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lectures/unpleasant.md

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## Overview
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This lecture builds on concepts and issues introduced in our lecture on {doc}`money supplies and price levels<money_inflation>`.
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This lecture builds on concepts and issues introduced in {doc}`money_inflation`.
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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
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on currency.
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And the larger is $T$, the higher is the gross-of-interest government deficit that must be financed
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by printing money at times $t \geq T$.
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These outcomes are the essential finding of Sargent and Wallace's **unpleasant monetarist arithmetic** {cite}`sargent1981`.
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```{tip}
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Please read our lecture on {doc}`money supplies and price levels<money_inflation>` before diving into this lecture.
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```
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These outcomes are the essential finding of Sargent and Wallace's "unpleasant monetarist arithmetic" {cite}`sargent1981`.
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That lecture described supplies and demands for money that appear in lecture.
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It also characterized the steady state equilibrium from which we work backwards in this lecture.
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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.
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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.
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## Setup
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## An open market operation at $t=0$
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Following {cite:t}`sargent1981`, we analyze consequences of a central bank policy that
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Following Sargent and Wallace {cite}`sargent1981`, we analyze consequences of a central bank policy that
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uses an open market operation to lower the price level in the face of a persistent fiscal
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deficit that takes the form of a positive $g$.
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## Algorithm (pseudo code)
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Now let's describe a computational algorithm in more detail in the form of a description
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that constitutes ''pseudo code'' because it approaches a set of instructions we could provide to a
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that constitutes pseudo code because it approaches a set of instructions we could provide to a
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Python coder.
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To compute an equilibrium, we deploy the following algorithm.
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* Compute $R_u, p_T$ from formulas {eq}`eq:up_steadyquadratic` and {eq}`eq:LafferTstationary` above
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* Compute a new estimate of $p_0$, call it $\widehat p_0$, from equation {eq}`eq:allts` above
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* Compute a new estimate of $p_0$, call it $\widehat p_0$, from equation {eq}`eq:allts` above
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* Note that the preceding steps define a mapping
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Notice that the slope of $p_0$ as a function of $m_0$ is constant.
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This outcome indicates that our model verifies a quantity theory of money outcome,
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something that {cite:t}`sargent1981` purposefully built into their model to justify
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something that Sargent and Wallace {cite}`sargent1981` purposefully built into their model to justify
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the adjective *monetarist* in their title.
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plot_path([80, 100], msm)
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```
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{numref}`fig:unpl1` summarizes outcomes of two experiments that convey messages of {cite:t}`sargent1981`.
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{numref}`fig:unpl1` summarizes outcomes of two experiments that convey messages of Sargent and Wallace {cite}`sargent1981`.
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* An open market operation that reduces the supply of money at time $t=0$ reduces the price level at time $t=0$
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