From dfd749d560a6db74d43af1e40c947d982d8ee05c Mon Sep 17 00:00:00 2001
From: Humphrey Yang <39026988+HumphreyYang@users.noreply.github.com>
Date: Mon, 5 Feb 2024 10:27:16 +1100
Subject: [PATCH] Add Admonitions for Migrated Lectures (#340)
* update admonitions for migrated lectures
* update migration admonition for schelling
* remove contents tags
* migrate complex_and_trig lecture
* update migration taf for scalar_dynam
* update config and intersphinx
* merge main and fix pv intersphinx
---
lectures/_config.yml | 27 +-
lectures/_toc.yml | 1 +
lectures/complex_and_trig.md | 530 +++++++++++++++++++++++++++++++++++
lectures/geom_series.md | 6 +-
lectures/lp_intro.md | 6 +
lectures/pv.md | 2 +-
lectures/scalar_dynam.md | 6 +
lectures/schelling.md | 6 +
lectures/short_path.md | 6 +
9 files changed, 586 insertions(+), 4 deletions(-)
create mode 100644 lectures/complex_and_trig.md
diff --git a/lectures/_config.yml b/lectures/_config.yml
index 4e8c9093..6b99e06b 100644
--- a/lectures/_config.yml
+++ b/lectures/_config.yml
@@ -35,7 +35,7 @@ latex:
targetname: quantecon-python-intro.tex
sphinx:
- extra_extensions: [sphinx_multitoc_numbering, sphinxext.rediraffe, sphinx_exercise, sphinx_togglebutton, sphinx_proof, sphinx_tojupyter]
+ extra_extensions: [sphinx_multitoc_numbering, sphinxext.rediraffe, sphinx_exercise, sphinx_togglebutton, sphinx.ext.intersphinx, sphinx_proof, sphinx_tojupyter]
config:
# false-positive links
linkcheck_ignore: ['https://doi.org/https://doi.org/10.2307/1235116']
@@ -68,6 +68,31 @@ sphinx:
binderhub_url : https://mybinder.org # The URL of the BinderHub (e.g., https://mybinder.org)
colab_url : https://colab.research.google.com
thebe : false # Add a thebe button to pages (requires the repository to run on Binder)
+ intersphinx_mapping:
+ pyprog:
+ - https://python-programming.quantecon.org/
+ - null
+ intro:
+ - https://intro.quantecon.org/
+ - null
+ dle:
+ - https://quantecon.github.io/lecture-dle/
+ - null
+ dps:
+ - https://quantecon.github.io/lecture-dps/
+ - null
+ eqm:
+ - https://quantecon.github.io/lecture-eqm/
+ - null
+ stats:
+ - https://quantecon.github.io/lecture-stats/
+ - null
+ tools:
+ - https://quantecon.github.io/lecture-tools-techniques/
+ - null
+ dynam:
+ - https://quantecon.github.io/lecture-dynamics/
+ - null
mathjax3_config:
tex:
macros:
diff --git a/lectures/_toc.yml b/lectures/_toc.yml
index 04d3b875..8a9ba039 100644
--- a/lectures/_toc.yml
+++ b/lectures/_toc.yml
@@ -14,6 +14,7 @@ parts:
- caption: Essential Tools
numbered: true
chapters:
+ - file: complex_and_trig
- file: linear_equations
- file: eigen_I
- file: intro_supply_demand
diff --git a/lectures/complex_and_trig.md b/lectures/complex_and_trig.md
new file mode 100644
index 00000000..a8ebb1bb
--- /dev/null
+++ b/lectures/complex_and_trig.md
@@ -0,0 +1,530 @@
+---
+jupytext:
+ text_representation:
+ extension: .md
+ format_name: myst
+kernelspec:
+ display_name: Python 3
+ language: python
+ name: python3
+---
+
+(complex_and_trig)=
+```{raw} html
+
+```
+
+```{index} single: python
+```
+
+# Complex Numbers and Trigonometry
+
+```{admonition} Migrated lecture
+:class: warning
+
+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).
+```
+
+## Overview
+
+This lecture introduces some elementary mathematics and trigonometry.
+
+Useful and interesting in its own right, these concepts reap substantial rewards when studying dynamics generated
+by linear difference equations or linear differential equations.
+
+For example, these tools are keys to understanding outcomes attained by Paul
+Samuelson (1939) {cite}`Samuelson1939` in his classic paper on interactions
+between the investment accelerator and the Keynesian consumption function, our
+topic in the lecture {doc}`Samuelson Multiplier Accelerator `.
+
+In addition to providing foundations for Samuelson's work and extensions of
+it, this lecture can be read as a stand-alone quick reminder of key results
+from elementary high school trigonometry.
+
+So let's dive in.
+
+### Complex Numbers
+
+A complex number has a **real part** $x$ and a purely **imaginary part** $y$.
+
+The Euclidean, polar, and trigonometric forms of a complex number $z$ are:
+
+$$
+z = x + iy = re^{i\theta} = r(\cos{\theta} + i \sin{\theta})
+$$
+
+The second equality above is known as **Euler's formula**
+
+- [Euler](https://en.wikipedia.org/wiki/Leonhard_Euler) contributed many other formulas too!
+
+The complex conjugate $\bar z$ of $z$ is defined as
+
+$$
+\bar z = x - iy = r e^{-i \theta} = r (\cos{\theta} - i \sin{\theta} )
+$$
+
+The value $x$ is the **real** part of $z$ and $y$ is the
+**imaginary** part of $z$.
+
+The symbol $| z |$ = $\sqrt{\bar{z}\cdot z} = r$ represents the **modulus** of $z$.
+
+The value $r$ is the Euclidean distance of vector $(x,y)$ from the
+origin:
+
+$$
+r = |z| = \sqrt{x^2 + y^2}
+$$
+
+The value $\theta$ is the angle of $(x,y)$ with respect to the real axis.
+
+Evidently, the tangent of $\theta$ is $\left(\frac{y}{x}\right)$.
+
+Therefore,
+
+$$
+\theta = \tan^{-1} \Big( \frac{y}{x} \Big)
+$$
+
+Three elementary trigonometric functions are
+
+$$
+\cos{\theta} = \frac{x}{r} = \frac{e^{i\theta} + e^{-i\theta}}{2} , \quad
+\sin{\theta} = \frac{y}{r} = \frac{e^{i\theta} - e^{-i\theta}}{2i} , \quad
+\tan{\theta} = \frac{y}{x}
+$$
+
+We'll need the following imports:
+
+```{code-cell} ipython
+import matplotlib.pyplot as plt
+plt.rcParams["figure.figsize"] = (11, 5) #set default figure size
+import numpy as np
+from sympy import (Symbol, symbols, Eq, nsolve, sqrt, cos, sin, simplify,
+ init_printing, integrate)
+```
+
+### An Example
+
+Consider the complex number $z = 1 + \sqrt{3} i$.
+
+For $z = 1 + \sqrt{3} i$, $x = 1$, $y = \sqrt{3}$.
+
+It follows that $r = 2$ and
+$\theta = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} = 60^o$.
+
+Let's use Python to plot the trigonometric form of the complex number
+$z = 1 + \sqrt{3} i$.
+
+```{code-cell} python3
+# Abbreviate useful values and functions
+π = np.pi
+
+
+# Set parameters
+r = 2
+θ = π/3
+x = r * np.cos(θ)
+x_range = np.linspace(0, x, 1000)
+θ_range = np.linspace(0, θ, 1000)
+
+# Plot
+fig = plt.figure(figsize=(8, 8))
+ax = plt.subplot(111, projection='polar')
+
+ax.plot((0, θ), (0, r), marker='o', color='b') # Plot r
+ax.plot(np.zeros(x_range.shape), x_range, color='b') # Plot x
+ax.plot(θ_range, x / np.cos(θ_range), color='b') # Plot y
+ax.plot(θ_range, np.full(θ_range.shape, 0.1), color='r') # Plot θ
+
+ax.margins(0) # Let the plot starts at origin
+
+ax.set_title("Trigonometry of complex numbers", va='bottom',
+ fontsize='x-large')
+
+ax.set_rmax(2)
+ax.set_rticks((0.5, 1, 1.5, 2)) # Less radial ticks
+ax.set_rlabel_position(-88.5) # Get radial labels away from plotted line
+
+ax.text(θ, r+0.01 , r'$z = x + iy = 1 + \sqrt{3}\, i$') # Label z
+ax.text(θ+0.2, 1 , '$r = 2$') # Label r
+ax.text(0-0.2, 0.5, '$x = 1$') # Label x
+ax.text(0.5, 1.2, r'$y = \sqrt{3}$') # Label y
+ax.text(0.25, 0.15, r'$\theta = 60^o$') # Label θ
+
+ax.grid(True)
+plt.show()
+```
+
+## De Moivre's Theorem
+
+de Moivre's theorem states that:
+
+$$
+(r(\cos{\theta} + i \sin{\theta}))^n =
+r^n e^{in\theta} =
+r^n(\cos{n\theta} + i \sin{n\theta})
+$$
+
+To prove de Moivre's theorem, note that
+
+$$
+(r(\cos{\theta} + i \sin{\theta}))^n = \big( re^{i\theta} \big)^n
+$$
+
+and compute.
+
+## Applications of de Moivre's Theorem
+
+### Example 1
+
+We can use de Moivre's theorem to show that
+$r = \sqrt{x^2 + y^2}$.
+
+We have
+
+$$
+\begin{aligned}
+1 &= e^{i\theta} e^{-i\theta} \\
+&= (\cos{\theta} + i \sin{\theta})(\cos{(\text{-}\theta)} + i \sin{(\text{-}\theta)}) \\
+&= (\cos{\theta} + i \sin{\theta})(\cos{\theta} - i \sin{\theta}) \\
+&= \cos^2{\theta} + \sin^2{\theta} \\
+&= \frac{x^2}{r^2} + \frac{y^2}{r^2}
+\end{aligned}
+$$
+
+and thus
+
+$$
+x^2 + y^2 = r^2
+$$
+
+We recognize this as a theorem of **Pythagoras**.
+
+### Example 2
+
+Let $z = re^{i\theta}$ and $\bar{z} = re^{-i\theta}$ so that $\bar{z}$ is the **complex conjugate** of $z$.
+
+$(z, \bar z)$ form a **complex conjugate pair** of complex numbers.
+
+Let $a = pe^{i\omega}$ and $\bar{a} = pe^{-i\omega}$ be
+another complex conjugate pair.
+
+For each element of a sequence of integers $n = 0, 1, 2, \ldots, $.
+
+To do so, we can apply de Moivre's formula.
+
+Thus,
+
+$$
+\begin{aligned}
+x_n &= az^n + \bar{a}\bar{z}^n \\
+&= p e^{i\omega} (re^{i\theta})^n + p e^{-i\omega} (re^{-i\theta})^n \\
+&= pr^n e^{i (\omega + n\theta)} + pr^n e^{-i (\omega + n\theta)} \\
+&= pr^n [\cos{(\omega + n\theta)} + i \sin{(\omega + n\theta)} +
+ \cos{(\omega + n\theta)} - i \sin{(\omega + n\theta)}] \\
+&= 2 pr^n \cos{(\omega + n\theta)}
+\end{aligned}
+$$
+
+### Example 3
+
+This example provides machinery that is at the heard of Samuelson's analysis of his multiplier-accelerator model {cite}`Samuelson1939`.
+
+Thus, consider a **second-order linear difference equation**
+
+$$
+x_{n+2} = c_1 x_{n+1} + c_2 x_n
+$$
+
+whose **characteristic polynomial** is
+
+$$
+z^2 - c_1 z - c_2 = 0
+$$
+
+or
+
+$$
+(z^2 - c_1 z - c_2 ) = (z - z_1)(z- z_2) = 0
+$$
+
+has roots $z_1, z_1$.
+
+A **solution** is a sequence $\{x_n\}_{n=0}^\infty$ that satisfies
+the difference equation.
+
+Under the following circumstances, we can apply our example 2 formula to
+solve the difference equation
+
+- the roots $z_1, z_2$ of the characteristic polynomial of the
+ difference equation form a complex conjugate pair
+- the values $x_0, x_1$ are given initial conditions
+
+To solve the difference equation, recall from example 2 that
+
+$$
+x_n = 2 pr^n \cos{(\omega + n\theta)}
+$$
+
+where $\omega, p$ are coefficients to be determined from
+information encoded in the initial conditions $x_1, x_0$.
+
+Since
+$x_0 = 2 p \cos{\omega}$ and $x_1 = 2 pr \cos{(\omega + \theta)}$
+the ratio of $x_1$ to $x_0$ is
+
+$$
+\frac{x_1}{x_0} = \frac{r \cos{(\omega + \theta)}}{\cos{\omega}}
+$$
+
+We can solve this equation for $\omega$ then solve for $p$ using $x_0 = 2 pr^0 \cos{(\omega + n\theta)}$.
+
+With the `sympy` package in Python, we are able to solve and plot the
+dynamics of $x_n$ given different values of $n$.
+
+In this example, we set the initial values: - $r = 0.9$ -
+$\theta = \frac{1}{4}\pi$ - $x_0 = 4$ -
+$x_1 = r \cdot 2\sqrt{2} = 1.8 \sqrt{2}$.
+
+We first numerically solve for $\omega$ and $p$ using
+`nsolve` in the `sympy` package based on the above initial
+condition:
+
+```{code-cell} python3
+# Set parameters
+r = 0.9
+θ = π/4
+x0 = 4
+x1 = 2 * r * sqrt(2)
+
+# Define symbols to be calculated
+ω, p = symbols('ω p', real=True)
+
+# Solve for ω
+## Note: we choose the solution near 0
+eq1 = Eq(x1/x0 - r * cos(ω+θ) / cos(ω), 0)
+ω = nsolve(eq1, ω, 0)
+ω = float(ω)
+print(f'ω = {ω:1.3f}')
+
+# Solve for p
+eq2 = Eq(x0 - 2 * p * cos(ω), 0)
+p = nsolve(eq2, p, 0)
+p = float(p)
+print(f'p = {p:1.3f}')
+```
+
+Using the code above, we compute that
+$\omega = 0$ and $p = 2$.
+
+Then we plug in the values we solve for $\omega$ and $p$
+and plot the dynamic.
+
+```{code-cell} python3
+# Define range of n
+max_n = 30
+n = np.arange(0, max_n+1, 0.01)
+
+# Define x_n
+x = lambda n: 2 * p * r**n * np.cos(ω + n * θ)
+
+# Plot
+fig, ax = plt.subplots(figsize=(12, 8))
+
+ax.plot(n, x(n))
+ax.set(xlim=(0, max_n), ylim=(-5, 5), xlabel='$n$', ylabel='$x_n$')
+
+# Set x-axis in the middle of the plot
+ax.spines['bottom'].set_position('center')
+ax.spines['right'].set_color('none')
+ax.spines['top'].set_color('none')
+ax.xaxis.set_ticks_position('bottom')
+ax.yaxis.set_ticks_position('left')
+
+ticklab = ax.xaxis.get_ticklabels()[0] # Set x-label position
+trans = ticklab.get_transform()
+ax.xaxis.set_label_coords(31, 0, transform=trans)
+
+ticklab = ax.yaxis.get_ticklabels()[0] # Set y-label position
+trans = ticklab.get_transform()
+ax.yaxis.set_label_coords(0, 5, transform=trans)
+
+ax.grid()
+plt.show()
+```
+
+### Trigonometric Identities
+
+We can obtain a complete suite of trigonometric identities by
+appropriately manipulating polar forms of complex numbers.
+
+We'll get many of them by deducing implications of the equality
+
+$$
+e^{i(\omega + \theta)} = e^{i\omega} e^{i\theta}
+$$
+
+For example, we'll calculate identities for
+
+$\cos{(\omega + \theta)}$ and $\sin{(\omega + \theta)}$.
+
+Using the sine and cosine formulas presented at the beginning of this
+lecture, we have:
+
+$$
+\begin{aligned}
+\cos{(\omega + \theta)} = \frac{e^{i(\omega + \theta)} + e^{-i(\omega + \theta)}}{2} \\
+\sin{(\omega + \theta)} = \frac{e^{i(\omega + \theta)} - e^{-i(\omega + \theta)}}{2i}
+\end{aligned}
+$$
+
+We can also obtain the trigonometric identities as follows:
+
+$$
+\begin{aligned}
+\cos{(\omega + \theta)} + i \sin{(\omega + \theta)}
+&= e^{i(\omega + \theta)} \\
+&= e^{i\omega} e^{i\theta} \\
+&= (\cos{\omega} + i \sin{\omega})(\cos{\theta} + i \sin{\theta}) \\
+&= (\cos{\omega}\cos{\theta} - \sin{\omega}\sin{\theta}) +
+i (\cos{\omega}\sin{\theta} + \sin{\omega}\cos{\theta})
+\end{aligned}
+$$
+
+Since both real and imaginary parts of the above formula should be
+equal, we get:
+
+$$
+\begin{aligned}
+\cos{(\omega + \theta)} = \cos{\omega}\cos{\theta} - \sin{\omega}\sin{\theta} \\
+\sin{(\omega + \theta)} = \cos{\omega}\sin{\theta} + \sin{\omega}\cos{\theta}
+\end{aligned}
+$$
+
+The equations above are also known as the **angle sum identities**. We
+can verify the equations using the `simplify` function in the
+`sympy` package:
+
+```{code-cell} python3
+# Define symbols
+ω, θ = symbols('ω θ', real=True)
+
+# Verify
+print("cos(ω)cos(θ) - sin(ω)sin(θ) =",
+ simplify(cos(ω)*cos(θ) - sin(ω) * sin(θ)))
+print("cos(ω)sin(θ) + sin(ω)cos(θ) =",
+ simplify(cos(ω)*sin(θ) + sin(ω) * cos(θ)))
+```
+
+### Trigonometric Integrals
+
+We can also compute the trigonometric integrals using polar forms of
+complex numbers.
+
+For example, we want to solve the following integral:
+
+$$
+\int_{-\pi}^{\pi} \cos(\omega) \sin(\omega) \, d\omega
+$$
+
+Using Euler's formula, we have:
+
+$$
+\begin{aligned}
+\int \cos(\omega) \sin(\omega) \, d\omega
+&=
+\int
+\frac{(e^{i\omega} + e^{-i\omega})}{2}
+\frac{(e^{i\omega} - e^{-i\omega})}{2i}
+\, d\omega \\
+&=
+\frac{1}{4i}
+\int
+e^{2i\omega} - e^{-2i\omega}
+\, d\omega \\
+&=
+\frac{1}{4i}
+\bigg( \frac{-i}{2} e^{2i\omega} - \frac{i}{2} e^{-2i\omega} + C_1 \bigg) \\
+&=
+-\frac{1}{8}
+\bigg[ \bigg(e^{i\omega}\bigg)^2 + \bigg(e^{-i\omega}\bigg)^2 - 2 \bigg] + C_2 \\
+&=
+-\frac{1}{8} (e^{i\omega} - e^{-i\omega})^2 + C_2 \\
+&=
+\frac{1}{2} \bigg( \frac{e^{i\omega} - e^{-i\omega}}{2i} \bigg)^2 + C_2 \\
+&= \frac{1}{2} \sin^2(\omega) + C_2
+\end{aligned}
+$$
+
+and thus:
+
+$$
+\int_{-\pi}^{\pi} \cos(\omega) \sin(\omega) \, d\omega =
+\frac{1}{2}\sin^2(\pi) - \frac{1}{2}\sin^2(-\pi) = 0
+$$
+
+We can verify the analytical as well as numerical results using
+`integrate` in the `sympy` package:
+
+```{code-cell} python3
+# Set initial printing
+init_printing()
+
+ω = Symbol('ω')
+print('The analytical solution for integral of cos(ω)sin(ω) is:')
+integrate(cos(ω) * sin(ω), ω)
+```
+
+```{code-cell} python3
+print('The numerical solution for the integral of cos(ω)sin(ω) \
+from -π to π is:')
+integrate(cos(ω) * sin(ω), (ω, -π, π))
+```
+
+### Exercises
+
+```{exercise}
+:label: complex_ex1
+
+We invite the reader to verify analytically and with the `sympy` package the following two equalities:
+
+$$
+\int_{-\pi}^{\pi} \cos (\omega)^2 \, d\omega = \pi
+$$
+
+$$
+\int_{-\pi}^{\pi} \sin (\omega)^2 \, d\omega = \pi
+$$
+```
+
+```{solution-start} complex_ex1
+:class: dropdown
+```
+
+Let's import symbolic $\pi$ from `sympy`
+
+```{code-cell} ipython3
+# Import symbolic π from sympy
+from sympy import pi
+```
+
+```{code-cell} ipython3
+print('The analytical solution for the integral of cos(ω)**2 \
+from -π to π is:')
+
+integrate(cos(ω)**2, (ω, -pi, pi))
+```
+
+```{code-cell} ipython3
+print('The analytical solution for the integral of sin(ω)**2 \
+from -π to π is:')
+
+integrate(sin(ω)**2, (ω, -pi, pi))
+```
+
+```{solution-end}
+```
diff --git a/lectures/geom_series.md b/lectures/geom_series.md
index 3d9bfe70..af7ec7b8 100644
--- a/lectures/geom_series.md
+++ b/lectures/geom_series.md
@@ -25,8 +25,10 @@ kernelspec:
# Geometric Series for Elementary Economics
-```{contents} Contents
-:depth: 2
+```{admonition} Migrated lecture
+:class: warning
+
+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).
```
## Overview
diff --git a/lectures/lp_intro.md b/lectures/lp_intro.md
index 8f2c02af..26896607 100644
--- a/lectures/lp_intro.md
+++ b/lectures/lp_intro.md
@@ -12,6 +12,12 @@ kernelspec:
(lp_intro)=
# Linear Programming
+```{admonition} Migrated lecture
+:class: warning
+
+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).
+```
+
In this lecture, we will need the following library. Install [ortools](https://developers.google.com/optimization) using `pip`.
```{code-cell} ipython3
diff --git a/lectures/pv.md b/lectures/pv.md
index c9923531..7bdfa0fc 100644
--- a/lectures/pv.md
+++ b/lectures/pv.md
@@ -82,7 +82,7 @@ A system of equations like {eq}`eq:Euler1` is an example of a linear **differen
There are powerful mathematical methods available for solving such systems and they are well worth
studying in their own right, being the foundation for the analysis of many interesting economic models.
-For an example, see {doc}`Samuelson multiplier-accelerator `
+For an example, see {doc}`Samuelson multiplier-accelerator `
In this lecture, we'll solve system {eq}`eq:Euler1` using matrix multiplication and matrix inversion, basic tools from linear algebra introduced in {doc}`linear equations and matrix algebra `.
diff --git a/lectures/scalar_dynam.md b/lectures/scalar_dynam.md
index bfb1274e..864cb54d 100644
--- a/lectures/scalar_dynam.md
+++ b/lectures/scalar_dynam.md
@@ -20,6 +20,12 @@ kernelspec:
(scalar_dynam)=
# Dynamics in One Dimension
+```{admonition} Migrated lecture
+:class: warning
+
+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).
+```
+
## Overview
In this lecture we give a quick introduction to discrete time dynamics in one dimension.
diff --git a/lectures/schelling.md b/lectures/schelling.md
index 94dd0987..1bdc9004 100644
--- a/lectures/schelling.md
+++ b/lectures/schelling.md
@@ -28,6 +28,12 @@ kernelspec:
```{index} single: Models; Schelling's Segregation Model
```
+```{admonition} Migrated lecture
+:class: warning
+
+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).
+```
+
## Outline
In 1969, Thomas C. Schelling developed a simple but striking model of racial
diff --git a/lectures/short_path.md b/lectures/short_path.md
index d9eb0771..68ab9e97 100644
--- a/lectures/short_path.md
+++ b/lectures/short_path.md
@@ -23,6 +23,12 @@ kernelspec:
```{index} single: Dynamic Programming; Shortest Paths
```
+```{admonition} Migrated lecture
+:class: warning
+
+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).
+```
+
## Overview
The shortest path problem is a [classic problem](https://en.wikipedia.org/wiki/Shortest_path) in mathematics and computer science with applications in
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