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Copy file name to clipboardExpand all lines: lectures/laffer_adaptive.md
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## Overview
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This lecture studies stationary and dynamic **Laffer curves** in the inflation tax rate in a non-linear version of the model studied in this XXXX lecture.
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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`.
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As in lecture XXXXX, this lecture uses the log-linear version of the demand function for money that Cagan {cite}`Cagan` used in his classic paper in place of the linear demand function used in this XXXX lecture.
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As in the lecture {doc}`money_inflation`, this lecture uses the log-linear version of the demand function for money that Cagan {cite}`Cagan` used in his classic paper in place of the linear demand function used in this lecture {doc}`money_inflation`.
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But now, instead of assuming ''rational expectations'' in the form of ''perfect foresight'',
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we'll adopt the ''adaptive expectations'' assumption used by Cagan {cite}`Cagan`.
Copy file name to clipboardExpand all lines: lectures/money_inflation.md
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## Overview
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This lecture describes a theory of price level variations that consists of two components
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This lecture extends and modifies the model in this lecture {doc}`cagan_ree` by modifying the
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law of motion that governed the supply of money.
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In particular, this lecture describes a theory of price level variations that consists of two components
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* a demand function for money
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* a law of motion for the supply of money
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Our model equates the demand for money to the supply at each time $t \geq 0$.
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Equality between those demands and supply gives in a **dynamic** model in which money supply
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and price level **sequences** are simultaneously determined by a special set of simultaneous linear
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equations.
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and price level **sequences** are simultaneously determined by a special set of simultaneous linear equations.
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These equations take the form of what are often called vector linear **difference equations**.
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In this lecture, we'll roll up our sleeves and solve those equations in a couple of different ways.
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As we'll see, Python is good at solving them.
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(One of the methods for solving vector linear difference equations will take advantage of a decomposition of a matrix that is studied in this lecture {doc}`eigen_I`.)
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In this lecture we will encounter these concepts
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* an **inflation tax** that a government gathers by printing paper or electronic money
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* a dynamic **Laffer curve** in the inflation tax rate that has two stationary equilibria
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* perverse dynamics under rational expectations in which the system converges to the higher stationary inflation tax rate
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* a peculiar comparative stationary-state analysis connected with that stationary inflation rate that assert that inflation can be **reduced** by running **higher** government deficits
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The same qualitive outcomes prevail in this lecture {doc}`money_inflation_nonlinear` that studies a nonlinear version of the model in this lecture.
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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.
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That lecture will show that
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* replacing rational expectations with adaptive expectations leaves the two stationary inflation rates unchanged, but that $\ldots$
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* it reverse the pervese dynamics by making the **lower** stationary inflation rate the one to which the system typically converges
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* a more plausible comparative dynamic outcome emerges in which now inflation can be **reduced** by running **lower** government deficits
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This outcome will be used to justify a selection of a stationary inflation rate that underlies the analysis of unpleasant monetarist arithmetic to be studies in this lecture {doc}`unpleasant`.
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We'll use theses tools from linear algebra:
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* matrix multiplication
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* matrix inversion
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* eigenvalues and eigenvectors of a matrix
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Let's start with some imports:
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```{code-cell} ipython3
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import numpy as np
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import matplotlib.pyplot as plt
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from matplotlib.ticker import MaxNLocator
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plt.rcParams['figure.dpi'] = 300
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from collections import namedtuple
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```
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## Demand for and Supply of Money
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So two steady states typically exist.
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## Some Code
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Let's start with some imports:
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```{code-cell} ipython3
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import numpy as np
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import matplotlib.pyplot as plt
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from matplotlib.ticker import MaxNLocator
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plt.rcParams['figure.dpi'] = 300
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from collections import namedtuple
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```
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Let's set some parameter values and compute possible steady state rates of return on currency $\bar R$, the seigniorage maximizing rate of return on currency, and an object that we'll discuss later, namely, an initial price level $p_0$ associated with the maximum steady state rate of return on currency.
Copy file name to clipboardExpand all lines: lectures/money_inflation_nonlinear.md
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## Overview
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This lecture studies stationary and dynamic **Laffer curves** in the inflation tax rate in a non-linear version of the model studied in this XXXX lecture.
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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`.
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This lecture uses the log-linear version of the demand function for money that Cagan {cite}`Cagan`
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used in his classic paper in place of the linear demand function used in this XXXX lecture.
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used in his classic paper in place of the linear demand function used in this lecture {doc}`money_inflation`.
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That change requires that we modify parts of our analysis.
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