IC Project 2P

Lorenz and Gini : 

Lorenz and Gini Angel Vera Lepe. A01063075 Omar Jair Casillas Ceron. A01062990 Larry Zanmar Sanchez Castelan. A01062991 Karen Monserrat Rodriguez Solis. A00343622

Who are those and why we care? : 

Who are those and why we care? Max Otto Lorenz was an American economist who developed the Lorenz curve in 1905 to describe income inequalities. He was active in both publishing and teaching and was at various times employed by the U.S. Census Bureau, the U.S. Bureau of Railway Economics, the U.S. Bureau of Statistics and the U.S. Interstate Commerce Commission. Corrado Gini was an Italian statistician demographer and sociologist who developed the Gini coefficient, a measure of the income inequality in a society. Gini was also a leading fascist theorist and ideologue who wrote The Scientific Basis of Fascism in 1927.

Their work… : 

Their work… Every point on the Lorenz curve represents a statement like "the bottom 20% of all households have 10% of the total income.". A perfectly equal income distribution would be one in which every person has the same income. In this case, the bottom "N"% of society would always have "N"% of the income. This can be depicted by the straight line "y" = "x"; called the "line of perfect equality.“ By contrast, a perfectly unequal distribution would be one in which one person has all the income and everyone else has none. In that case, the curve would be at "y" = 0 for all "x" < 100%, and "y" = 100% when "x" = 100%. This curve is called the "line of perfect inequality.“

Slide 4: 

The Gini coefficient is the area between the line of perfect equality and the observed Lorenz curve, as a percentage of the area between the line of perfect equality and the line of perfect inequality. The higher the coefficient, the more unequal the distribution is. All of this is a graphical representation of the cumulative distribution function of the empirical probability distribution of wealth; it is a graph showing the proportion of the distribution assumed by the bottom y% of the values. Many economists consider it to be a measure of social inequality.

How do we do it? : 

How do we do it? For a population of size n, with a sequence of values yi, i = 1 to n, that are indexed in non-decreasing order (yi ≤ yi+1), the Lorenz curve is the continuous piecewise linear function connecting the points (Fi , Li), i = 0 to n, where F0 = 0, L0 = 0, and for i = 1 to n: For a discrete probability function f(y), let yi, i = 1 to n, be the points with non-zero probabilities indexed in increasing order ( yi < yi+1). The Lorenz curve is the continuous piecewise linear function connecting the points ( Fi , Li ), i = 0 to n, where F0 = 0, L0 = 0, and for i = 1 to n:

Slide 6: 

For a probability density function f(x) with the cumulative distribution function F(x), the Lorenz curve L(F(x)) is given by: For a cumulative distribution function F(x) with inverse x(F), the Lorenz curve L(F) is given by: The inverse x(F) may not exist because the cumulative distribution function has jump discontinuities or intervals of constant values. However, the previous formula can still apply by generalizing the definition of x(F): x(F1) = inf {y : F(y) ≥ F1}

Why is this important? : 

Why is this important? The concept is useful in studies of biodiversity, where cumulative proportion of species is plotted against cumulative proportion of individuals. It is commonly used as a measure of inequality of income or wealth. It has, however, also found application in the study of inequalities in disciplines as diverse as health science, ecology, and chemistry.

Part II: Practice : 

Part II: Practice After understanding what a Lorenz Curve and a Gini Coefficient are, we’re going to try doing those two using data of the income distribution for the United States for the year 1978:

Slide 9: 

How do we do this? First we graph the ideal income, which is a straight line expressing that for zero households there’s zero income, for 20 households there’s 20% income, etc. After that we calculate the real income, using this process: 1st – Do the graph.

Slide 10: 

2nd – Calculate the Gini coefficient like this: We know, from what we investigated before, there’s an A-area (the actual Gini coefficient) and a B-area. Now, we realize that the total area of the graph is 1, so a perfect income could be 0.5. That means that the Gini coefficient would never be higer than 0.5. Now, we look for the formula we need to calculate the Gini coefficient.

Slide 11: 

Unfortunately, we found that we have to solve for B first. We could do this using this formula: After doing all of that, our result for B= 0.3132 Now, using that number, we solve for A:

Part III: Analysis for several years : 

Part III: Analysis for several years There are some expressions that may be related to this project and we’d like to analyze them: The rich get richer while the poor get poorer. The first line means that money makes the circumstances provide more opportunities to grow for those who have lots of money because poor people cannot afford things that could help them to improve their economy, for example having a good initial investment for a business-project. While the rich people have money to invest in whatever they want including business projects. We could say a synonym of this phrase is: You need money to make money.

Slide 13: 

A rising tide lifts all boats. Any kind of event, no matter if it is good or bad, will affect everybody. For example if a crisis shows up it will screw everybody. Maybe not in the same way or intensity. Because the poor people will stop eating and the rich people will stop buying unnecessary things and will sell one of their 10 houses but will still eat like everyday.

Slide 14: 

Now we are given the data of the in the United States for 30+ years. Using the same steps that we used before, we made up a table and worked on it:

Slide 15: 

And now. Calculations to get the Gini% for each year 1968 1982

Slide 16: 

1992 2001

Part IV: Analysis for several countries : 

Part IV: Analysis for several countries So now, after 2 parts of calculating Gini coefficients for several years and places. We have to calculate some more Gini coefficients. 2 to be more specific. And so we should think we’re going to get bored of more of the same formulas. But guess what?! WE WON’T!. Because this time the data we have to consider has no equal x.

Challenging table : 

Challenging table

Slide 19: 

Some guys started this project and their calculation of the area under Lorenz curve using left hand sum and right hand sum. But left hand sum and right hand sum can only be used for equal x data tables. Since we didn’t use LHS and RHS to get the area of B and we used our awesome formula… …which considers xo, x1 and so on. We don’t care nor need equal x. So we can still do the same calculations… and yup get bored.

Part V: Comparing each country : 

Part V: Comparing each country

Slide 21: 

Now… talking about the real life, the organizations that collect, analyze and report this information are only two: the United Nations (UN) and the Central Intelligence Agency (CIA). The map you saw before shows us the differences in national income equality around the world by the national Gini coefficient, this number is between 0 and 1, where 0 corresponds with perfect equality and 1 corresponds with perfect inequality. Meaning this, the quality of the income distribution inside domestic territory. Each color in the map represents 1 of the 10 ranges in which the Gini coefficient is classified; so each country has a different color, depending of its level of income equality.

Slide 22: 

What can we see? Mexico is found in a middle position, meaning that our inocome distribution is not as bad as other countries, but still, those countries have less development than Mexico so, we are the last in the developed range.

Conclusions? : 

Conclusions? Apparently the Gini coefficient actually helps for something in real life, and it indeed is important (no matter how boring it turned to be). There are two ways of calculating the Gini coefficient, but we believe that the hard barely understandable one is also the most precise one.

References : 

References Source: U.S. Bureau of Census, Current Population Reports, P-60, No. 121, "Money Income in 1978 of Households in the United States," Washington, D.C.: U.S. Government Printing Office, 1980. Data taken from cover. (Data are before taxes.) http://mercury.soas.ac.uk/users/sm97/teaching_intro_qm_notes_gini_coefficient.htm http://translate.google.com.mx/translate?hl=es&langpair=en%7Ces&u=http://people.hofstra.edu/geotrans/eng/ch4en/meth4en/ch4m1en.html http://en.wikipedia.org/wiki/List_of_countries_by_income_equality http://en.allexperts.com/e/m/ma/max_o._lorenz.htm http://www.britannica.com/EBchecked/topic/233928/Corrado-Gini http://cursos.itesm.mx/webapps/portal/frameset.jsp?tab_id=_2_1&url=%2Fwebapps%2Fblackboard%2Fexecute%2Flauncher%3Ftype%3DCourse%26id%3D_259455_1%26url%3D