A Lorenz curve is an econometric tool that models the distribution of wealth, income, or other resources. If the members of a population are ordered from least wealth to greatest wealth, then each point (x, y) on the Lorenz curve signifies that the bottom x% of the population holds y% of the total wealth. In a population where everyone's wealth is equal, the bottom x% own exactly x% of the total. The resulting Lorenz curve for this scenario is a straight line, known as the line of total equality. When the distribution or resources is unequal, y is less than x. The area of the gap between the line of total equality and the Lorenz curve the basis for the Gini coefficient of inequality. A Lorenz curve L(x) is defined on the interval [0, 1] and satisfies the following properties: L(x) is continuous on [0, 1] L(0) = 0 and L(1) = 1 (The bottom 0% own 0% of the wealth, and the bottom 100% own 100% of the wealth.) L(x) is increasing on [0, 1] L'(x) is non-decreasing on [0, 1] (A Lorenz curve cannot be concave down on a subinterval of [0, 1], however, it can be linear in places. For example, the Lorenz curve derived from discrete data is a piecewise linear function.) If a population is large enough, its Lorenz curve can be approximated by a smooth function such as a polynomial or exponential equation, rather than a piecewise linear function. Here are some examples of Lorenz curve equations: For instance, consider the polynomial Lorenz curve L(x) = 0.050975x + 0.949025x. This particular equation satisfies Pareto's 80/20 rule, i.e., the top 20% are responsible for 80% of the wealth, and the bottom 80% are responsible for 20%. You can verifiy this by evaluating L(0.8), which equals 0.2. 1. a) Show that the Gini index G is twice the area between the Lorenz curve and the line y=x, i.e. 1 = 2 [ ()] 0 b) What is the value of G for a perfectly egalitarian society (everybody has the same income)? What is the value of G for a perfectly totalitarian society (a single person receives all the income?) 2. The following table (derived from data supplied by the US Census Bureau) shows values of the Lorenz function for income distribution in the United States for the year 2008 x L(x) 0.0 0.000 0.2 0.034 0.4 0.120 0.6 0.267 0.8 0.500 1.0 1.000 a) What percentage of the total US income was received by the richest 20% of the population in 2008? b) Use excel to fit a quadratic function to the data in the table. Graph the data points and the quadratic function. Is the quadratic model a reasonable fit? c) Use the quadratic model for the Lorenz function to estimate the Gini index for the United States in 2008 3. a) Using the US Census Bureau, create similar tables for the years 1970, 1980, 1990, 2000, 2010, 2015 b) Use quadratic models to compute Gini indexes for the years. c) Use a power model ( = ) to compute Gini indexes for the years