Problem 23 - 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 A, Trapezoidal Rule

Step-by-Step Solution

Answer

The Gini coefficient for Country A is 0.36.

Step 1: Identify the Data Points for Country A

List the given data points and the missing data points for Country A. The data points are: For \(x = 0, L(x) = 0\), \(x = 20, L(x) = 5\), \(x = 40, L(x) = 20\), \(x = 60, L(x) = 30\), \(x = 80, L(x) = 55\), and \(x = 100, L(x) = 100\). These points will be used to approximate the area under the Lorenz curve using the Trapezoidal Rule.

Step 2: Understand the Trapezoidal Rule

The Trapezoidal Rule is used to estimate the area under a curve by dividing it into trapezoids. The formula for the Trapezoidal Rule is given by: \[\text{Area} = \frac{h}{2} \left( f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n) \right)\]where \(h\) is the width of each segment along the \(x\)-axis (here \(h = 20\)) and \(f(x_i)\) are the function values at the endpoints and in between.

Step 3: Apply the Trapezoidal Rule

Apply the Trapezoidal Rule to calculate the area under the Lorenz curve for Country A. Use the given data:\[h = 20\]\[\text{Area} = \frac{20}{2} \left( 0 + 2(5 + 20 + 30 + 55) + 100 \right)\]\[= 10 \times (0 + 220 + 100) = 10 \times 320 = 3200\]The total area under a perfect equality line (identity line) is 5000 (computed as the triangular area \(0.5 \times 100 \times 100\)).

Step 4: Calculate the Coefficient of Inequality for Country A

The coefficient of inequality, often represented as the Gini coefficient, can be calculated with the formula: \[G = \frac{\text{Area of the inequality}}{\text{Area of the whole} (0.5 \times 100 \times 100)} = \frac{5000 - 3200}{5000}\]\[G = \frac{1800}{5000} = 0.36\]

Step 5: Conclusion

The Gini coefficient for Country A, calculated using the Trapezoidal Rule as an approximation method, is 0.36. This number represents the inequality distribution of income in Country A, where 0 indicates perfect equality and 1 indicates maximum inequality.

Key Concepts

Lorenz Curve

The Lorenz Curve is a graphical representation used to show the distribution of a quantity, such as income or wealth, across a population. It is an essential concept for understanding economic inequality. In the Lorenz Curve, the x-axis typically represents the cumulative percentage of households, arranged from the poorest to the richest, while the y-axis represents the cumulative percentage of the total income or wealth that these households possess. In a perfectly equal distribution, the curve is a straight diagonal line known as the line of equality.
The further the actual Lorenz curve is from this line, the greater the inequality. To draw a Lorenz Curve, one plots the cumulative income share against the cumulative population share using income data collected for different population quantiles. This graphical tool is vital for economists to visualize and quantify inequality within an economy, enabling a comprehensive assessment of the society's income distribution.

Trapezoidal Rule

The Trapezoidal Rule is an approximation method for calculating the area under a curve, which in the context of the Lorenz Curve, helps in estimating the total income shared among the different population groups. When plotting the Lorenz Curve, one needs to determine the total area below it to calculate economic inequality, such as the Gini coefficient.
The Trapezoidal Rule simplifies this task by dividing the area into a series of trapezoids rather than relying on complex calculations of irregular shapes. The formula for this method is:

• Divide the area under the curve into sections (trapezoids) with equal bases.
• Apply the formula \( ext{Area} = \frac{h}{2} \left( f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n) \right) \), \( h \) being the width of the segments and \( f(x) \) representing the function values.

Using the Trapezoidal Rule allows for a practical and accurate approximation method for economists to compute areas beneath curves such as the Lorenz Curve, which reflects the distribution of income in a society. This step is crucial in assessing economic data and determining measures of inequality.

Coefficient of Inequality

The Coefficient of Inequality, often known as the Gini Coefficient, is a statistical measure used to gauge the extent of inequality of a distribution, primarily income or wealth. It is derived from the Lorenz Curve and calculated as the ratio of the area that lies between the line of equality (the perfect equality line) and the Lorenz Curve over the total area under the line of equality.
The formula is: \( G = \frac{A}{A + B} \), where \( A \) is the area between the Lorenz Curve and the line of equality, and \( B \) is the area under the Lorenz Curve. The coefficient ranges from 0 to 1, where:

• 0 indicates perfect equality (everyone has the same income).
• 1 indicates maximum inequality (one person has all the income).

Understanding the Coefficient of Inequality provides insight into the income distribution dynamics in an economy, enabling policymakers and researchers to evaluate how income is shared and to develop strategies to reduce economic disparities.

Income Distribution

Income distribution refers to how the nation’s total income is divided amongst its population. It is a crucial factor in special analyses concerning economic health, social equity, and policy-making. The disparities in income distribution can signal varying degrees of economic inequality, which is often measured using tools like the Lorenz Curve and Gini Coefficient.
An equitable distribution suggests most individuals receive relatively similar shares of the total income, fostering societal stability and economic development. Contrastingly, wide disparities often lead to discussions on social justice, welfare policy, and economic reforms. Various influences on income distribution include:

• Education and qualifications.
• Job opportunities and market demand.
• Government policies and taxation.

By assessing income distribution, experts can determine an economy's social structure, identify groups that may require support, and implement targeted measures to promote fairness and equality within the country.