Delta i , \cdots
+
+at times :math:`0, 1, 2, \ldots`.
+
+**Remark 2** Let :math:`g_t` be an exogenous sequence of government
+expenditures.
+
+If we generalize the model so that the national income identity
+becomes
+
+.. math:: c_t + i_t + g_t  = y_t
+
+then a version of the preceding argument shows that the **government
+expenditures multiplier** is also :math:`\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.
+
+.. Dongchen: can you think of some simple Python things to add to
+.. illustrate basic concepts, maybe the idea of a "difference equation" and how we solve it?
+
+
+Example: Interest Rates and Present Values
+==========================================
+
+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 :math:`t = 0, 1, 2, \ldots`
+indexes time.
+
+We let :math:`r \in (0,1)` be a one-period **net nominal interest rate**
+
+-  if the nominal interest rate is :math:`5` percent,
+   then :math:`r= .05`
+
+A one-period **gross nominal interest rate** :math:`R` is defined as
+
+.. math:: R = 1 + r \in (1, 2)
+
+-  if :math:`r=.05`, then :math:`R = 1.05`
+
+**Remark:** The gross nominal interest rate :math:`R` is an **exchange
+rate** or **relative price** of dollars at between times :math:`t` and
+:math:`t+1`. The units of :math:`R` are dollars at time :math:`t+1` per
+dollar at time :math:`t`.
+
+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 :math:`x` dollars to you today, you pay me :math:`R x`
+   dollars tomorrow.
+
+-  This means that you borrowed :math:`x` dollars for me at a gross
+   interest rate :math:`R` and a net interest rate :math:`r`.
+
+We assume that the net nominal interest rate :math:`r` is fixed over
+time, so that :math:`R` is the gross nominal interest rate at times
+:math:`t=0, 1, 2, \ldots`.
+
+Two important geometric sequences are
+
+.. math::
+  :label: geom1
+
+  1, R, R^2, \cdots
+
+and
+
+.. math::
+  :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.
+
+Accumulation
+------------
+
+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, :math:`1` dollar invested at time :math:`0` pays interest
+   :math:`r` dollars after one period, so we have :math:`r+1 = R`
+   dollars at time\ :math:`1`
+
+-  at time :math:`1` we reinvest :math:`1+r =R` dollars and receive interest
+   of :math:`r R` dollars at time :math:`2` plus the *principal*
+   :math:`R` dollars, so we receive :math:`r R + R = (1+r)R = R^2`
+   dollars at the end of period :math:`2`
+
+-  and so on
+
+Evidently, if we invest :math:`x` dollars at time :math:`0` and
+reinvest the proceeds, then the sequence
+
+.. math:: x , xR , x R^2, \cdots
+
+tells how our account accumulates at dates :math:`t=0, 1, 2, \ldots`.
+
+Discounting
+-----------
+
+Geometric sequence :eq:`geom2` tells us how much future dollars are worth in terms of today's dollars.
+
+Remember that the units of :math:`R` are dollars at :math:`t+1` per
+dollar at :math:`t`.
+
+It follows that
+
+-  the units of :math:`R^{-1}` are dollars at :math:`t` per dollar at :math:`t+1`
+
+-  the units of :math:`R^{-2}` are dollars at :math:`t` per dollar at :math:`t+2`
+
+-  and so on; the units of :math:`R^{-j}` are dollars at :math:`t` per
+   dollar at :math:`t+j`
+
+So if someone has a claim on :math:`x` dollars at time :math:`t+j`, it
+is worth :math:`x R^{-j}` dollars at time :math:`t` (e.g., today).
+
+
+
+Application to Asset Pricing
+----------------------------
+
+A **lease** requires a payments stream of :math:`x_t` dollars at
+times :math:`t = 0, 1, 2, \ldots` where
+
+.. math::  x_t = G^t x_0
+
+where :math:`G = (1+g)` and :math:`g \in (0,1)`.
+
+Thus, lease payments increase at :math:`g` percent per period.
+
+For a reason soon to be revealed, we assume that :math:`G < R`.
+
+The **present value** of the lease is
+
+.. math::
+
+    \begin{aligned} p_0  & = x_0 + x_1/R + x_2/(R^2) + \ddots \\
+                     & = x_0 (1 + G R^{-1} + G^2 R^{-2} + \cdots ) \\
+                     & = x_0 \frac{1}{1 - G R^{-1}} \end{aligned}
+
+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
+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
+
+<<<<<<< HEAD
+and the fact that :math:`r` is small to approximate
+=======
+and the fact that :math:`r` is small to aproximate
+>>>>>>> a67f8a797020d585a745fcb8dc22b6b4ef322d1c
+:math:`\frac{1}{1+r} \approx 1 - r`.
+
+Use this approximation to write :math:`p_0` as
+
+.. math::
+
+    \begin{aligned}
+    p_0 &= x_0 \frac{1}{1 - G R^{-1}} \\
+    &= x_0 \frac{1}{1 - (1+g) (1-r) } \\
+    &= x_0 \frac{1}{1 - (1+g - r - rg)} \\
+    & \approx x_0 \frac{1}{r -g }
+   \end{aligned}
+
+where the last step uses the approximation :math:`r g \approx 0`.
+
+The approximation
+
+.. math:: p_0 = \frac{x_0 }{r -g }
+
+is known as the **Gordon formula** for the present value or current
+price of an infinite payment stream :math:`x_0 G^t` when the nominal
+one-period interest rate is :math:`r` and when :math:`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 :math:`x_t` for :math:`t= 1,2, \dots,T`, where again
+
+.. math:: x_t = G^t x_0
+
+The present value of this lease is:
+
+.. math:: \begin{aligned} \begin{split}p_0&=x_0 + x_1/R  + \dots +x_T/R^T \\ &= x_0(1+GR^{-1}+\dots +G^{T}R^{-T}) \\ &= \frac{x_0(1-G^{T+1}R^{-(T+1)})}{1-GR^{-1}}  \end{split}\end{aligned}
+
+Applying the Taylor series to :math:`R^{-(T+1)}` about :math:`r=0` we get:
+
+.. math:: \frac{1}{(1+r)^{T+1}}= 1-r(T+1)+\frac{1}{2}r^2(T+1)(T+2)+\dots \approx 1-r(T+1)
+
+Similarly, applying the Taylor series to :math:`G^{T+1}` about :math:`g=0`:
+
+.. math:: (1+g)^{T+1} = 1+(T+1)g(1+g)^T+(T+1)Tg^2(1+g)^{T-1}+\dots \approx 1+ (T+1)g
+
+Thus, we get the following approximation:
+
+.. math:: p_0 =\frac{x_0(1-(1+(T+1)g)(1-r(T+1)))}{1-(1-r)(1+g) }
+
+Expanding:
+
+.. math::  \begin{aligned} p_0 &=\frac{x_0(1-1+(T+1)^2 rg -r(T+1)+g(T+1))}{1-1+r-g+rg}  \\&=\frac{x_0(T+1)((T+1)rg+r-g)}{r-g+rg} \\ &\approx \frac{x_0(T+1)(r-g)}{r-g}+\frac{x_0rg(T+1)}{r-g}\\ &= x_0(T+1) + \frac{x_0rg(T+1)}{r-g}  \end{aligned}
+
+We could have also approximated by removing the second term
+:math:`rgx_0(T+1)` when :math:`T` is relatively small compared to
+:math:`1/(rg)` to get :math:`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 :math:`T`, and :math:`g` and :math:`r` in Julia.
+
+First we plot the true finite stream present-value after computing it
+below
+
+.. code-block:: julia
+
+    # True present value of a finite lease
+    function finite_lease_pv(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))
+    end
+
+    # First approximation for our finite lease
+    finite_lease_pv_approx_f(T, g, r, x_0) = x_0 * (T + 1) + x_0 * r * g * (T + 1) / (r - g)
+
+    # Second approximation for our finite lease
+    finite_lease_pv_approx_s(T, g, r, x_0) = x_0 * (T + 1)
+
+    # Infinite lease
+    function infinite_lease(g, r, x_0)
+        G = 1 + g
+        R = 1 + r
+        return x_0 / (1 - G * R^(-1))
+    end
+
+
+Now that we have test run our functions, we can plot some outcomes.
+
+First we study the quality of our approximations
+
+.. code-block:: julia
+
+    g = 0.02
+    r = 0.03
+    x_0 = 1
+    T_max = 50
+    T = 0:T_max
+    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)
+
+    plot(T, y_1, label="True T-period Lease PV", title= "Finite Lease Present Value T Periods Ahead", xlabel = "T Periods Ahead", ylabel = "Present Value, p₀")
+    plot!(T, y_2, label="T-period Lease First-order Approx.")
+    plot!(T, y_3, label="T-period Lease First-order Approx. adj.")
+    plot!(legend = :topleft)
+
+.. code-block:: julia
+    :class: test
+
+    @testset begin
+    @test y_1[4] ≈ 3.942123696037542
+    @test y_2[4] ≈ 4.24
+    @test y_3[4] ≈ 4
+    end
+
+Evidently our approximations perform well for small values of :math:`T`.
+
+However, holding :math:`g` and r fixed, our approximations deteriorate as :math:`T` increases.
+
+Next we compare the infinite and finite duration lease present values
+over different lease lengths :math:`T`.
+
+.. code-block:: julia
+
+    # Convergence of infinite and finite
+    T_max = 1000
+    T = 0:T_max
+    y_1 = finite_lease_pv.(T, g, r, x_0)
+    y_2 = ones(T_max+1) .* infinite_lease(g, r, x_0)
+    plot(T, y_1, label="T-period lease PV", title= "Infinite and Finite Lease Present Value T Periods Ahead", xlabel = "T Periods Ahead", ylabel = "Present Value, p0")
+    plot!(T, y_2, linestyle = :dash, label="Infinite lease PV")
+    plot!(legend = :bottomright)
+
+.. code-block:: julia
+    :class: test
+
+    @testset begin
+       @test y_1[4] ≈ 3.942123696037542
+       @test y_2[4] ≈ 103.00000000000004
+    end
+
+
+The above graphs shows how as duration :math:`T \rightarrow +\infty`,
+the value of a lease of duration :math:`T` approaches the value of a
+perpetual lease.
+
+Now we consider two different views of what happens as :math:`r` and
+:math:`g` covary
+
+.. code-block:: julia
+
+    T_max = 10
+    T=0:T_max
+
+    # r >> g, much bigger than g
+    r = 0.9
+    g = 0.4
+    plot(finite_lease_pv.(T, g, r, x_0), label="r >> g", title= "Value of lease of length T", xlabel = "T periods ahead", ylabel = "Present Value, p0")
+
+    # r > g
+    r = 0.5
+    g = 0.4
+    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
+    plot!(finite_lease_pv.(T, g, r, x_0), label="r ~ g", color="orange")
+
+    # r < g
+    r = 0.4
+    g = 0.5
+    plot!(finite_lease_pv.(T, g, r, x_0), label="r < g", color="red")
+    plot!(legend = :topleft)
+
+The above graphs 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.
+
+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!
+
+.. code-block:: julia
+
+    T = 3
+    r = 0.01:0.005:0.985
+    g = 0.011:0.005:0.986
+    N = length(r)
+
+    rr = reshape(repeat(r, N), N, N)'
+    gg = repeat(g, 1, N)
+    z = finite_lease_pv.(T, gg, rr, x_0)
+    plot(r, g, z, st=:surface, c=:coolwarm,title= "Three Period Lease PV with varying g and r", xlabel = "r", ylabel = "g")
+
+.. code-block:: julia
+    :class: test
+
+    @testset begin
+        @test z[4] ≈ 4.096057303642319
+    end
+
+.. We can use a little calculus to study how the present value :math:`p_0`
+.. of a lease varies with :math:`r` and :math:`g`.
+..
+.. We will use a library called `SymPy <https://www.sympy.org/>`__.
+..
+.. 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
+..
+.. .. code-block:: julia
+..
+..     # Creates algebraic symbols that can be used in an algebraic expression
+..     g, r, x0 = symbols("g, r, x0")
+..     G = (1 + g)
+..     R = (1 + r)
+..     p0 = x0 / (1 - G * R ^ (-1))
+..     print("Our formula is:")
+..     p0
+..
+.. .. code-block:: julia
+..
+..     print("dp0 / dg is:")
+..     dp_dg = diff(p0, g)
+..     dp_dg
+..
+.. .. code-block:: julia
+..
+..     print("dp0 / dr is:")
+..     dp_dr = diff(p0, r)
+..     dp_dr
+
+..     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.
+
+..     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.
+
+Back to the Keynesian Multiplier
+================================
+
+We will now go back to the case of the Keynesian multiplier and plot the
+time path of :math:`y_t`, given that consumption is a constant fraction
+of national income, and investment is fixed.
+
+.. code-block:: julia
+
+    # Function that calculates a path of y
+    function calculate_y(i, b, g, T, y_init)
+        y = zeros(T+1)
+        y[1] = i + b * y_init + g
+        for t = 2:(T+1)
+            y[t] = b * y[t-1] + i + g
+        end
+        return y
+    end
+
+    # 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
+
+    plot(0:T, calculate_y(i_0, b, g_0, T, y_init), title= "Path of Aggregate Output Over Time", xlabel = "t", label = "yt")
+
+    # Output predicted by geometric series
+    hline!([i_0 / (1 - b) + g_0 / (1 - b)], linestyle=:dash, seriestype="hline", label = "Predicted")
+    plot!(legend = :bottomright)
+
+.. code-block:: julia
+    :class: test
+
+    @testset begin
+        @test calculate_y(i_0, b, g_0, T, y_init)[4] ≈ 1.4444444444444444
+    end
+
+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
+
+.. code-block:: julia
+
+    # Changing fraction of consumption
+    b_0 = 1/3
+    b_1 = 2/3
+    b_2 = 5/6
+    b_3 = 0.9
+
+    plt = plot(title= "Changing Consumption as a Fraction of Income", xlabel = "t", ylabel = "yt")
+    x = 0:T
+    for b in [b_0, b_1, b_2, b_3]
+        y = calculate_y(i_0, b, g_0, T, y_init)
+        plot!(plt, x, y, label=string("b= ",round(b,digits=2)))
+    end
+    plot!(plt, legend = :bottomright)
+
+
+Increasing the marginal propensity to consumer :math:`b` increases the
+path of output over time
+
+.. below we need to set b=0.9 which does not happen in python
+.. since python changes value by reference in the for loop
+
+.. code-block:: julia
+
+    x = 0:T
+    b = 0.9
+    y_0 = calculate_y(i_0, b, g_0, T, y_init)
+    l = @layout [a ; b]
+
+    # Changing initial investment:
+    i_1 = 0.4
+    y_1 = calculate_y(i_1, b, g_0, T, y_init)
+    plt_1 = plot(x,y_0, label = "i=0.3", linestyle= :dash, title= "An Increase in Investment on Output", xlabel = "t", ylabel = "y_t")
+    plot!(plt_1, x, y_1, label = "i=0.4")
+    plot!(plt_1, legend = :bottomright)
+
+    # Changing government spending
+    g_1 = 0.4
+    y_1 = calculate_y(i_0, b, g_1, T, y_init)
+    plt_2 = plot(x, y_0, label = "g=0.3", linestyle= :dash, title= "An Increase in Investment on Output", xlabel = "t", ylabel = "y_t")
+    plot!(plt_2, x, y_1, label="g=0.4", linestyle=:dash)
+    plot!(plt_2, legend = :bottomright)
+
+    plot(plt_1, plt_2, layout = l)
+
+.. code-block:: julia
+    :class: test
+
+    @testset begin
+        @test y_1[2] ≈ 1.33
+        @test y_1[10] ≈ 4.5592509193
+    end
+
+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.
diff --git a/source/rst/tools_and_techniques/index.rst b/source/rst/tools_and_techniques/index.rst
index 03d98c53..5cd1c0a4 100644
--- a/source/rst/tools_and_techniques/index.rst
+++ b/source/rst/tools_and_techniques/index.rst
@@ -19,6 +19,7 @@ tools and techniques.
 .. toctree::
     :maxdepth: 2
 
+    geom_series
     linear_algebra
     orth_proj
     lln_clt