# Material for the Course on Inequality

Here is the video that we discussed during the first course,

References for the course are the following,

For similar course lectures notes, see Emmanuel Flachaire’s ECON-473 webpage, as well as Michel Lubrano’s notes.

For the project, some references in French,

and in English

There is also the following report (one can focus only on a chapter of the report)

# Inequality, Welfare and Poverty

The short graduate course on Inequality, Welfare and Poverty will start next week. Slides available are online.

# Inequalities and Poverty #1

This Monday, we will start the graduate course on inequalities and poverty. Slides are below

# Inequalities and Quantile Regression

In the course on inequality measure, we’ve seen how to compute various (standard) inequality indices, based on some sample of incomes (that can be binned, in various categories). On Thursday, we discussed the fact that incomes can be related to different variables (e.g. experience), and that comparing income inequalities between coutries can be biased, if they have very different age structures.

So we’ve seen quantile regressions. I can mention some old slides (used in a crash course at McGill three years ago)., as well as a more technical discussion on ties, and non-unicity of the regression line.

In order to illustrate, consider  the following dataset

> salary <- read.table("http://data.princeton.edu/wws509/datasets/salary.dat",header=TRUE)
> plot(salary$yd,salary$sl)
> abline(lm(sl~yd,data=salary),col="blue")

We have here the stndard regression line, obtained using ordinary least squares. Here we have the expected income given the experience. But we can also use a quantile regression,

$Q_\tau(Y\vert\boldsymbol{X})=\boldsymbol{X}^{\text{\sffamily T}}\boldsymbol{\beta}$

> library(quantreg)
> Q10 <- rq(sl~yd,data=salary,tau=.1)
> Q90 <- rq(sl~yd,data=salary,tau=.9)
> abline(Q10,col="red")
> abline(Q90,col="purple")

A classical tool to describe inequalities is the ratio of the 90% quantile over the 10% quantile (among so many others,

> ratio9010 = function(age){
+   predict(Q90,newdata=data.frame(yd=age))/
+   predict(Q10,newdata=data.frame(yd=age))
+ }

For instance, among people with 5 years of experience, there is an inequality index of

> ratio9010(5)
1.401749

while for people with 30 years of experience, it would be

> ratio9010(30)
1.9488

If we plot the evolution of this 90-10 ratio, as a function of the experience, we get the following increasing trend

> A=0:30
> plot(A,Vectorize(ratio9010)(A),type="l",ylab="90-10 quantile ratio")

So clearly, comparing inequalitis ceteris paribus between two groups, should be performed carefully, and probably including some covariates.

# Visualizing Inequalities in a 3-Person Economy

Yesterday, in the course on inequalities, I mentioned (too) briefly the 3-person Economy. I wanted to spend some time in a short post on visualisations of inequalities in such a context. As mentioned in the slides,  it is possible to use a ternary plot representation. In the case where we believe that the scale independence principle makes sense, i.e. $I(\lambda\boldsymbol{x})=I(\boldsymbol{x})$. A distribution of incomes can be represented as a barycenter in an equilateral triangle (also called de Finetti diagram). The midpoint is the equal situation: the three agents share the same wealth. Because of the scale independence property, we can look at distribution of wealth on the simplex. A wealth distribution is a vector $\boldsymbol{\omega}=(\omega_A,\omega_B,\omega_C)$ where each component is one of the (red) distance below. A is on top of the triangle, and the vertical distance is proportional to the wealth of A. The closer to the bottom line, the poorer A is.

To visualize this distribution of wealth, we can use the trifield package. To add the point and the three segments, the code is

tripoint=function(s){
p=s/sum(s)
p1=c(0,s[2]+s[1]/2,s[3]+s[1]/2)/sum(s)
p2=c(s[1]+s[2]/2,0,s[3]+s[2]/2)/sum(s)
p3=c(s[1]+s[3]/2,s[2]+s[3]/2,0)/sum(s)
C <- abc2xy(matrix(p,1,3))
points(C,pch=19,col="red",cex=2)
C1 <- abc2xy(matrix(p1,1,3))
C2 <- abc2xy(matrix(p2,1,3))
C3 <- abc2xy(matrix(p3,1,3))
segments(C1[1],C1[2],C[1],C[2],lwd=2,col="red")
segments(C2[1],C2[2],C[1],C[2],lwd=2,col="red")
segments(C3[1],C3[2],C[1],C[2],lwd=2,col="red")
}

For instance, to visualize the equal case (inequality indices are defined as a distance to this situation)

tripoint(c(1,1,1))

For a case where there is inequality, use for instance

tripoint(c(1,2,3))

# Inequalities, course 3

Tomorrow, we will discuss inequality indices, from a statistical perspective, and also an axiomatic point of view. In order to illustrate, we will use to following dataset,

> income <- read.csv("http://www.vcharite.univ-mrs.fr/pp/lubrano/cours/fes96.csv",sep=";",header=FALSE)\$V1

Slides can be found online. Since it is the first year I give this course, all comments are welcome…

# Welfare, Inequality and Poverty

This week, we will start the crash course on Welfare, Inequality and Poverty. I will upload the slides soon. Reference for the course are the following,

See also Emmanuel Flachaire’s ECON-473 webpage, as well as Michel Lubrano’s notes. In the introductionary course, I will also mention Le Monde, 2012 (on poverty), with the pdf. An interesting video is based on Norton & Ariely, 2011

# Income distribution and Tour de France

A few days ago, Jean-François Mignot published an interesting article entitled Tour de France 2014 : pourquoi le vainqueur gagne 100 fois plus que le 10e. In this article, we have the following graph, with the income of the cyclist, as a function of his final ranking (the data where downloaded from http://sportbuzzbusiness.fr/)

> bike=read.csv(
+ "http://freakonometrics.free.fr/tourdefrance.csv",
+ xlab="(Final) rank",ylab="Bonus")
> library(ineq)
[1] 0.910461