Problem 32 - Applied Calculus

Question

The following tables give the distribution of family income in the United States: Exercise 31 is for the year 1977 and Exercise 32 is for \(1989 .\) Use the procedure described in the Graphing Calculator Exploration on the previous page to find the Lorenz function of the form \(x^{n}\) for the data. Then find the Gini index. If you do both problems, did family income become more concentrated or less concentrated from 1977 to \(1989 ?\) $$ \begin{array}{cc} \hline \begin{array}{c} \text { Proportion } \\ \text { (Lowest) } \\ \text { of Families } \end{array} & \begin{array}{c} \text { Proportion of } \\ \text { Income (1989) } \end{array} \\ 0.20 & 0.04 \\ 0.40 & 0.14 \\ 0.60 & 0.29 \\ 0.80 & 0.51 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Calculate \( n \) using points, average them, and use \( G = 1 - \frac{2}{1 + n} \) for the Gini index.

Step 1: Understand the data

We have data for proportions of families and their corresponding proportions of income for 1989. The goal is to find the Lorenz function of the form \( L(x) = x^n \), which represents income distribution, and then calculate the Gini index from it.

Step 2: Use Lorenz curve relationship

For a Lorenz function \( L(x) = x^n \), we equate the given points \((0.2, 0.04), (0.4, 0.14), (0.6, 0.29), (0.8, 0.51)\) to find the value of \( n \). Solve these equations to get the best fit for \( n \).

Step 3: Solve for "n"

We assume \( L(0.2) = 0.04 \), resulting in \( 0.2^n = 0.04 \). Solving gives \( n = \frac{\log 0.04}{\log 0.2} \). Calculate similar equations for the other points and average the values of \( n \) obtained for the best approximation.

Step 4: Average out "n"

Calculate \( n \) for all given points and find the average \( n \). This provides a single value of \( n \) that describes the entire Lorenz curve.

Step 5: Calculate Gini index

The Gini index is given by \( G = 1 - \frac{2}{1 + n} \). Substitute the averaged \( n \) into this formula to find the Gini index for 1989.

Key Concepts

Gini Index

The Gini Index is a measure of statistical dispersion intended to represent the income inequality within a nation or a group. It is a number between 0 and 1, where 0 represents perfect equality (everyone has the same income) and 1 represents perfect inequality (where a single individual has all the income).
The Gini Index is calculated using the Lorenz curve, which graphically represents income distribution. In a Lorenz curve, we plot the cumulative percentage of total income received by the bottom x% of the population. The Gini Index is effectively the area between the line of perfect equality (a 45-degree line) and the Lorenz curve, divided by the total area under the line of perfect equality.
This index helps us understand not only how wealth is distributed but also the gap between the rich and the poor. By calculating the Gini index from our Lorenz curve approximation, we get a clearer picture of income concentration in the year of study.

Income Distribution

Income Distribution refers to how a nation's total GDP is spread amongst its population. Analyzing income distribution helps us understand economic inequality and societal wealth variance. It is a crucial concept that influences economic policy decisions.
In our exercise, the income distribution data is presented in proportions, reflecting how much income is controlled by a certain percentage of families. For example, if 20% of families receive only 4% of total income, it suggests high inequality since a small portion of families have access to limited resources.
By using these proportions in conjunction with the Lorenz curve model, we can visualize how fairly or unfairly income is divided. It allows for a more quantitative assessment of whether the income gap is growing or shrinking over specified periods, such as from 1977 to 1989 in this case.

Mathematical Modeling

Mathematical modeling plays a pivotal role in understanding and predicting various complex systems, including economic phenomena like income distribution. In the context of the Lorenz curve, we use mathematical equations to fit a curve to our data points representing how income is distributed among different population groups.
For our exercise, we assume a simple model where the Lorenz function takes the form of a power function, specifically, \( L(x) = x^n \). By solving for \( n \) using logarithmic transformations, we are effectively using a mathematical model to best fit the observed data points.
This model allows us to make structured predictions and draw conclusions about income concentration trends over time. It also serves as a base to compute critical economic indicators like the Gini index, helping us simplify and interpret complex societal issues.

Graphing Calculator Explorations

Graphing calculators provide an interactive way to visualize complex mathematical relationships and statistics. When dealing with income distribution and the Lorenz curve, these tools become exceptionally valuable.
In this context, a graphing calculator can help by plotting data points accurately and allowing us to determine the best-fit line or curve for these points. By inputting our data from 1989, we can explore different values of \( n \) until we find the one that best approximates our curve.
Additionally, calculators can speed up lengthy computations, such as logarithmic transformations, helping us solve equations more efficiently. This exploration not only illustrates the income distribution visually but also enhances our understanding of how theoretical models align with real-world data trends.