Problem 25 - Calculus Single Variable

Question

Income data for three countries are given in the following tables. In each table, \(x\) represents a percentage, and \(L(x)\) is the corresponding value of the Lorenz function, as described in Example \(3 .\) In each of Exercises \(23-27,\) use the specified approximation method to estimate the coefficient of inequality for the indicated country. (The values \(L(0)=0\) and \(L(100)=100\) are not included in the tables, but they should be used.) $$ \begin{array}{|c|r|c|c|c|}\hline x & 20 & 40 & 60 & 80 \\\\\hline L(x) & 5 & 20 & 30 & 55\\\\\hline\end{array}$$ Income Data, Country A $$\begin{array}{|c|c|c|c|}\hline x & 25 & 50 & 75 \\\\\hline L(x) & 15 & 25 & 40 \\\\\hline\end{array}$$ Income Data, Country B $$\begin{array}{|c|r|r|r|r|r|r|r|r|r|}\hline \boldsymbol{x} & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 & 90 \\\\\hline \boldsymbol{L}(\boldsymbol{x}) & 4 & 8 & 14 & 22 & 32 & 42 & 56 & 70 & 82 \\\\\hline\end{array}$$ Income Data, Country \(\mathbf{C}\) Country C, Trapezoidal Rule

Step-by-Step Solution

Answer

The Gini coefficient for Country C is 0.12.

Step 1: Understand the Lorenz Function and Coefficient of Inequality

The Lorenz function, denoted as \( L(x) \), represents the cumulative distribution of income. The coefficient of inequality, Gini coefficient, measures income inequality. It is calculated by finding the area between the line of equality and the Lorenz curve, then dividing by the area under the line of equality (0.5).

Step 2: Prepare the Data for Calculation

For Country C, use \( L(0) = 0 \) and \( L(100) = 100 \) along with given data. The \( x \) values are 0, 10, 20, ..., 100, and the corresponding \( L(x) \) values are 0, 4, 8, ..., 100.

Step 3: Apply the Trapezoidal Rule

The trapezoidal rule approximates the integral of \( L(x) \) over the interval \([0, 1]\). The formula is \[ \int_0^1 L(x) \, dx = \frac{h}{2} \left[f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n)\right] \]. Here, \( f(x_i) = L(x_i)/100 \) and \( h = \frac{1}{10} = 0.1 \).

Step 4: Calculate the Area Under the Lorenz Curve

Using the trapezoidal rule, calculate the area under the Lorenz curve for Country C. \[ \text{Area} = \frac{0.1}{2} [0 + 2(4+8+14+22+32+42+56+70+82) + 100] \] \[ = 0.05 [0 + 2 \times 330 + 100] \] \[ = 0.05 [660 + 100] = 0.05 \times 760 = 38 \].

Step 5: Calculate the Gini Coefficient

The Gini coefficient is calculated as \( G = 0.5 - \text{Area under Lorenz curve} \). Hence, \( G = 0.5 - 0.38 = 0.12 \).

Key Concepts

Lorenz Function

The Lorenz function, represented as \(L(x)\), is a fundamental concept in assessing income distribution. It provides a graphical representation of the proportion of total income received by any given percentage of the population. Imagine lining up everyone in the country from the poorest to the richest. The Lorenz curve helps us see how income is spread across this spectrum.
For a specific percentage of the population \(x\), \(L(x)\) corresponds to the cumulative share of income earned by the lowest \(x\)% of the population. If income were perfectly equally distributed, the Lorenz curve would coincide with the line of equality, which is a 45-degree line where \(L(x) = x\).

• A curve below the line of equality signifies unequal distribution.
• The more bowed the Lorenz curve, the greater the degree of income inequality.

Understanding the Lorenz function is crucial because it lays the groundwork for calculating the Gini Coefficient, an index that quantifies income inequality.

Income Inequality

Income inequality is a measure of how unevenly income is distributed within a population. High income inequality implies that a large portion of a country’s income is in the hands of a small percentage of people. The Gini Coefficient is the most widely used metric to assess this inequality, providing a scalar representation of how far a society's income distribution deviates from perfect equality.

To calculate it, we use the Lorenz curve, as a visual tool to compare observed income distribution against perfect equality. The Gini Coefficient is calculated using the area between the Lorenz curve and the line of equality. In mathematical terms, it is equal to twice the area between these two curves: if \(A\) is the area between the line of equality and the Lorenz curve, and \(B\) is the area under the Lorenz curve, then the Gini Coefficient \(G\) is given by:
\[ G = \frac{A}{A+B} \] This relationship means a Gini Coefficient of 0 indicates perfect equality, where every person has exactly the same income. In contrast, a coefficient of 1 shows extreme inequality, with all income earned by a single individual.

Trapezoidal Rule

The calculation of areas under curves like the Lorenz curve often requires methods of numerical integration, especially when datasets provide only discrete values. The Trapezoidal Rule is a simple yet effective technique for approximating these integrations, particularly when dealing with large datasets.
The core idea is to approximate the region under the curve by dividing it into trapezoids rather than more complex shapes. For each interval between adjacent points, the curve is approximated by a straight line connecting them, forming a trapezoid. The area of each trapezoid is then calculated and summed to estimate the total area under the curve.
For a function \(f(x)\) sampled at points \(x_0, x_1, ..., x_n\) with a constant width \(h\) between these points, the Trapezoidal Rule is expressed mathematically as:
\[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{2} \left( f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right) \]In the context of calculating the Gini Coefficient, we use this rule to determine the area under the Lorenz curve from the discrete income data provided, allowing for an effective assessment of inequality.