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\begin{document}
\title{Econ 340: Week 1}
\author[1]{Ennio E. Piano}%
\affil[1]{George Mason University}%
\vspace{-1em}
\date{\today}
\begingroup
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\section*{Introduction}
{\label{389528}}
\subsection*{General information}
{\label{763108}}
\begin{itemize}
\tightlist
\item
Instructor: Ennio E. Piano,~\emph{Department of
Economics~}\&~\emph{F.A. Hayek Advanced Program in Politics Philosophy
and Economics, George Mason University}
\item
Email: \href{mailto:epiano@gmu.edu}{\nolinkurl{epiano@gmu.edu}}
\item
Class schedule: Thursday, 4:30-7:10 PM, Buchanan Hall D005
\item
Office hours: Thursday, 9:00-10:30 AM, Buchanan Hall, D137-7 (this is
in the Hayek Program Office)
\end{itemize}
\subsection*{Course structure and
objectives}
{\label{448036}}
\begin{itemize}
\tightlist
\item
13 lectures over three months and a half
\item
Main goal: familiarizing yourself with the basic tools of mathematics
as they are used by economists.
\item
Focus on the economics, not the mathematics. The mathematic is
instrumental to the economic argument one wishes to communicate.
\item
Most of the class will cover basic calculus. If we have time we will
also cover integration (unlikely). Basic calculus is what you need to
know to understand most papers in economics.
\item
We will mainly look at~\emph{microeconomic applications}. Why?
Contemporary macro is basically microeconomics ``addressed in a loud
voice'' (Richard Wagner)
\item
Textbook: Sydsaeter et al.~\emph{Essential Mathematics for Economic
Analysis~}(any edition after the second would work, available at the
bookshop on campus)
\item
Notes: The last two weeks we will review some important applications
of the math learned throughout the semester, as they will be
prominently features on the final exam. Plus: a good exercise to
review the content of the class. I will provide my own notes with the
derivation of important models.
\end{itemize}
\subsection*{Tentative outline}
{\label{904366}}
\begin{itemize}
\tightlist
\item
See syllabus
\end{itemize}
\subsection*{Grading}
{\label{778279}}
\begin{itemize}
\tightlist
\item
10\%: Homework (two sets: \#1 to be distributed on September 21st and
due October 5th; \#2 to be distributed on November 16th and due on
November 30th).~\textbf{Bring a physical copy to class. Do not send me
an electronic copy. I do not accept late work under ANY CIRCUMSTANCES.
Also: no make up work.}
\item
40\%: Midterm (3-5 problems depending on difficulty)
\item
40\%: Final (3-5 problems depending on difficulty, cumulative)
\item
10\%: Class participation (class participation is~\emph{extremely
important} for a class like this one. Mathematics is hard to grasp and
master and requires continuous involvement (George Polya). You will be
asked to come to the whiteboard and attempt to do an exercise.
Everyone in the class will have to do it. \textbf{You will not be
graded based on whether you know the answer, but on whether you
participate in the collective effort to find one.}
\item
For this reason, class attendance is strongly recommended although it
is not required.~
\end{itemize}
\section*{Why mathematical economics?}
{\label{258223}}
\subsection*{Why economics?}
{\label{276312}}
\begin{itemize}
\tightlist
\item
``There is only one social science\ldots{}'' (Gary Becker)
\item
Simple tools to explain the social world
\item
From purposefulness to utility maximization to the law of demand
\item
The theory of everything
\end{itemize}
\subsection*{Why mathematics?}
{\label{350619}}
\begin{itemize}
\tightlist
\item
``The objective of mathematicians is to discover and to communicate
certain truths.'' (Daniel Solow)
\item
The ``clarity argument''
(e.g.,~\href{https://paulromer.net/clear-writing-produces-clearer-thoughts/}{Paul
Romer})
\end{itemize}
\begin{quote}
Good mathematical theory is valuable because it encourages clear writing
and thereby produces clearer thoughts in the mind of the author. It
encourages clear writing by limiting the vocabulary that the author
uses. The mathematical equations depend on a~limited set of symbols. In
good theory, each symbol is tightly bound to specific, precisely defined
word or phrase from the vocabulary. The combination of the equations and
the words used in the theory give the words and symbols precise meaning.
The prose in good theory does not pull in vaguely defined terms that are
not in this vocabulary.
\end{quote}
\subsection*{Example (also from Romer)}
{\label{247595}}
\begin{itemize}
\tightlist
\item
Say we want to investigate the relationship between two economic
variables: human capital and knowledge
\item
We start with an abstract conception of the relationship:
\end{itemize}
\(H\rightarrow A \rightarrow H\)
\begin{itemize}
\tightlist
\item
This relationship represents the English sentence:~
\end{itemize}
\begin{quote}
Someone with human capital can produce knowledge that is codified in the
text of a message and send it to someone else, who reads the message and
stores the insight as human capital
\end{quote}
\begin{itemize}
\tightlist
\item
Say we want to include the time dimension implied by this sentence in
the mathematical formulation. How do we do this?
\end{itemize}
\(H_t\rightarrow A_{t+1}\rightarrow H_{t+2}\)
\begin{itemize}
\tightlist
\item
You can make it even more clear by specifying the relationship implied
by the arrows
\end{itemize}
\(H_t\rightarrow^{w} A_{t+1}\rightarrow^{r} H_{t+2}\)
\section*{Algebra}
{\label{351800}}
\subsection*{The real numbers}
{\label{925091}}
\begin{itemize}
\tightlist
\item
The set of all real numbers, ``R,'' includes the sets of natural
numbers, integers, rational, and irrational numbers.
\end{itemize}
\subsubsection*{Natural numbers}
{\label{885856}}
\begin{itemize}
\tightlist
\item
Numbers we can actually count in nature (positive integers)
\item
1, 2, 3, 4, \ldots{}
\item
We can classify natural numbers as even (2, 4, 6, \ldots{}) and odd
(1, 3, 5, \ldots{}). (What about zero? Zero is an even integer, but it
is neither positive nor negative)
\item
The symbol for the natural numbers is simply ``N.'' When we mean to
include the zero we write "N\textsubscript{0}" while when we want to
clearly exclude the zero we write "N\textsubscript{+}".
\end{itemize}
\subsubsection*{Integers}
{\label{679130}}
\begin{itemize}
\tightlist
\item
The set of all natural numbers, zero, and the negative of all natural
numbers (negative integers)
\item
0,~\(\pm\)1,~\(\pm\)2, \ldots{}
\item
The number line
\item
The symbol for the set of all integers is ``Z''
\end{itemize}
\subsubsection*{Rational numbers}
{\label{125392}}
\begin{itemize}
\tightlist
\item
The set of all numbers that can be expressed ``rationally,'' that is,
that can be expressed as a fraction of two integers
\item
\(\frac{a}{b}\), where a, b are integers.
Important:~\(\frac{a}{0}\) is not defined for
any~\(a \in R\)
\item
The set of rational numbers is represented by the symbol ``Q''
\end{itemize}
\subsubsection*{Irrational numbers}
{\label{869338}}
\begin{itemize}
\tightlist
\item
The set of irrational numbers is the set of all numbers that cannot be
expressed as fractions
\item
\(2^{.5}\) and~\(\pi\) are examples of irrational
numbers
\end{itemize}
\subsubsection*{Getting familiar with the language of
math}
{\label{153315}}
\begin{itemize}
\tightlist
\item
We say that 1 is a member of N, which is a subset of Z etc.
~if~\(1\in N \subset Z \subset Q \subset R\)
\end{itemize}
\subsubsection*{Examples}
{\label{616942}}
\subsection*{}
{\label{282494}}
\subsection*{Integers powers}
{\label{282494}}
\subsubsection*{Positive integer to the power of another positive
integer}
{\label{877105}}
\begin{itemize}
\tightlist
\item
~\(a^n=a\cdot a\cdot ... \cdot a\) (multiply an integer number by itself for n times)
\item
\(a^0=1, \forall a \in \mathbb{R}\) and~\(a\neq 0\).~
\item
\(0^0\)is undefined.
\item
\(a^1=a, a^2=a\cdot a\) and so forth
\end{itemize}
\subsubsection*{Negative integer to the power of a positive
integer}
{\label{549316}}
\begin{itemize}
\tightlist
\item
\(a^{-n}=1/a^{n}\). Mind that this is undefined
for~\(a=0\).
\end{itemize}
\subsubsection*{Rational number to the power of an
integer}
{\label{754364}}
\begin{itemize}
\tightlist
\item
\((\frac{p}{q})^n=\frac{p}{q}\cdot \frac{p}{q} \cdot ... \cdot\frac{p}{q}\) and also~\((\frac{p}{q})^n=\frac{p^n}{q^n}\)
\end{itemize}
\subsection*{Properties of powers}
{\label{561871}}
\begin{itemize}
\tightlist
\item
Powers with the same base are multiplied by adding the
exponents:~\(a^r\cdot a^s=a^{r+s}\)
\item
A power of a power is solved by multiplying the
exponents:~\((a^r)^s=a^{r\cdot s}\)
\item
These rules also applies to divisions. Thus~\(a^r\div a^s= a^{r-s}\)
\item
Also, the power of a product is equal to the product of the
powers:~\((a\cdot b)^{r}=a^r\cdot b^r\)
\item
Do example 1 p6
\end{itemize}
\subsection*{Compound interest}
{\label{456353}}
\begin{itemize}
\tightlist
\item
The rule of compound interest is very important in economic
applications (from intertemporal choice theory development economics)
\item
Say we have \$100. If we put this money into our saving account we get
an interest of 1.5\% a month. What do we have after one year? After 5
years? after 30 years?
\item
General formula:~\(x_t=x_0(1+r)^t\)
\item
So, after one year we get~\(x_1=100(1.015)^1=101.5\)
\item
After five years we get~\(x_1=100(1.015)^5\approx 107.73\)
\item
After forty years:~\(x_1=100(1.015)^{40}\approx 181.40\)
\item
Of course, the growth factor (the element within the parentheses) can
also imply a negative rate of growth:~\(x_t=x_0(1-r)^t\)
\end{itemize}
\subsection*{Why do we have negative
exponents?}
{\label{330519}}
\begin{itemize}
\tightlist
\item
We can use negative exponents to answer the question: how much money
should I have deposited, say, five years ago in order to get \$110
today?
\item
\(x_0=x_t(1=r)^{-t}\)
\item
Note that this is just obtained by rearranging the formula above!
\end{itemize}
\subsection*{Rules of algebra}
{\label{221929}}
\begin{itemize}
\tightlist
\item
I trust that you guys all know the rules of algebra. If you need to
refresh your memory the book does a pretty good job. If you have any
doubts/difficulties, you can email me or come to office hours.
\end{itemize}
\subsection*{Fractional powers}
{\label{740779}}
\subsubsection*{Square roots}
{\label{375298}}
\begin{itemize}
\tightlist
\item
The inverse of the square of a real number, we can write a square root
in many ways:~\(a^{.5}\equiv a^{1/2} \equiv \sqrt{a}\)
\item
I personally prefer the first as it makes multiplication of powers
more intuitive
\item
Of course, a square root is only valide for~\(a\geq 0\).
\item
The same rules we saw for integer powers also apply for fractional
powers (product rule, ratio rule)
\item
Important:~\(\sqrt{a+b}\neq \sqrt{a}+\sqrt{b}\) !
\item
Also important:~\(a^2=\pm \sqrt{a^2}\)
\item
Examples
\end{itemize}
\subsubsection*{Nth roots}
{\label{180798}}
\begin{itemize}
\tightlist
\item
We call~\(a^{1/n}\) the nth root of~\(a\)
\item
Note that~\((a^{1/n})^n=a^{1/n\cdot n}=a\)
\item
Do example 1 p23
\item
Also note that~\(a^{p/q}\equiv (a^{1/q})^p\)
\item
Do example 4 p25
\end{itemize}
\subsection*{Inequalities}
{\label{702203}}
\begin{itemize}
\tightlist
\item
We say that a is larger than b if~\(a>b\)
\item
We say that a is larger or equal to b if~\(a\geq b\)
\item
These implies the following rules
\item
If~\(a, b>0 \Longrightarrow a+b>0\)
\item
Similarly, if~\(a, b>0 \Longrightarrow a\cdot b>0\)
\item
Also, if~\(a>b \Longrightarrow a-b>0\)
\item
We can also prove that if~\(a>b \Longrightarrow a+c>b+c\)
\item
Proof: a-b= a+c-(b+c), which implies that, for a\textgreater{}b,
a-b\textgreater{}0. But then~a+c-(b+c)\textgreater{}0 and
rearranging,~a+c\textgreater{}b+c. QED
\end{itemize}
\subsubsection*{Properties of
inequalities}
{\label{313220}}
\begin{enumerate}
\tightlist
\item
If~\(a>b\) and~\(b>c\),
then~\(a>c\)
\item
If~\(a>b\) and~\(c>0\),
then~\(ac>bc\). But
if~\(c<0\),~\(acb\) and~\(c>d\),
then~\(a+c>b+d\)
\end{enumerate}
\begin{itemize}
\tightlist
\item
Proof of 3:~\(a+d>b+d\) and~\(c+a>d+a\). Combining the
two we get~\(a+c>a+d>b+d \Longrightarrow a+c>b+d\)
\end{itemize}
\subsubsection*{Sign diagrams}
{\label{840588}}
\begin{itemize}
\tightlist
\item
Skip the sign diagrams
\end{itemize}
\subsubsection*{Double inequalities}
{\label{957982}}
\begin{itemize}
\tightlist
\item
When we have two valid inequalities we can rewrite them as a double
inequality.
\item
Examples
\end{itemize}
\subsection*{Intervals and absolute
values}
{\label{864260}}
\begin{itemize}
\tightlist
\item
The \textbf{open} interval~\(a0\)
\item
Do example 1 p33
\item
If~\(|x|