Problem 61 - Calculus Early Transcendentals

Question

A Lorenz curve is given by \(y=L(x),\) where \(0 \leq x \leq 1\) represents the lowest fraction of the population of a society in terms of wealth and \(0 \leq y \leq 1\) represents the fraction of the total wealth that is owned by that fraction of the society. For example, the Lorenz curve in the figure shows that \(L(0.5)=0.2,\) which means that the lowest \(0.5(50 \%)\) of the society owns \(0.2(20 \%)\) of the wealth. a. A Lorenz curve \(y=L(x)\) is accompanied by the line \(y=x\) called the line of perfect equality. Explain why this line is given this name. b. Explain why a Lorenz curve satisfies the conditions \(L(0)=0, L(1)=1,\) and \(L^{\prime}(x) \geq 0\) on [0,1] c. Graph the Lorenz curves \(L(x)=x^{p}\) corresponding to \(p=1.1,1.5,2,3,4 .\) Which value of \(p\) corresponds to the most equitable distribution of wealth (closest to the line of perfect equality)? Which value of \(p\) corresponds to the least equitable distribution of wealth? Explain. d. The information in the Lorenz curve is often summarized in a single measure called the Gini index, which is defined as follows. Let \(A\) be the area of the region between \(y=x\) and \(y=L(x)\) (see figure) and let \(B\) be the area of the region between \(y=L(x)\) and the \(x\) -axis. Then the Gini index is $$G=\frac{A}{A+B} . \text { Show that } G=2 A=1-2 \int_{0}^{1} L(x) d x$$ e. Compute the Gini index for the cases \(L(x)=x^{p}\) and \(p=1.1,1.5,2,3,4\) f. What is the smallest interval \([a, b]\) on which values of the Gini index lie for \(L(x)=x^{p}\) with \(p \geq 1 ?\) Which endpoints of \([a, b]\) correspond to the least and most equitable distribution of wealth? g. Consider the Lorenz curve described by \(L(x)=5 x^{2} / 6+x / 6\) Show that it satisfies the conditions \(L(0)=0, L(1)=1,\) and \(L^{\prime}(x) \geq 0\) on \([0,1] .\) Find the Gini index for this function.

Step-by-Step Solution

Answer

Question: Explain why the line of perfect equality has the name and provide examples of why a Lorenz curve satisfies the given conditions. Answer: The line of perfect equality (\(y = x\)) is so named because it represents a completely equal distribution of wealth, i.e., every individual in the society has the same amount of wealth. If \(x\) is the fraction of the population, then \(y = x\) means that the same fraction of the population owns the same fraction of the total wealth. For example, if 20% of the population (\(x=0.2\)) owns 20% of the total wealth (\(y=0.2\)), we have a perfect equal distribution. A Lorenz curve satisfies the conditions \(L(0)=0\), \(L(1)=1\), and \(L^{\prime}(x)\geq 0\) on \([0,1]\) due to the following reasons: 1. \(L(0) = 0\): When the fraction of the population is 0, i.e., no people, there's no wealth owned by that fraction. 2. \(L(1) = 1\): When the entire population is considered (100% of the population, \(x = 1\)), they own 100% of the total wealth (\(y = 1\)). 3. \(L^{\prime}(x) \geq 0\): The derivative being greater than or equal to 0 means that the Lorenz curve is always increasing. This property reflects the fact that as we consider larger portions of the society, their total wealth will not decrease.

Step 1: a. Explain why the line of perfect equality has the name

The line \(y = x\) is called the line of perfect equality because it represents a completely equal distribution of wealth, i.e., every individual in the society has the same amount of wealth. If \(x\) is the fraction of the population, then \(y = x\) means that the same fraction of the population owns the same fraction of the total wealth. For example, if 20% of the population (\(x=0.2\)) owns 20% of the total wealth (\(y=0.2\)), we have a perfect equal distribution.

Step 2: b. Explain why a Lorenz curve satisfies the conditions

The conditions \(L(0)=0\), \(L(1)=1\), and \(L^{\prime}(x)\geq 0\) on \([0,1]\) hold due to the following reasons: 1. \(L(0) = 0\): When the fraction of the population is 0, i.e., no people, there's no wealth owned by that fraction. 2. \(L(1) = 1\): When the entire population is considered (100% of the population, \(x = 1\)), they own 100% of the total wealth (\(y = 1\)). 3. \(L^{\prime}(x) \geq 0\): The derivative being greater than or equal to 0 means that the Lorenz curve is always increasing. This property reflects the fact that as we consider larger portions of the society, their total wealth will not decrease.

Step 3: c. Graph the Lorenz curves

To graph the Lorenz curves for \(L(x) = x^p\) with \(p = 1.1,1.5,2,3,4\), simply plot the function for each value of \(p\) on the domain \([0,1]\). You should notice that as the value of \(p\) increases, the curve gets farther away from the line of perfect equality (\(y=x\)). Therefore, of the given values of \(p\), the most equitable distribution of wealth corresponds to \(p=1.1\). The least equitable distribution of wealth corresponds to \(p=4\).

Step 4: d. Gini index formula derivation

The Gini index is given by the formula \(G=\frac{A}{A+B}\), where \(A\) and \(B\) are the areas between the Lorenz curve and the diagonal (line of perfect equality) and between the Lorenz curve and the x-axis, respectively. If we combine these areas, we get the total area under the diagonal, which is a right triangle of base 1 and height 1. Thus, we can write \(A+B=\frac{1}{2}\). Then, to express the Gini index in terms of the integral of the Lorenz curve, we rewrite our equation as \(A = \frac{1}{2} - B\). We can find the area \(B\) by integrating the Lorenz curve over the interval \([0,1]\). Therefore, \(B = \int_0^1 L(x) dx\). Substituting this expression into the equation for \(A\), we get \(A = \frac{1}{2} - \int_0^1 L(x) dx\). Now, substituting the expression for \(A\) into the Gini index formula, we find that \(G = 1 - 2 \int_0^1 L(x) dx\).

Step 5: e. Compute the Gini index for different values of \(p\)

We use the derived formula for the Gini index: \(G = 1 - 2\int_0^1 L(x) dx = 1 - 2\int_0^1 x^p dx\). To compute the Gini index for each value of \(p\), integrate the Lorenz curve for \(p=1.1, 1.5, 2, 3, 4\) on the interval \([0,1]\) and then compute \(G\). 1. For \(p=1.1\): \(G = 1 - 2\int_0^1 x^{1.1} dx = 1 - 2\left[ \frac{x^{2.1}}{2.1} \right]_0^1 = 1 - \frac{2}{2.1} = \frac{1}{10}\) 2. For \(p=1.5\): \(G = 1 - 2\int_0^1 x^{1.5} dx = 1 - 2\left[ \frac{x^{2.5}}{2.5} \right]_0^1 = 1 - \frac{2}{2.5} = \frac{1}{5}\) 3. For \(p=2\): \(G = 1 - 2\int_0^1 x^{2} dx = 1 - 2\left[\frac{x^3}{3}\right]_0^1 = 1 - \frac{2}{3}\) 4. For \(p=3\): \(G = 1 - 2\int_0^1 x^{3} dx = 1 - 2\left[\frac{x^4}{4}\right]_0^1 = 1 - \frac{1}{2}\) 5. For \(p=4\): \(G = 1 - 2\int_0^1 x^{4} dx = 1 - 2\left[\frac{x^5}{5}\right]_0^1 = 1 - \frac{2}{5}\)

Step 6: f. Smallest interval of Gini index for \(L(x)=x^p\) with \(p\geq 1\)

We determine the smallest interval [a, b] in which the Gini index lies for \(L(x) = x^p\) with \(p\geq 1\) by looking at the limits of the Gini index as \(p\) approaches 1 and infinity. Find the limit as p approaches 1: $$\lim_{p\to1} G(p) = \lim_{p\to1} \left( 1 - 2\int_0^1 x^p dx \right) = 1 - 2\int_0^1 x^1 dx = 1 - 1 = 0$$ Find the limit as p approaches infinity: $$\lim_{p\to\infty} G(p) = \lim_{p\to\infty} \left( 1 - 2\int_0^1 x^p dx \right) = 1$$ Thus, the smallest interval in which the Gini index lies for \(L(x) = x^p\) with \(p\geq 1\) is \([0, 1]\). The least equitable distribution of wealth corresponds to \(G=1\), while the most equitable corresponds to \(G=0\).

Step 7: g. Conditions for \(L(x)= \frac{5x^2}{6}+\frac{x}{6}\) and Gini index

To satisfy the conditions \(L(0)=0\), \(L(1)=1\), and \(L'(x)\geq 0\) on \([0, 1]\), we verify these properties for the given Lorenz curve, \(L(x)=\frac{5x^2}{6}+\frac{x}{6}\): 1. \(L(0)=\frac{5(0)^2}{6}+\frac{0}{6}=0\) 2. \(L(1)=\frac{5(1)^2}{6}+\frac{1}{6}=1\) 3. \(L'(x)=\frac{10x}{6}+\frac{1}{6}= \frac{10x+1}{6}\), which is non-negative on the interval \([0, 1]\). All three conditions are satisfied. To find the Gini index for this function, use the Gini index formula and calculate the integral: $$G = 1 - 2\int_0^1 \left(\frac{5x^2}{6}+\frac{x}{6}\right) dx = 1 - 2\left[\frac{5x^3}{18}+\frac{x^2}{12}\right]_0^1 = 1 - \frac{7}{9} = \frac{2}{9}$$ Thus, the Gini index for this Lorenz curve is \(\frac{2}{9}\).

Key Concepts

Gini Index

The Gini Index is a key measure of inequality in a society's wealth distribution. It turns complex wealth distribution patterns into a single number. This index ranges from 0 to 1, where:

• A Gini index of 0 indicates perfect equality, meaning everyone has the same wealth.
• A Gini index of 1 suggests perfect inequality, where all wealth belongs to one person.

The Gini Index is calculated using the Lorenz Curve, specifically by finding the area between the actual Lorenz curve and the line of perfect equality. This area, denoted as "A," is compared to the total area under the line of equality ("A + B"). The Gini Index formula is: \[G = \frac{A}{A+B} = 1 - 2 \int_{0}^{1} L(x) \, dx\] where \(L(x)\) is the Lorenz Curve function. This formula shows how the Gini Index directly relates to calculus integration, as it requires integrating the curve to determine areas. Calculating the Gini Index enables us to compare different wealth distributions easily, providing a clear indicator of economic inequality.

Wealth Distribution

Wealth distribution refers to how wealth is divided among members of a society. It is a crucial factor in understanding societal fairness and economic health. The Lorenz Curve is a useful graphical tool for visualizing wealth distribution.
The Lorenz Curve represents the accumulated share of wealth possessed by the bottom x% of the population. For example, if a curve shows that 50% of the population owns only 30% of the wealth, it indicates a level of inequality.
Factors affecting wealth distribution include:

• Income disparities
• Inherited wealth
• Educational opportunities
• Economic policies

Analyzing wealth distribution helps policymakers identify and address economic disparities. Different patterns of wealth distribution can either indicate a healthy, balanced economy or highlight significant inequalities that may need corrective action. The Lorenz Curve thus serves as a starting point for more detailed economic analysis.

Calculus Integration

Calculus integration plays a vital role in determining the Gini Index from the Lorenz Curve, enabling the measurement of inequality in wealth distribution. The area under the Lorenz Curve can be computed using definite integrals.
Here's how calculus integration is involved:

• To find the area under the Lorenz Curve \(y = L(x)\), integrate the function from 0 to 1: \(\int_{0}^{1} L(x) \, dx\).
• This integral calculates the area between the Lorenz curve and the x-axis, referred to as "B."
• Subtracting this value from the area under the line of equality (which is 0.5 for a normalized graph) helps find the area "A," crucial for the Gini Index.

These steps highlight calculus integration's importance in economics, where it aids in quantifying economic concepts like inequality. Understanding these mathematical tools allows analysts to interpret and apply data meaningfully, leading to informed decision-making in policy and resource allocation.