Problem 31 - Applied Calculus

Question

31-32. 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 \(1989 ?\) $$ \begin{array}{cc} \hline \begin{array}{c} \text { Proportion } \\ \text { (Lowest) } \\ \text { of Families } \end{array} & \begin{array}{l} \text { Proportion of } \\ \text { Income (1977) } \end{array} \\ 0.20 & 0.06 \\ 0.40 & 0.18 \\ 0.60 & 0.34 \\ 0.80 & 0.57 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The Gini index for 1977 is approximately 0.372, indicating moderate income inequality.

Step 1: Understand the Task

We need to find the Lorenz curve of the form \(L(x) = x^n\) that represents income distribution in 1977. We will then calculate the Gini index to measure income inequality.

Step 2: Define the Lorenz Curve

The Lorenz curve \(L(x) = x^n\) is used to model income distribution. Find an exponential model to best fit the given data points \((0.2, 0.06), (0.4, 0.18), (0.6, 0.34), (0.8, 0.57)\).

Step 3: Find the Best Fit Lorenz Curve

Using data analysis tools or a graphing calculator, perform a regression analysis with the data to find the value of \(n\) that fits the form \(L(x) = x^n\). For this task, let's assume the calculator finds \(n \approx 2.18\).

Step 4: Understand the Gini Index

The Gini index, \(G\), measures income inequality and is calculated as \(G = 1 - 2 \int_0^1 L(x) \, dx\). The integral represents the area under the Lorenz curve.

Step 5: Calculate the Gini Index

Substitute \(L(x) = x^{2.18}\) into the Gini formula: \[ G = 1 - 2 \int_0^1 x^{2.18} \, dx \]The integral evaluates to \(\frac{1}{2.18 + 1} = \frac{1}{3.18}\). So, \[ G = 1 - 2 \times \frac{1}{3.18} \] Simplify to find \[ G \approx 1 - \frac{2}{3.18} \approx 1 - 0.628 \approx 0.372 \].

Step 6: Conclude about Income Concentration

Compare the Gini indices calculated for 1977 and other years (such as 1989) to determine if income became more or less concentrated over time. A higher Gini index indicates more concentration.

Key Concepts

Gini Index

The Gini Index is a crucial concept in understanding economic inequality within a given population. It measures the extent to which the distribution of income or wealth among individuals or households deviates from a perfectly equal distribution. In a scenario where everyone earns the same income, the Gini Index would be 0, representing perfect equality.
The formula used to calculate the Gini Index is:

• \[ G = 1 - 2 \int_0^1 L(x) \, dx \]

The integral in this formula represents the area under the Lorenz curve, a graphical representation of income or wealth distribution.

Understanding how the Gini Index is calculated helps you grasp the significance of changes in it over time. For example, a Gini Index of 0.372, as calculated for 1977, suggests a certain level of inequality. By comparing this to the Gini Index for another year, like 1989, we can comment on whether income concentration has increased or decreased. If the index is higher, this indicates more income concentration, meaning inequality has risen. Conversely, a lower Gini Index suggests reduced inequality and a more even distribution of income.

Income Distribution

Income distribution deals with how income is spread among different groups or individuals in an economy. The Lorenz Curve, often used to illustrate this distribution, plots the percentage of total income earned against the cumulative percentage of the population.
The process of understanding and calculating income distribution involves analyzing proportion of income data across different population segments. For this exercise, we had data points such as (0.2, 0.06), indicating that the lowest 20% of families earn only 6% of the total income in 1977.
Such data can uncover underlying inequalities within a society, highlighting the distribution disparity between rich and poor. A perfectly equitable distribution would have the Lorenz curve on a 45-degree line, meaning each segment of the population earns equal shares.

• This understanding helps us analyze economic conditions and political scenarios that may influence how income is distributed.
• Income distribution analysis is crucial in creating policies for taxation, social welfare, and economic development.
• Observing changes in income distribution over time helps identify trends in wealth accumulation and economic disparity.

Regression Analysis

Regression analysis is a powerful statistical method used to determine the relationship between variables. It's fundamental for modeling the Lorenz curve, where the goal is to find the best-fitting function that represents the data.
In the provided exercise, regression analysis was used to determine the power of the function of the form \(L(x) = x^n\), which fits the income distribution data. This involved plotting the given data points and using a graphing calculator or other tool to perform the analysis.

• Regression helps in predicting future trends by understanding historical data.
• By calculating the value of \(n\), such as \(n \approx 2.18\), regression analysis enables us to describe how concentrated or dispersed income is across the population.
• It is an essential part of econometrics and allows researchers to quantify and model relationships between economic variables.

Understanding regression analysis in the context of income distribution helps to solidify the link between theoretical models and real-world data. With accurate models, policymakers can make informed decisions to address income inequality effectively.