Extended Technology Application Business: Distribution of Wealth Goto pagee Lorenz Functions and the m run a - t The distribution of wealth within a population is of great interest to many economists and sociologists. Let y = f(x) represent the percentage of wealth owned by x percent of the population, with x and y expressed as decimals between 0 and 1. The assumptions are that 0% of the population owns 0% of the wealth and that 100% of the population owns 100% of the wealth. With these requirements in place, the Lorenz function is defined to be any continuous, increasing and concave upward function connecting the points (0,0) and (1, 1), which represent the two extremes. The function is named for economist Max Otto Lorenz (18801962), who developed these concepts as a graduate student in 19051906. If the collective wealth of a society is equitably distributed among its population, we would observe that y (1, 1) y ,3 Line of 0 g equality Elf: v: u t: u E E 3 83 f(x) (0,0) x (0.0) x Percentage Percentage of population of population (1, 1) \"x% of the population owns 36% of the wealth,\" and this is modeled by the function f (x) = x, where 0 S x E 1. This is an example of a Lorenz function that is often called the line of equality. In many societies, the distribution of wealth is not equitable. For example, the Lorenz function f(x) = x3 would represent a society in which a large percentage of the population owns a small percentage of the wealth. For example, in this society, we observe that f(0.7) = 0.7"3 = 0.343, meaning that 70% of the population owns just 34.3% of the wealth, with the implication that the other 30% owns the remaining 65.7% of the wealth. In the graphs below, we see the line of equality in the left-most graph, and increasingly inequitable distributions as we move to the right. y (1. 1) y (1, 1) Percentage of wealth Percentage of wealth x) I (x) n .0 o v n .0 o v x x Percentage of population Percentage of population Most equitable distribution Less equitable distribution Note that the area between the line of equality and the 2. Suppose the Lorenz function for a country is given graph of the Lorenz function f(x) is small if the distribu- by f ( x) = x5, 0sx=1. tion of wealth is close to equitable and is large when the a) What percentage of the wealth is owned by 60% distribution is very unequitable. The Gini coefficient of the population? named for the Italian statistician and demographer b) Calculate the Gini index. Corrado Gini, 1884-1965) is a measure of the difference between the actual distribution of wealth in a society and Regression for Determining Lorenz Functions the ideal distribution represented by the line of equality. If data exist on the distribution of wealth in a society, a It is the ratio of the area between the line of equality and Lorenz function can be determined using regression. the graph of the Lorenz function to the area below the line of equality and above the x-axis. In the figure that EXERCISES follows, the Gini coefficient is represented by the formula 3. The data in the table show the amount of wealth dis- tributed within a population. Gini coefficient = A +B 1, 1) X 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Line of 0.0178 0.06 0.122 0.201 0.297 0.409 0.536 0.677 0.833 equality of wealth Percentag -Lorenz function, f(x) B (0, 0 Percentage of population The area A is found by calculating the area between two curves, Jo (x - f(x)) dx, where x is the line of equality and f(x) is the Lorenz function for a particular society. We observe that A + B is a triangle with area 2(1)(1) = 2. Thus, the Gini coefficient can be written as an integral: ( x - f( x ) ) do Gini coefficient = A A + B - 2 60 ( x - s( * ) ) dox . For the most equitable distribution of wealth, the Gini coefficient would be 0, since there would be no dif- ference (area) between the graph of the Lorenz function Use regression to determine a power function that and the line of equality; for the most inequitable distribu- best fits these data. (Note: Entering the point (0, 0) tion of wealth, the Gini coefficient would be 1. Often, the may cause an error message to appear. However, the Gini coefficient is multiplied by 100 to give the Gini point (1, 1) should be entered along with the rest of index: a Gini coefficient of 0.34 gives a Gini index of 34. the data.) a) Express the Lorenz function in the form EXERCISES f(x) = x". The coefficient should be 1, so you may have to do some rounding 1. Suppose the Lorenz function for a country is given b) Determine the Gini coefficient and the Gini by f ( x) = x, 0