diff --git a/lectures/unpleasant.md b/lectures/unpleasant.md index f5232b82..c3592872 100644 --- a/lectures/unpleasant.md +++ b/lectures/unpleasant.md @@ -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. @@ -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. @@ -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. @@ -114,11 +112,11 @@ $$ (eq:openmarketconstraint) This equation says that the government (e.g., the central bank) can *decrease* $m_0$ relative to $\check m_0$ by *increasing* $B_{-1}$ relative to $\check B_{-1}$. -This is a version of a standard constraint on a central bank's **open market operations** in which it expands the stock of money by buying government bonds from the public. +This is a version of a standard constraint on a central bank's [**open market operations**](https://www.federalreserve.gov/monetarypolicy/openmarket.htm) in which it expands the stock of money by buying government bonds from the public. ## An open market operation at $t=0$ -Following Sargent and Wallace (1981), we analyze consequences of a central bank policy that +Following Sargent and Wallace {cite}`sargent1981`, we analyze consequences of a central bank policy that uses an open market operation to lower the price level in the face of a persistent fiscal deficit that takes the form of a positive $g$. @@ -242,18 +240,20 @@ $$ 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\} $$ (eq:pTformula) -**Remark:** +```{prf:remark} We can verify the equivalence of the two formulas on the right sides of {eq}`eq:pTformula` by recalling that $R_u$ is a root of the quadratic equation {eq}`eq:up_steadyquadratic` that determines steady state rates of return on currency. +``` ## Algorithm (pseudo code) Now let's describe a computational algorithm in more detail in the form of a description -that constitutes ''pseudo code'' because it approaches a set of instructions we could provide to a +that constitutes pseudo code because it approaches a set of instructions we could provide to a Python coder. To compute an equilibrium, we deploy the following algorithm. +```{prf:algorithm} Given *parameters* include $g, \check m_0, \check B_{-1}, \widetilde R >1, T $. We define a mapping from $p_0$ to $\widehat p_0$ as follows. @@ -280,8 +280,7 @@ $$ * 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 @@ -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 @@ -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 $\bar g_t$. Let's start with some imports: @@ -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 @@ -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 @@ -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 $t=0$