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\title{Income inequality}
\date{\today}

\begin{document}

\ 

\vspace{-.25in}

\maketitle

\vspace{-.2in}

In 1906 Italian economist Vilfredo Pareto observed that the wealthiest 20\% of Italian land-owners controlled 80\% of the land.
Since Pareto first stated this 80-20 rule, it has been found to apply (approximately) to many different situations: about 80\% of sales come from 20\% of customers, about 80\% of stock market growth comes from 20\% of stocks, about 80\% of health care spending is on 20\% of patients, and so on.
This rule has come to be known as the \href{http://en.wikipedia.org/wiki/Pareto_principle}{\textbf{Pareto principle}} (or the 80-20 rule).

\prob{
Suppose wealth distribution in the country of Extremistan\footnote{the names Extremistan and Mediocristan come from Nassim Nicholas Taleb's book, \textbf{\href{http://www.betterworldbooks.com/product/detail/The-Black-Swan--The-Impact-of-the-Highly-Improbable-9780141034591}{The Black Swan}}} follows the Pareto principle strictly: the richest 20\% have 80\% of the wealth.
Moreover, among the richest 20\% of the population the Pareto principle still applies: the richest 20\% of this group have 80\% of the wealth of this group (that's 80\% of the 80\% of all wealth this group controls).
This means that the richest 4\% ($0.2 \times 0.2$) of all Extremistanis have 64\% ($0.8 \times 0.8$) of all the wealth.
\begin{enumerate}[a) ]
%\item The richest 4\% of Extremistanis have what percent of the wealth?
\item Apply the Pareto principle again to find the percentage of all wealth owned by the richest $0.8\%$.

\vspace{.75in}

\item The Pareto principle also applies to the poorest 80\% of Extremistanis (who have 20\% of all the wealth).  
What percent of the wealth do the poorest 64\% have?

\vspace{.75in}

\end{enumerate}
}

\prob{
In Mediocristan, wealth is normally distributed with a mean of $6$ and a standard deviation of $1$ (units are 10,000 Mediocristani dollars).
The proportion of wealth in the hands of the poorest $100p\%$ of Mediocristanis is then
\begin{equation} \label{lorenz}
L(p) = p-\frac{1}{6}\left( \frac{1}{\sqrt{2\pi}} \right) e^{-\frac{1}{2}[z(p)]^2} 
\end{equation} 
where $z(p)$ is the \pth percentile of the standard normal distribution (\texttt{qnorm(p)} in \textbf{R}).
Note that this means that the proportion of wealth owned by the richest $100(1-p)\%$ is then $1-L(p)$.
Compare the percentage of Mediocristan's wealth controlled by the poorest 80\%, poorest 64\%, richest 20\%, and richest 4\% of Mediocristanis to the wealth of their counterparts in Extremistan.
}


\vfill


\newpage


\dfn{
The \href{http://en.wikipedia.org/wiki/Lorenz_curve}{\textbf{Lorenz curve}} of a wealth distribution is the curve describing the proportion of wealth owned by the poorest 100$p$\% of the population.
If the random variable $X$ is the wealth of a random person in the population, then the Lorenz curve is the graph of \[
L(p) = \frac{ \int_{-\infty}^{x(p)} x f(x) dx }{E(X)}
\]
where $f(x)$ is the PDF of $X$ and $x(p)$ is the \pth percentile of $X$.
The function in $L(p)$ in the previous problem (equation \ref{lorenz}) is the Lorenz curve for Mediocristan.   
}


As Vilfredo Pareto observed, the real world is much more like Extremistan than Mediocristan.
The probability distributions that capture this kind of behavior \href{http://www.randomservices.org/random/special/Pareto.html}{\textbf{Pareto distributions}}.

\dfn{
The PDF of a Pareto distribution with parameter $\alpha>1$ is \[
f(x) = \begin{cases} \alpha x^{-(\alpha + 1)} & \text{if } x > 1 \\ 0 & \text{elsewhere} \end{cases} \]
Note that different values of $\alpha$ give different versions of the Pareto principle:
the 80-20 rule corresponds to a parameter of $\alpha \approx 1.16096$.
}

\prob{
The goal of the next problem is to calculate the Lorenz curve for a Pareto distribution.
To do this, you'll first need to find a formula for $x(p)$, the \pth percentile.
Do this by solving the following equation for $x(p)$: \[ p = \int_1^{x(p)} \alpha x^{-(\alpha + 1)} dx \]
}

\vfill

\newpage

\prob{
Suppose that income in some country follows a Pareto distribution with parameter $\alpha > 1$.
\begin{enumerate}[a)]
\item Find a formula for the Lorenz curve $L(p)$ (this will depend on $\alpha$).
\item Graph the Lorenz curve for $\alpha= 1.1$ and $\alpha = 1.5$ on a single set of axes (I recommend using graphing software such as \href{https://www.desmos.com/calculator}{Desmos}; remember $p$ is a proportion, which means the domain for $L(p)$ is the interval $[0,1]$).
\item Calculate $L(0.8)$ for both values of $\alpha$ and explain what these numbers mean (e.g. if $\alpha=1.1$, the poorest 80\% have \dots).
\end{enumerate}
}

\vfill

\newpage

%\prob{
%Calculate $\lim_{\alpha \to \infty}L(p)$ and $\lim_{\alpha \to 1^+} L(p)$ for the general Lorenz curve you found in problem 3 and explain what these limits mean.
%}
%
%\vfill

\dfn{
The \textbf{\href{https://en.wikipedia.org/wiki/Gini_coefficient}{Gini coefficient}} of a distribution is defined to be $1-2B$ where $B$ is the area under the Lorenz curve: $B = \int_0^1 L(p) dp$.
}

The Pareto distribution has a Gini coefficient that approaches $1$ as $\alpha$ approaches $1$ from the right;
this corresponds to all the wealth in Extremistan being in the hands of just one person (\textbf{maximum inequality}).
The Gini coefficient approaches $0$ as $\alpha$ increases to $\infty$;
this corresponds to a completely equal distribution of wealth (\textbf{minimum inequality}).
People (at places like the \href{http://data.worldbank.org/indicator/SI.POV.GINI}{World Bank} and the \href{https://www.cia.gov/the-world-factbook/field/gini-index-coefficient-distribution-of-family-income/country-comparison}{CIA}) actually care about Gini coefficients and use them as a way of measuring income inequality. 
Wikipedia's list of countries ordered by Gini coefficient is a good summary: \url{http://en.wikipedia.org/wiki/List_of_countries_by_income_equality}.
 
\prob{
Each Gini coefficient corresponds to a different value of the parameter $\alpha$ in the Pareto distribution.
And each value of $\alpha$ corresponds to a different version of the Pareto principle.
For example, the World Bank estimate of the Gini coefficient for US family income in 2016 was $0.411$.
This corresponds to a parameter of $\alpha \approx 1.7165$ and a 65-35 rule (approximately).
%Each of you will be given 2 countries (request your favorites).
%Your job is to look up the CIA estimate of the Gini coefficient for income of both countries and calculate the corresponding Pareto parameters. 
%Finally, find the version of the Pareto principle that holds in each country: is it an 80-20 rule, a 65-35 rule, or something else?
Choose your favorite country (other than the US) and
look up the the Gini coefficient for income for your country.
The calculate the corresponding Pareto parameter $\alpha$ and
Use this to find the proportion of the country's wealth owned by the poorest $80\%$, $L(0.8)$.
}

\chal{
Find the version of the 80-20 rule that holds for your country: is it an 80-20 rule, a 65-35 rule, or something else?
}

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\end{document}