Problem 47 - Calculus Single Variable

Question

Several values of the Lorenz function \(L\) have been tabulated (refer to Example 2). Use trapezoidal approximations to estimate the coefficient of inequality that corresponds to the given data. (Note: The tables represent partitions that are not uniform. Also, the data points (0,0) and (100,100) have not been included in the tables but should be used in the calculations.) $$ \begin{array}{|c|r|r|r|r|r|r|} \hline \boldsymbol{x} & 16 & 28 & 51 & 75 & 88 & 97 \\ \hline \boldsymbol{L}(\boldsymbol{x}) & 3 & 8 & 24 & 46 & 69 & 88 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The coefficient of inequality is approximately 0.3932.

Step 1: Add Boundary Points

First, add the boundary points (0, 0) and (100, 100) to the given table. This is because the Lorenz curve starts at the origin and ends at (100, 100). So the updated table is: \(x = [0, 16, 28, 51, 75, 88, 97, 100]\) and \(L(x) = [0, 3, 8, 24, 46, 69, 88, 100]\).

Step 2: Understand Trapezoidal Rule

The trapezoidal rule approximates the area under a curve by dividing it into trapezoids. The area of each trapezoid is calculated using \(\text{Area} = \frac{1}{2}(b-a)(f(b)+f(a))\) for two points \((a, f(a))\) and \((b, f(b))\).

Step 3: Calculate Each Trapezoid Area

Calculate the area of each trapezoid formed with consecutive points in the table:\[\begin{align*}\text{Area}_1 &= \frac{1}{2}(16-0)(0 + 3) = \frac{1}{2}(16)(3) = 24,\\text{Area}_2 &= \frac{1}{2}(28-16)(3 + 8) = \frac{1}{2}(12)(11) = 66,\\text{Area}_3 &= \frac{1}{2}(51-28)(8 + 24) = \frac{1}{2}(23)(32) = 368,\\text{Area}_4 &= \frac{1}{2}(75-51)(24 + 46) = \frac{1}{2}(24)(70) = 840,\\text{Area}_5 &= \frac{1}{2}(88-75)(46 + 69) = \frac{1}{2}(13)(115) = 747.5,\\text{Area}_6 &= \frac{1}{2}(97-88)(69 + 88) = \frac{1}{2}(9)(157) = 706.5,\\text{Area}_7 &= \frac{1}{2}(100-97)(88 + 100) = \frac{1}{2}(3)(188) = 282.\end{align*}\]

Step 4: Sum of Trapezoid Areas

Sum up all the calculated trapezoid areas to get the total area under the Lorenz curve:\[\text{Total Area} = 24 + 66 + 368 + 840 + 747.5 + 706.5 + 282 = 3034.\]

Step 5: Calculate Total Possible Area

The total possible area under the line of equality (straight line from (0,0) to (100,100)) is calculated as the area of a triangle: \(\frac{1}{2}\times \text{Base}\times \text{Height} = \frac{1}{2}\times 100\times 100 = 5000.\)

Step 6: Calculate Coefficient of Inequality

The coefficient of inequality (Gini Coefficient) is calculated as follows: \[\text{Inequality Coefficient} = 1 - \frac{\text{Total Area under Lorenz Curve}}{\text{Total Possible Area}} = 1 - \frac{3034}{5000} = 1 - 0.6068 = 0.3932.\]

Key Concepts

Lorenz curve

The Lorenz curve is a graphical representation used to illustrate the distribution of wealth or income within a population. It is useful for showing inequality and highlighting the disparity between different segments of a society. The curve is shaped by plotting the cumulative percentage of total income earned (vertical axis) against the cumulative percentage of the population (horizontal axis).

• The line of equality, a 45-degree line, represents perfect equality where everyone has the same income.
• The more the Lorenz curve bends away from this line, the greater the inequality.

It's important to know that while the Lorenz curve provides a visual cue to inequality, its interpretation requires additional mathematical measurements for precise assessments.

Gini coefficient

The Gini coefficient is a measure derived from the Lorenz curve, quantifying inequality. It is expressed as a number between 0 and 1, where 0 represents perfect equality, and 1 represents perfect inequality. The Gini coefficient is calculated by examining the area between the line of equality and the Lorenz curve, compared to the total area beneath the line of equality.

• If the Gini coefficient is low, the distribution of income is relatively equal.
• A high Gini coefficient denotes significant inequality.
• It is a widely used metric because it condenses complex income distribution into a single value.

This coefficient helps policymakers and analysts assess economic conditions and understand how wealth is distributed within a society.

inequality measurement

Measuring inequality is crucial in addressing economic disparities and devising effective policies. Different methods, such as the Lorenz curve and Gini coefficient, provide insights into inequality levels within a society. Each of these methods offers unique advantages for understanding the distribution of resources:

• The Lorenz curve provides a visual representation, making it easy to comprehend disparities quickly.
• The Gini coefficient quantifies these disparities into a single number for simpler comparison across regions or time periods.

While these tools are valuable, they are not exhaustive. Comprehensive inequality analysis may involve additional metrics such as the Hoover index or Theil index, tailored to specific investigative needs.

numerical integration

Numerical integration is a mathematical tool used to estimate the area under a curve when an exact integral is difficult or impossible to calculate analytically. In contexts like measuring areas described by Lorenz curves, methods such as the trapezoidal rule are employed.

• The trapezoidal rule relies on breaking down the curve into trapezoids, whose areas are easier to calculate.
• This approach involves summing the areas of each trapezoid, providing an approximation of the total area under the curve.
• It is particularly useful when data points are not uniformly distributed, which is a common occurrence in real-world data.

By accurately estimating these areas, numerical integration helps in computing valuable metrics like the Gini coefficient, giving critical insights into economic inequality.