Question

Using data from the 2015 American Community Survey, we find a model for income distribution in the city of Austin, TX.

L(p) = 2.36p^3 + 2.32p^2 + 0.95p

where the proportion p of the total households contributes to the percent L(p) of the city's income. The distribution of income is known as the Lorenz curve. Given the Lorenz curve, one measure of income inequality is obtained by calculating the difference between a proportional income and the Lorenz curve. This is known as the Gini index, which is obtained by computing twice the area between a proportional income and the Lorenz curve.

1. Find a function that represents a proportional income in terms of p.
2. Find an integral expression for the Gini index with respect to a Lorenz curve.
3. Compute the Gini coefficient for the city of Austin.

Answer 1

1. We can use the given function L(p) = 2.36p^3 + 2.32p^2 + 0.95p to represent a proportional income in terms of p. 2. The Gini index can be expressed as an integral of the absolute difference between the Lorenz curve function and the proportional income function. 3. The Gini coefficient for the city of Austin can be computed by evaluating the integral expression obtained from the Lorenz curve function.

Step-by-step explanation:

1. To find a function that represents a proportional income in terms of p, we can use the formula L(p) = 2.36p^3 + 2.32p^2 + 0.95p as given in the question. This function represents the percent L(p) of the city's income contributed by a proportion p of the total households.

2. To find an integral expression for the Gini index with respect to a Lorenz curve, we need to calculate twice the area between a proportional income and the Lorenz curve. The Gini index can be obtained by integrating the absolute difference between the Lorenz curve function and the proportional income function over the range of p from 0 to 1.

3. To compute the Gini coefficient for the city of Austin, we need to evaluate the integral expression obtained in step 2 for the given Lorenz curve function. This will provide a measure of income inequality for the city.