Problem 29 - Applied Calculus

Question

Find the Gini index for the given Lorenz curve. $$ L(x)=\frac{x+x^{2}+x^{3}}{3} $$

Step-by-Step Solution

Answer

The Gini index is \( \frac{5}{18} \).

Step 1: Understanding the Gini Index Formula

The Gini index (or Gini coefficient) is a measure of inequality, derived using the Lorenz curve. The Gini index is calculated with the integral formula: \[ G = 1 - 2 imes ext{Area under the Lorenz curve} \] The goal is to integrate the Lorenz curve \( L(x) \) over the interval [0, 1].

Step 2: Set Up the Integral for the Area Under the Curve

To find the area under the Lorenz curve, we will compute: \[ ext{Area} = \int_0^1 L(x) \, dx \] Substitute the given function into the integral: \[ \int_0^1 \frac{x + x^2 + x^3}{3} \, dx \]

Step 3: Solve the Integral

Calculate the integral: \[ \int_0^1 \frac{x + x^2 + x^3}{3} \, dx = \frac{1}{3} \int_0^1 (x + x^2 + x^3) \, dx \]Split the integral: \[ = \frac{1}{3} \left( \int_0^1 x \, dx + \int_0^1 x^2 \, dx + \int_0^1 x^3 \, dx \right) \]Compute each part: \[ \int_0^1 x \, dx = \left[ \frac{x^2}{2} \right]_0^1 = \frac{1}{2} \]\[ \int_0^1 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^1 = \frac{1}{3} \]\[ \int_0^1 x^3 \, dx = \left[ \frac{x^4}{4} \right]_0^1 = \frac{1}{4} \]

Step 4: Combine Integration Results

Combine the results of the split integral and apply the \( \frac{1}{3} \): \[ \frac{1}{3} \left( \frac{1}{2} + \frac{1}{3} + \frac{1}{4} \right) \] Calculate inside the parentheses first:\[ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{6}{12} + \frac{4}{12} + \frac{3}{12} = \frac{13}{12} \]Then multiply by \( \frac{1}{3} \): \[ \frac{1}{3} \times \frac{13}{12} = \frac{13}{36} \]

Step 5: Calculate the Gini Index

With the area under the Lorenz curve known, compute the Gini index using: \[ G = 1 - 2 \times \frac{13}{36} \]Perform the multiplication and subtraction: \[ 1 - \frac{26}{36} = 1 - \frac{13}{18} = \frac{5}{18} \]

Step 6: Conclusion

The Gini index, which is a measure of inequality based on the given Lorenz curve, comes out to be \( \frac{5}{18} \). This represents the level of inequality depicted by the curve.

Key Concepts

Lorenz Curve

The Lorenz curve is a graphical representation often used to illustrate income or wealth distribution within a given population. It is a fundamental tool for understanding economic inequality. The x-axis of a Lorenz curve represents the cumulative share of the population, while the y-axis represents the cumulative share of income or wealth.
This curve is particularly valuable because it visually demonstrates the degree of inequality. To interpret the curve:

• Perfect equality would be shown by a 45-degree line (also known as the line of equality), where each segment of the population earns an equal portion of the total income.
• Any deviation below this line indicates the presence of inequality, with the curvature's distance from the line indicating the inequality's magnitude.

Understanding and interpreting Lorenz curves is crucial for economists and policymakers aiming to assess economic conditions and plan interventions.

Inequality Measurement

Measuring inequality involves using specific tools and indices like the Gini index. The Gini index is derived from the Lorenz curve and quantifies inequality, offering a single value that summarizes income distribution differences.
The Gini index ranges between 0 and 1:

• An index of 0 indicates perfect equality, where all individuals or households receive exactly the same income.
• An index of 1 or 100% represents perfect inequality, where one individual or household has all the income, and others have none.

The calculated Gini index for the given Lorenz curve \( L(x) = \frac{x + x^2 + x^3}{3} \) signifies the inequality represented by the curve. This measure helps compare the economic inequality in different populations, facilitating better insights into wealth and income disparities.

Integral Calculus

Integral calculus plays a pivotal role in calculating areas under curves, such as the Lorenz curve, essential for deriving the Gini index. The integral calculates the cumulative portion of income illustrating how wealth or income is distributed across a population. The process of finding the area under a curve involves:

• Setting up the definite integral over the desired interval, in this case, from 0 to 1.
• Performing the integration operation for each term within the Lorenz function.
• Combining and calculating the results to find the whole area under the Lorenz curve.

For the Lorenz function \( L(x) = \frac{x + x^2 + x^3}{3} \),integral calculus was used to partition and evaluate each mathematical component, leading to an accurate Gini index calculation. Understanding these mathematical operations provides a deeper comprehension of economic inequality analysis.