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[unpleasant] Address Comments on Styles and Improve Citation Style #495

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[unpleasant] Address Comments on Styles and Improve Citation Style #495

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add admonitions, unbold emphasis, and clean the citation style
HumphreyYang Jul 1, 2024
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address comments
HumphreyYang Jul 4, 2024
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38 changes: 19 additions & 19 deletions lectures/unpleasant.md
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Expand Up @@ -16,7 +16,7 @@ kernelspec:
## Overview


This lecture builds on concepts and issues introduced in our lecture on **Money Supplies and Price Levels**.
This lecture builds on concepts and issues introduced in {doc}`money_inflation`.

That lecture describes stationary equilibria that reveal a [*Laffer curve*](https://en.wikipedia.org/wiki/Laffer_curve) in the inflation tax rate and the associated stationary rate of return
on currency.
Expand All @@ -34,20 +34,18 @@ The critical **money-to-bonds** ratio stabilizes only at time $T$ and afterwards
And the larger is $T$, the higher is the gross-of-interest government deficit that must be financed
by printing money at times $t \geq T$.

These outcomes are the essential finding of Sargent and Wallace's **unpleasant monetarist arithmetic** {cite}`sargent1981`.

**Reader's Guide:** Please read our lecture on Money Supplies and Price levels before diving into this lecture.
These outcomes are the essential finding of Sargent and Wallace's "unpleasant monetarist arithmetic" {cite}`sargent1981`.

That lecture described supplies and demands for money that appear in lecture.

It also characterized the steady state equilibrium from which we work backwards in this lecture.

In addition to learning about ''unpleasant monetarist arithmetic", in this lecture we'll learn how to implement a **fixed point** algorithm for computing an initial price level.
In addition to learning about "unpleasant monetarist arithmetic", in this lecture we'll learn how to implement a [*fixed point*](https://en.wikipedia.org/wiki/Fixed_point_(mathematics)) algorithm for computing an initial price level.


## Setup

Let's start with quick reminders of the model's components set out in our lecture on **Money Supplies and Price Levels**.
Let's start with quick reminders of the model's components set out in {doc}`money_inflation`.

Please consult that lecture for more details and Python code that we'll also use in this lecture.

Expand Down Expand Up @@ -79,7 +77,7 @@ where $\gamma_1 > \gamma_2 > 0$.

## Monetary-Fiscal Policy

To the basic model of our lecture on **Money Supplies and Price Levels**, we add inflation-indexed one-period government bonds as an additional way for the government to finance government expenditures.
To the basic model of {doc}`money_inflation`, we add inflation-indexed one-period government bonds as an additional way for the government to finance government expenditures.

Let $\widetilde R > 1$ be a time-invariant gross real rate of return on government one-period inflation-indexed bonds.

Expand Down Expand Up @@ -114,11 +112,11 @@ $$ (eq:openmarketconstraint)
This equation says that the government (e.g., the central bank) can *decrease* $m_0$ relative to
$\check m_0$ by *increasing* $B_{-1}$ relative to $\check B_{-1}$.

This is a version of a standard constraint on a central bank's **open market operations** in which it expands the stock of money by buying government bonds from the public.
This is a version of a standard constraint on a central bank's [**open market operations**](https://www.federalreserve.gov/monetarypolicy/openmarket.htm) in which it expands the stock of money by buying government bonds from the public.

## An open market operation at $t=0$

Following Sargent and Wallace (1981), we analyze consequences of a central bank policy that
Following Sargent and Wallace {cite}`sargent1981`, we analyze consequences of a central bank policy that
uses an open market operation to lower the price level in the face of a persistent fiscal
deficit that takes the form of a positive $g$.

Expand Down Expand Up @@ -242,18 +240,20 @@ $$
p_T = \frac{m_0}{\gamma_1 - \overline g - \gamma_2 R_u^{-1}} = \gamma_1^{-1} m_0 \left\{\frac{1}{R_u-\lambda} \right\}
$$ (eq:pTformula)

**Remark:**
```{prf:remark}
We can verify the equivalence of the two formulas on the right sides of {eq}`eq:pTformula` by recalling that
$R_u$ is a root of the quadratic equation {eq}`eq:up_steadyquadratic` that determines steady state rates of return on currency.
```

## Algorithm (pseudo code)

Now let's describe a computational algorithm in more detail in the form of a description
that constitutes ''pseudo code'' because it approaches a set of instructions we could provide to a
that constitutes pseudo code because it approaches a set of instructions we could provide to a
Python coder.

To compute an equilibrium, we deploy the following algorithm.

```{prf:algorithm}
Given *parameters* include $g, \check m_0, \check B_{-1}, \widetilde R >1, T $.

We define a mapping from $p_0$ to $\widehat p_0$ as follows.
Expand All @@ -280,8 +280,7 @@ $$

* Compute $R_u, p_T$ from formulas {eq}`eq:up_steadyquadratic` and {eq}`eq:LafferTstationary` above

* Compute a new estimate of $p_0$, call it $\widehat p_0$, from equation {eq}`eq:allts` above

* Compute a new estimate of $p_0$, call it $\widehat p_0$, from equation {eq}`eq:allts` above

* Note that the preceding steps define a mapping

Expand All @@ -298,7 +297,7 @@ p_{0,j+1} = (1-\theta) {\mathcal S}(p_{0,j}) + \theta p_{0,j},
$$

where $\theta \in [0,1)$ is a relaxation parameter.

```

## Example Calculations

Expand All @@ -318,7 +317,7 @@ That leaves the public with less currency but more government interest-bearing b

Since the public has less currency (it's supply has diminished) it is plausible to anticipate that the price level at time $0$ will be driven downward.

But that is not the end of the story, because this ''open market operation'' at time $0$ has consequences for future settings of $m_{t+1}$ and the gross-of-interest government deficit $\bar g_t$.
But that is not the end of the story, because this **open market operation** at time $0$ has consequences for future settings of $m_{t+1}$ and the gross-of-interest government deficit $\bar g_t$.
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Let's start with some imports:
Expand All @@ -329,7 +328,7 @@ import matplotlib.pyplot as plt
from collections import namedtuple
```

Now let's dive in and implement our ''pseudo code'' in Python.
Now let's dive in and implement our pseudo code in Python.

```{code-cell} ipython3
# Create a namedtuple that contains parameters
Expand Down Expand Up @@ -395,14 +394,15 @@ def compute_fixed_point(m0, p0_guess, model, θ=0.5, tol=1e-6):

return p0
```

Let's look at how price level $p_0$ in the stationary $R_u$ equilibrium depends on the initial
money supply $m_0$.

Notice that the slope of $p_0$ as a function of $m_0$ is constant.

This outcome indicates that our model verifies a ''quantity theory of money'' outcome,
This outcome indicates that our model verifies a quantity theory of money outcome,
something that Sargent and Wallace {cite}`sargent1981` purposefully built into their model to justify
the adjective **monetarist** in their title.
the adjective *monetarist* in their title.


```{code-cell} ipython3
Expand Down Expand Up @@ -494,7 +494,7 @@ mystnb:
plot_path([80, 100], msm)
```

{numref}`fig:unpl1` summarizes outcomes of two experiments that convey messages of {cite}`sargent1981`.
{numref}`fig:unpl1` summarizes outcomes of two experiments that convey messages of Sargent and Wallace {cite}`sargent1981`.

* An open market operation that reduces the supply of money at time $t=0$ reduces the price level at time $t=0$

Expand Down