Problem 27 - 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 Simpson's Rule

Step-by-Step Solution

Answer

The estimated Gini coefficient for Country C using Simpson's Rule is approximately 24.27.

Step 1: Understanding Simpson's Rule

Simpson's Rule is a numerical method used to approximate the integral of a function. It is particularly useful for estimating the area under a curve given discrete data points. For Country C, where there are 9 subintervals (between every pair of points in the table), we divide the overall interval \(0 \, \text{to} \, 100\) into 5 subintervals of 20 each. This is necessary to apply Simpson's Rule, which requires an even number of segments.

Step 2: Review Data for Country C

We use the provided data:\[x: 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100\L(x): 0, 4, 8, 14, 22, 32, 42, 56, 70, 82, 100\]Ensure the endpoints are included as boundary conditions.

Step 3: Using Simpson's Rule Formula

Simpson's Rule formula for the interval from \(0 \, \text{to} \, 100\) with \(n = 10\): \[\frac{h}{3} [L(x_0) + 4L(x_1) + 2L(x_2) + 4L(x_3) + \cdots + 4L(x_9) + L(x_{10})]\]Here, \ h = 10 \, (\text{span between data points})\.

Step 4: Applying Coefficients and Summing

Substitute the Lorenz values into the formula:\[\frac{10}{3} [0 + 4(4) + 2(8) + 4(14) + 2(22) + 4(32) + 2(42) + 4(56) + 2(70) + 4(82) + 100]\]This involves calculating contributions from each Lorenz value using the weights in Simpson's Rule.

Step 5: Calculate Integral Value

Calculate each term: - Boundary terms: \(0 + 100 = 100\)- Terms with \(4\): \(4(4 + 14 + 32 + 56 + 82) = 4 \times 188 = 752\)- Terms with \(2\): \(2(8 + 22 + 42 + 70) = 2 \times 142 = 284\)- Sum them: \(100 + 752 + 284 = 1136\)\[\frac{10}{3} \times 1136 = 3786.6667\]

Step 6: Calculate Gini Coefficient

The Gini coefficient is calculated as follows: \[\text{Gini} = \,1 - \, \frac{1}{50} \times 3786.6667\]\[= 1 - 75.7333 = 24.267\]This value is then normalized by 100 to find the average distance from equality (area between diagonal and Lorenz curve).

Key Concepts

Simpson's Rule

Simpson's Rule is a numerical integration technique used to estimate the area under a curve. This method is particularly beneficial when dealing with discrete data points, as it provides a more accurate approximation than other simpler techniques, such as the Trapezoidal Rule. The beauty of Simpson's Rule lies in its ability to fit a series of parabolas through sections of the curve, resulting in a superior approximation.
To apply Simpson's Rule, it's crucial to have an even number of subintervals. This was addressed in the solution by dividing the overall interval for Country C (0 to 100) into five equal subintervals of 20. This requirement ensures that the process fits the parabolas correctly across the entire span, ultimately improving the estimation of the integral. The formula \[ \frac{h}{3} [L(x_0) + 4L(x_1) + 2L(x_2) + 4L(x_3) + \cdots + 4L(x_9) + L(x_{10})] \] is usedto calculate the integral, where \( h \) is the span between data points (in this case, 10 units). This application involves multiplying certain Lorenz values by 4, others by 2, all depending on their position in the subinterval sequence. This balancing act is key to obtaining a reliable approximation of the area under the curve.

Lorenz Function

The Lorenz function is a vital concept for understanding income distribution within a given population. It graphically represents the relationship between the cumulative percentage of income earned and the cumulative percentage of the population.
This function is instrumental in visualizing income inequality, with the curve itself illustrating how far a society is from perfect equality.
In the problem at hand, the Lorenz Functionis given as a set of values \( L(x) \) which pertain to certain percentage points \( x \). For Country C, these values serve as important inputs into the process of numerical integration via Simpson's Rule.

• When the Lorenz curve lies beneath the perfect equality line (which runs at 45 degrees from origin to maximum), it indicates inequality – typically in income distribution.
• The further away the Lorenz curve is from this perfect diagonal line, the greater the level of inequality represented.

Understanding the Lorenz Function is crucial when analyzing data on income distribution, and it serves as the foundational base for calculating the Gini coefficient – a numerical representation of inequality.

Numerical Integration

Numerical integration is a mathematical technique used to estimate the integral of a function when an analytical solution is difficult or impossible to determine. In practice, this process involves calculating the area under a curve, where the curve is defined by a set of data points instead of a continuous equation.
In the context of the exercise, numerical integration offers a way to evaluate the area under a Lorenz curve. Various methods exist for performing numerical integration, and Simpson's Rule is chosen in this case due to its improved accuracy for polynomial-like curves. This technique involves using a predefined procedure to approximate the integral by summing up areas of simpler shapes (like trapezoids or parabolas) that approximate the curve's segments. It balances approximation complexity with the practical need for precision.

• For Country C's data, Simpson's Rule is applied, transforming the given discrete Lorenz data points into a coherent approximation of the whole curve's area.
• Numerical integration is crucial here because the integral of the Lorenz function relates directly to calculating the Gini coefficient, a widely-used measure of inequality.

Income Inequality

Income inequality is a measure of how unevenly income is distributed within a population. Societies strive for equality where everyone receives an equal share, but real-world scenarios often reveal disparities due to various socioeconomic factors. The Gini coefficient is a fundamental concept used to quantify these disparities. It derives from the integration of the Lorenz function, translating the graphical representation of income distribution into a single, interpretable value. The coefficient ranges from 0 to 1, where 0 signifies perfect equality (everyone has the same income), and 1 indicates maximum inequality (one person has all the income, and everyone else has none).
In practice, the Gini coefficient provides insight into the severity of income inequality in a country. For example, a higher Gini coefficient suggests a wider income gap among its citizens. In this exercise, the computation of the Gini coefficient for Country C offers a quantitative analysis of its income distribution, highlighting how numerical integration of the Lorenz function's values reflects the reality of economic disparities.

• An understanding of income inequality allows policymakers to address socio-economic issues more effectively.
• Measuring and interpreting the Gini coefficient helps in making informed decisions when designing economic policies aimed at improving equity.