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lectures/_toc.yml

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- caption: Nonlinear Dynamics
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numbered: true
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chapters:
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- file: solow
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- file: scalar_dynam
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- file: solow
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- file: cobweb
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- file: olg
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- file: commod_price

lectures/scalar_dynam.md

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(scalar_dynam)=
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# Dynamics in One Dimension
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```{admonition} Migrated lecture
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:class: warning
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This lecture has moved from our [Intermediate Quantitative Economics with Python](https://python.quantecon.org/intro.html) lecture series and is now a part of [A First Course in Quantitative Economics](https://intro.quantecon.org/intro.html).
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```
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## Overview
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In this lecture we give a quick introduction to discrete time dynamics in one dimension.
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In economics many variables depend on their past values
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For example, it seems reasonable to believe that inflation last year with affects inflation this year.
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(Perhaps high inflation last year will lead people to demand higher wages to
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compensate, which will feed into higher prices this year.)
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Letting $\pi_t$ be inflation this year and $\pi_{t-1}$ be inflation last year, we
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can write this relationship in a general form as
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* In one-dimensional models, the state of the system is described by a single variable.
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* The variable is a number (that is, a point in $\mathbb R$).
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$$ \pi_t = f(\pi_{t-1}) $$
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While most quantitative models have two or more state variables, the
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one-dimensional setting is a good place to learn the foundations of dynamics
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and understand key concepts.
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where $f$ is some function describing the relationship between the variables.
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This equation is an example of one-dimensional discrete time dynamic system.
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In this lecture we cover the foundations of one-dimensional discrete time
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dynamics.
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(While most quantitative models have two or more state variables, the
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one-dimensional setting is a good place to learn foundations
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and understand key concepts.)
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Let's start with some standard imports:
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import numpy as np
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```
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## Some definitions
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This section sets out the objects of interest and the kinds of properties we study.
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### Example: a linear model
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One simple example of a dynamic system is when $S=\mathbb R$ and $g(x)=ax +
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b$, where $a, b$ are constants (sometimes they are referred as parameters).
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b$, where $a, b$ are constants (sometimes called ``parameters'').
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This leads to the **linear difference equation**
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When models are nonlinear, however, the situation can be quite different.
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For example, the law of motion for the Solow-Swan growth model, a simplified version of which is
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For example, in a later lecture {doc}`solow`, we will study the Solow-Swan growth model, which has dynamics
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```{math}
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:label: solow_lom2

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