Skip to content
New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Jump to bottom

[unpleasant] Update editorial suggestions #447

Merged
merged 5 commits into from
Jul 1, 2024
Merged

[unpleasant] Update editorial suggestions #447

Changes from all commits
Commits
Show all changes
5 commits
Select commit Hold shift + click to select a range
f4ccaef
[unpleasant] Update editorial suggestions
SylviaZhaooo May 22, 2024
da69308
update links between lectures
HumphreyYang Jun 27, 2024
414d880
Update unpleasant.md
SylviaZhaooo Jun 30, 2024
e47d444
Update unpleasant.md
SylviaZhaooo Jul 1, 2024
87b01e4
Update unpleasant.md
SylviaZhaooo Jul 1, 2024
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
62 changes: 28 additions & 34 deletions lectures/unpleasant.md
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -11,14 +11,14 @@ kernelspec:
name: python3
---

# Unpleasant Monetarist Arithmetic
# Some Unpleasant Monetarist Arithmetic

## Overview


This lecture builds on concepts and issues introduced in our lecture on **Money Supplies and Price Levels**.

That lecture describes stationary equilibria that reveal a **Laffer curve** in the inflation tax rate and the associated stationary rate of return
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.

In this lecture we study a situation in which a stationary equilibrium prevails after date $T > 0$, but not before then.
Expand Down Expand Up @@ -75,6 +75,8 @@ $$
b_t = \gamma_1 - \gamma_2 R_t^{-1} .
$$ (eq:up_bdemand)

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.
Expand All @@ -91,8 +93,8 @@ $$
Just before the beginning of time $0$, the public owns $\check m_0$ units of currency (measured in dollars)
and $\widetilde R \check B_{-1}$ units of one-period indexed bonds (measured in time $0$ goods); these two quantities are initial conditions set outside the model.

Notice that $\check m_0$ is a **nominal** quantity, being measured in dollar, while
$\widetilde R \check B_{-1}$ is a **real** quantity, being measured in time $0$ goods.
Notice that $\check m_0$ is a *nominal* quantity, being measured in dollars, while
$\widetilde R \check B_{-1}$ is a *real* quantity, being measured in time $0$ goods.


### Open market operations
Expand All @@ -109,8 +111,8 @@ $$
B_{-1} - \check B_{-1} = \frac{1}{p_0 \widetilde R} \left( \check m_0 - m_0 \right)
$$ (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 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.

Expand Down Expand Up @@ -152,15 +154,15 @@ running monetary and fiscal policies.

Here, by **fiscal policy** we mean the collection of actions that determine a sequence of net-of-interest government deficits $\{g_t\}_{t=0}^\infty$ that must be financed by issuing to the public either money or interest bearing bonds.

By **monetary policy** or **debt-management polcy**, we mean the collection of actions that determine how the government divides its portolio of debts to the public between interest-bearing parts (government bonds) and non-interest-bearing parts (money).
By **monetary policy** or **debt-management policy**, we mean the collection of actions that determine how the government divides its portolio of debts to the public between interest-bearing parts (government bonds) and non-interest-bearing parts (money).
SylviaZhaooo marked this conversation as resolved.
Show resolved Hide resolved

By an **open market operation**, we mean a government monetary policy action in which the government
(or its delegate, say, a central bank) either buys government bonds from the public for newly issued money, or sells bonds to the public and withdraws the money it receives from public circulation.

## Algorithm (basic idea)


We work backwards from $t=T$ and first compute $p_T, R_u$ associated with the low-inflation, low-inflation-tax-rate stationary equilibrium of our lecture on the dynamic Laffer curve for the inflation tax.
We work backwards from $t=T$ and first compute $p_T, R_u$ associated with the low-inflation, low-inflation-tax-rate stationary equilibrium in {doc}`money_inflation_nonlinear`.

To start our description of our algorithm, it is useful to recall that a stationary rate of return
on currency $\bar R$ solves the quadratic equation
Expand All @@ -171,7 +173,7 @@ $$ (eq:up_steadyquadratic)

Quadratic equation {eq}`eq:up_steadyquadratic` has two roots, $R_l < R_u < 1$.

For reasons described at the end of **this lecture**, we select the larger root $R_u$.
For reasons described at the end of {doc}`money_inflation`, we select the larger root $R_u$.


Next, we compute
Expand Down Expand Up @@ -211,7 +213,7 @@ We want to compute

$$
\begin{aligned}
p_0 & = \gamma_1^{-1} \left[ \sum_{j=0}^\infty \lambda^j m_{1+j} \right] \cr
p_0 & = \gamma_1^{-1} \left[ \sum_{j=0}^\infty \lambda^j m_{j} \right] \cr
& = \gamma_1^{-1} \left[ \sum_{j=0}^{T-1} \lambda^j m_{0} + \sum_{j=T}^\infty \lambda^j m_{1+j} \right]
\end{aligned}
$$
Expand Down Expand Up @@ -252,9 +254,9 @@ Python coder.

To compute an equilibrium, we deploy the following algorithm.

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

We define a mapping from $p_0$ to $p_0$ as follows.
We define a mapping from $p_0$ to $\widehat p_0$ as follows.

* Set $m_0$ and then compute $B_{-1}$ to satisfy the constraint on time $0$ **open market operations**

Expand Down Expand Up @@ -301,7 +303,7 @@ where $\theta \in [0,1)$ is a relaxation parameter.
## Example Calculations

We'll set parameters of the model so that the steady state after time $T$ is initially the same
as in our lecture on "Money and Inflation".
as in {doc}`money_inflation_nonlinear`

In particular, we set $\gamma_1=100, \gamma_2 =50, g=3.0$. We set $m_0 = 100$ in that lecture,
but now the counterpart will be $M_T$, which is endogenous.
Expand Down Expand Up @@ -469,25 +471,16 @@ def simulate(m0, model, length=15, p0_guess=1):
def plot_path(m0_arr, model, length=15):

fig, axs = plt.subplots(2, 2, figsize=(8, 5))

titles = ['$p_t$', '$m_t$', '$b_t$', '$R_t$']

for m0 in m0_arr:
paths = simulate(m0, msm, length=length)

axs[0, 0].plot(paths[0])
axs[0, 0].set_title('$p_t$')

axs[0, 1].plot(paths[1])
axs[0, 1].set_title('$m_t$')

axs[1, 0].plot(paths[2])
axs[1, 0].set_title('$b_t$')

axs[1, 1].plot(paths[3])
axs[1, 1].set_title('$R_t$')

axs[0, 1].hlines(model.m0_check, 0, length,
color='r', linestyle='--')
axs[0, 1].text(length*0.8, model.m0_check*0.9, '$\check{m}_0$')
paths = simulate(m0, model, length=length)
for i, ax in enumerate(axs.flat):
ax.plot(paths[i])
ax.set_title(titles[i])

axs[0, 1].hlines(model.m0_check, 0, length, color='r', linestyle='--')
axs[0, 1].text(length * 0.8, model.m0_check * 0.9, '$\check{m}_0$')
plt.show()
```

Expand All @@ -501,11 +494,12 @@ mystnb:
plot_path([80, 100], msm)
```

Figure {numref}`fig:unpl1` summarizes outcomes of two experiments that convey messages of
Sargent and Wallace's **unpleasant monetarist arithmetic** {cite}`sargent1981`.
{numref}`fig:unpl1` summarizes outcomes of two experiments that convey messages of {cite}`sargent1981`.

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

* The lower is the post-open-market-operation money supply at time $0$, lower is the price level at time $0$.

* An open market operation that reduces the post-open-market-operation money supply at time $0$ also **lowers** the rate of return on money $R_u$ at times $t \geq T$ because it brings a higher gross-of-interest government deficit that must be financed by printing money (i.e., levying an inflation tax) at time $t \geq T$.
* An open market operation that reduces the post open market operation money supply at time $0$ also *lowers* the rate of return on money $R_u$ at times $t \geq T$ because it brings a higher gross of interest government deficit that must be financed by printing money (i.e., levying an inflation tax) at time $t \geq T$.

* $R$ is important in the context of maintaining monetary stability and addressing the consequences of increased inflation due to government deficits. Thus, a larger $R$ might be chosen to mitigate the negative impacts on the real rate of return caused by inflation.