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[update_money_inflation_nonlinear] Editorial Suggestion (#448)
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* update_money_inflation_nonlinear

This updates the lecture money_inflation_nonlinear.md according to #446.

In particular, this includes
## Code
- Remove the grid.
- Change the legend for the [Laffer curve plot](https://intro.quantecon.org/money_inflation_nonlinear.html#laffer-curve-nonlinear) to "Laffer curve". Also, use default colors in `matplotlib`.

## Content
- Bold face is reserved for definitions and italics are used for emphases.
- Only the first word in the title should be capitalized.
- Migrate some of "This lecture studies" to  "We study ...".
- Maintain consistent use of bar and overline.
- (28.3) repeats (28.2). Consider removing (28.3) and adjusting the links to (28.2) if (28.3) is removed.
- Define all the notations like $\pi$ and $m$.
- "limiting values exists" -> exist
-  "...that studies a linear version of the model in this lecture`"  -> "that studies a linear version of the model in this lecture."
- In Equation (28.6) we have $x: x \ge 0$. Remove "x:", and put a curly bracket covering the equation after max to indicate the scope of max.

* update_money_inflation_nonlinear

Delete 'this lecture' before {doc}.
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longye-tian authored Jun 17, 2024
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Expand Up @@ -15,18 +15,18 @@ kernelspec:

## Overview

This lecture studies stationary and dynamic **Laffer curves** in the inflation tax rate in a non-linear version of the model studied in this lecture {doc}`money_inflation`.
We study stationary and dynamic *Laffer curves* in the inflation tax rate in a non-linear version of the model studied in {doc}`money_inflation`.

This lecture uses the log-linear version of the demand function for money that {cite}`Cagan`
used in his classic paper in place of the linear demand function used in this lecture {doc}`money_inflation`.
We use the log-linear version of the demand function for money that {cite}`Cagan`
used in his classic paper in place of the linear demand function used in {doc}`money_inflation`.

That change requires that we modify parts of our analysis.

In particular, our dynamic system is no longer linear in state variables.

Nevertheless, the economic logic underlying an analysis based on what we called ''method 2'' remains unchanged.

in this lecture we shall discover qualitatively similar outcomes to those that we studied in the lecture {doc}`money_inflation`.
We shall discover qualitatively similar outcomes to those that we studied in {doc}`money_inflation`.

That lecture presented a linear version of the model in this lecture.

Expand All @@ -35,17 +35,17 @@ As in that lecture, we discussed these topics:
* an **inflation tax** that a government gathers by printing paper or electronic money
* a dynamic **Laffer curve** in the inflation tax rate that has two stationary equilibria
* perverse dynamics under rational expectations in which the system converges to the higher stationary inflation tax rate
* a peculiar comparative stationary-state analysis connected with that stationary inflation rate that assert that inflation can be **reduced** by running **higher** government deficits
* a peculiar comparative stationary-state analysis connected with that stationary inflation rate that assert that inflation can be *reduced* by running *higher* government deficits

These outcomes will set the stage for the analysis of this lecture {doc}`laffer_adaptive` that studies a version of the present model that uses a version of "adaptive expectations" instead of rational expectations.
These outcomes will set the stage for the analysis of {doc}`laffer_adaptive` that studies a version of the present model that uses a version of "adaptive expectations" instead of rational expectations.

That lecture will show that

* replacing rational expectations with adaptive expectations leaves the two stationary inflation rates unchanged, but that $\ldots$
* it reverse the pervese dynamics by making the **lower** stationary inflation rate the one to which the system typically converges
* a more plausible comparative dynamic outcome emerges in which now inflation can be **reduced** by running **lower** government deficits
* it reverse the pervese dynamics by making the *lower* stationary inflation rate the one to which the system typically converges
* a more plausible comparative dynamic outcome emerges in which now inflation can be *reduced* by running *lower* government deficits

## The Model
## The model

Let

Expand All @@ -70,9 +70,9 @@ where $g$ is the part of government expenditures financed by printing money.
**Remark:** Please notice that while equation {eq}`eq:mdemand` is linear in logs of the money supply and price level, equation {eq}`eq:msupply` is linear in levels. This will require adapting the equilibrium computation methods that we deployed in {doc}`money_inflation`.
## Computing An Equilibrium Sequence
## Computing an equilibrium sequence
We'll deploy a method similar to **Method 2** used in {doc}`money_inflation`.
We'll deploy a method similar to *Method 2* used in {doc}`money_inflation`.
We'll take the time $t$ state vector to be $m_t, p_t$.
Expand All @@ -84,14 +84,7 @@ $$
\lambda \equiv \frac{\alpha}{1+ \alpha}
$$
Let's rewrite equations {eq}`eq:msupply` and {eq}`eq:mdemand`, respectively, as
$$
\exp(m_{t+1}) - \exp(m_t) = g \exp(p_t)
$$ (eq:msupply2)
and
Let's rewrite equation {eq}`eq:mdemand`, respectively, as
$$
p_t = (1-\lambda) m_{t+1} + \lambda p_{t+1}
Expand All @@ -103,34 +96,34 @@ We'll summarize our algorithm with the following pseudo-code.
* start for $m_0, p_0$ at time $t =0$
* solve {eq}`eq:msupply2` for $m_{t+1}$
* solve {eq}`eq:msupply` for $m_{t+1}$
* solve {eq}`eq:mdemand2` for $p_{t+1} = \lambda^{-1} p_t + (1 - \lambda^{-1}) m_{t+1}$
* compute $\pi_t = p_{t+1} - p_t$ and $\mu_t = m_{t+1} - m_t $
* compute the inflation rate $\pi_t = p_{t+1} - p_t$ and growth of money supply $\mu_t = m_{t+1} - m_t $
* iterate on $t$ to convergence of $\pi_t \rightarrow \overline \pi$ and $\mu_t \rightarrow \overline \mu$
It will turn out that
* if they exist, limiting values $\overline \pi$ and $\overline \mu$ will be equal
* if limiting values exists, there are two possible limiting values, one high, one low
* if limiting values exist, there are two possible limiting values, one high, one low
* for almost all initial log price levels $p_0$, the limiting $\overline \pi = \overline \mu$ is
the higher value
* for each of the two possible limiting values $\bar \pi$ ,there is a unique initial log price level $p_0$ that implies that $\pi_t = \mu_t = \bar \mu$ for all $t \geq 0$
* for each of the two possible limiting values $\overline \pi$ ,there is a unique initial log price level $p_0$ that implies that $\pi_t = \mu_t = \overline \mu$ for all $t \geq 0$
* this unique initial log price level solves $\log(\exp(m_0) + g \exp(p_0)) - p_0 = - \alpha \bar \pi $
* this unique initial log price level solves $\log(\exp(m_0) + g \exp(p_0)) - p_0 = - \alpha \overline \pi $
* the preceding equation for $p_0$ comes from $m_1 - p_0 = - \alpha \bar \pi$
* the preceding equation for $p_0$ comes from $m_1 - p_0 = - \alpha \overline \pi$
## Limiting Values of Inflation Rate
## Limiting values of inflation rate
We can compute the two prospective limiting values for $\bar \pi$ by studying the steady-state Laffer curve.
We can compute the two prospective limiting values for $\overline \pi$ by studying the steady-state Laffer curve.
Thus, in a **steady state**
Thus, in a *steady state*
$$
m_{t+1} - m_t = p_{t+1} - p_t = x \quad \forall t ,
Expand All @@ -147,7 +140,7 @@ $$ (eq:steadypi)
where we require that
$$
g \leq \max_{x: x \geq 0} \exp(-\alpha x) - \exp(-(1 + \alpha) x) ,
g \leq \max_{x \geq 0} \{\exp(-\alpha x) - \exp(-(1 + \alpha) x) \},
$$ (eq:revmax)
so that it is feasible to finance $g$ by printing money.
Expand All @@ -159,7 +152,7 @@ The right side of {eq}`eq:steadypi` is the quantity of time $t$ goods that the
Soon we'll plot the left and right sides of equation {eq}`eq:steadypi`.
But first we'll write code that computes a steady-state
$\bar \pi$.
$\overline \pi$.
Let's start by importing some libraries
Expand All @@ -182,7 +175,7 @@ CaganLaffer = namedtuple('CaganLaffer',
"λ",
"g" ])
# Create a CaganLaffer model
# Create a Cagan Laffer model
def create_model(α=0.5, m0=np.log(100), g=0.35):
return CaganLaffer(α=α, m0=m0, λ=α/(1+α), g=g)
Expand All @@ -191,7 +184,7 @@ model = create_model()
+++ {"user_expressions": []}
Now we write code that computes steady-state $\bar \pi$s.
Now we write code that computes steady-state $\overline \pi$s.
```{code-cell} ipython3
# Define formula for π_bar
Expand All @@ -208,9 +201,9 @@ def solve_π_bar(model, x0):
print(f'The two steady state of π are: {π_l, π_u}')
```
We find two steady state $\bar \pi$ values
We find two steady state $\overline \pi$ values.
## Steady State Laffer Curve
## Steady state Laffer curve
The following figure plots the steady state Laffer curve together with the two stationary inflation rates.
Expand All @@ -237,7 +230,7 @@ def plot_laffer(model, πs):
# Plot the function
plt.plot(x_values, y_values,
label=f'$exp((-{α})x) - exp(- (1- {α}) x)$')
label=f'Laffer curve')
for π, label in zip(πs, ['$\pi_l$', '$\pi_u$']):
plt.text(π, plt.gca().get_ylim()[0]*2,
label, horizontalalignment='center',
Expand All @@ -248,16 +241,15 @@ def plot_laffer(model, πs):
plt.xlabel('$\pi$')
plt.ylabel('seigniorage')
plt.legend()
plt.grid(True)
plt.show()
# Steady state Laffer curve
plot_laffer(model, (π_l, π_u))
```
## Associated Initial Price Levels
## Associated initial price levels
Now that we have our hands on the two possible steady states, we can compute two initial log price levels $p_0$, which as initial conditions, imply that $\pi_t = \bar \pi $ for all $t \geq 0$.
Now that we have our hands on the two possible steady states, we can compute two initial log price levels $p_0$, which as initial conditions, imply that $\pi_t = \overline \pi $ for all $t \geq 0$.
```{code-cell} ipython3
def solve_p0(p0, m0, α, g, π):
Expand Down Expand Up @@ -320,9 +312,9 @@ eq_g = lambda x: np.exp(-model.α * x) - np.exp(-(1 + model.α) * x)
print('eq_g == g:', np.isclose(eq_g(m_seq[-1] - m_seq[-2]), model.g))
```
## Slippery Side of Laffer Curve Dynamics
## Slippery side of Laffer curve dynamics
We are now equipped to compute time series starting from different $p_0$ settings, like those in this lecture {doc}`money_inflation`.
We are now equipped to compute time series starting from different $p_0$ settings, like those in {doc}`money_inflation`.
```{code-cell} ipython3
:tags: [hide-cell]
Expand Down Expand Up @@ -398,26 +390,26 @@ p0_bars = (p0_l, p0_u)
draw_iterations(p0s, model, line_params, p0_bars, num_steps=20)
```
Staring at the paths of price levels in {numref}`p0_path_nonlin` reveals that almost all paths converge to the **higher** inflation tax rate displayed in the stationary state Laffer curve. displayed in figure {numref}`laffer_curve_nonlinear`.
Staring at the paths of price levels in {numref}`p0_path_nonlin` reveals that almost all paths converge to the *higher* inflation tax rate displayed in the stationary state Laffer curve. displayed in figure {numref}`laffer_curve_nonlinear`.
Thus, we have reconfirmed what we have called the "perverse" dynamics under rational expectations in which the system converges to the higher of two possible stationary inflation tax rates.
Those dynamics are "perverse" not only in the sense that they imply that the monetary and fiscal authorities that have chosen to finance government expenditures eventually impose a higher inflation tax than required to finance government expenditures, but because of the following "counterintuitive" situation that we can deduce by staring at the stationary state Laffer curve displayed in figure {numref}`laffer_curve_nonlinear`:
* the figure indicates that inflation can be **reduced** by running **higher** government deficits, i.e., by raising more resources through printing money.
* the figure indicates that inflation can be *reduced* by running *higher* government deficits, i.e., by raising more resources through printing money.
```{note}
The same qualitive outcomes prevail in this lecture {doc}`money_inflation` that studies a linear version of the model in this lecture`.
The same qualitive outcomes prevail in {doc}`money_inflation` that studies a linear version of the model in this lecture.
```
We discovered that
* all but one of the equilibrium paths converge to limits in which the higher of two possible stationary inflation tax prevails
* there is a unique equilibrium path associated with "plausible" statements about how reductions in government deficits affect a stationary inflation rate
As in this lecture {doc}`money_inflation`,
As in {doc}`money_inflation`,
on grounds of plausibility, we again recommend selecting the unique equilibrium that converges to the lower stationary inflation tax rate.
As we shall see, we accepting this recommendation is a key ingredient of outcomes of the "unpleasant arithmetic" that we describe in lecture {doc}`unpleasant`.
As we shall see, we accepting this recommendation is a key ingredient of outcomes of the "unpleasant arithmetic" that we describe in {doc}`unpleasant`.
In lecture, {doc}`laffer_adaptive`, we shall explore how {cite}`bruno1990seigniorage` and others justified our equilibrium selection in other ways.
In {doc}`laffer_adaptive`, we shall explore how {cite}`bruno1990seigniorage` and others justified our equilibrium selection in other ways.

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