Problem 21 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A certain country's income distribution is described by the function $$ f(x)=\frac{15}{16} x^{2}+\frac{1}{16} x $$ a. Sketch the Lorentz curve for this function. b. Compute \(f(0.4)\) and \(f(0.9)\) and interpret your results.

Step-by-Step Solution

Answer

The Lorentz curve for this income distribution function is a convex curve, starting from the origin (0,0) and ending at point (1,1). For the given points, \(f(0.4) = 0.15\) and \(f(0.9) = 0.78375\). This means that the bottom 40% of the population holds 15% of the total income and the bottom 90% holds about 78.37% of the total income, illustrating income inequality within the country.

Step 1: 1. Identify the function type

The given function is: $$ f(x)=\frac{15}{16} x^{2}+\frac{1}{16} x $$ This is a quadratic function, with the coefficients given as \(\frac{15}{16}\) and \(\frac{1}{16}\).

Step 2: 2. Sketch the Lorentz curve

A Lorentz curve is a graphical representation of income inequality in a population. We can plot the Lorentz curve from the function by plotting the cumulative income share (x) against the cumulative population share (f(x)). To sketch the Lorentz curve, you would need graph paper or a suitable graphing software to plot the function for x in the range of 0 to 1 (0% to 100% of the cumulative population). The curve will be shaped as a convex curve, starting from the origin (0,0) and ending at point (1,1).

Step 3: 3. Calculate the function value at given points

Now, we will compute the function value at the given points \(x=0.4\) and \(x=0.9\) using the formula: $$ f(x)=\frac{15}{16} x^{2}+\frac{1}{16} x $$ For \(x=0.4\): $$ f(0.4)=\frac{15}{16} (0.4)^{2}+\frac{1}{16} (0.4) $$ $$ f(0.4)=0.15 $$ For \(x=0.9\): $$ f(0.9)=\frac{15}{16} (0.9)^{2}+\frac{1}{16} (0.9) $$ $$ f(0.9)=0.78375 $$

Step 4: 4. Interpret the results

By calculating the function values at the given points, we have the cumulative income share for the corresponding cumulative population shares. When \(x=0.4\), the function value \(f(0.4)=0.15\), which means that the cumulative income share of the bottom 40% of the population is 15%. When \(x=0.9\), the function value \(f(0.9)=0.78375\), which implies that the cumulative income share of the bottom 90% of the population is approximately 78.37%. These findings illustrate the income inequality within the country: the bottom 40% of the population holds only 15% of the total income, while the bottom 90% holds about 78.37% of the total income, leaving the top 10% to hold the remaining 21.63%.

Key Concepts

Income Distribution

Income distribution is a crucial concept in economics that looks at how a country’s total income is distributed among its population. This distribution can be highly uneven, reflecting various social, economic, and policy factors that influence how wealth is shared.
Consider a scenario where income distribution can be plotted on a graph. Here, a perfect equality scenario would see every portion of the population earning an equal share of income.

• A perfectly equal income distribution would result in a straight, diagonally-lining curve from the origin to the top right.
• In reality, income is often concentrated in the hands of a few, creating curves more bent below the line of equality.

Analyzing income distribution helps in understanding wider socio-economic disparities and informs the formulation of policies designed to promote equity.

Quadratic Function

Quadratic functions are polynomial functions of degree two and have a specific form: \( f(x) = ax^2 + bx + c \). In the given exercise, the function \( f(x) = \frac{15}{16} x^{2} + \frac{1}{16} x \) represents such a quadratic function.

• The coefficient \( \frac{15}{16} \) multiplies the \( x^2 \) term, shaping the parabola that describes our function.
• The other coefficient, \( \frac{1}{16} \), transforms the graph by slightly shifting its position.
• The absence of a constant term \( c \) ensures the curve passes through the origin.

This specific quadratic function is used in the original problem to model income distribution, as it provides a simple, yet effective way to illustrate how income might be distributed across a population. Understanding the shape and properties of quadratic functions is essential in plotting and interpreting curves like the Lorenz curve.

Cumulative Income Share

The concept of cumulative income share is essential when analyzing a Lorenz Curve. It represents the total amount of income earned by a certain percentage of the population. For instance, when we say that the bottom 40% of the population earns 15% of the income, this is a reflection of cumulative income share.

• It helps to visualize how wealth is distributed.
• The curve plots cumulative income share against cumulative population share.
• The entire graph is valuable for assessing disparities within an economy.

In mathematical terms, this involves calculating the value of a function at various points (such as \( x = 0.4 \) and \( x = 0.9 \)), and interpreting these points within the context of population and income dynamics.

Income Inequality Analysis

Income inequality analysis often uses graphical tools like the Lorenz Curve to visually express the degree of inequality in income distribution. The information derived from the Lorenz curve is usually compared to the line of equality to determine the extent of inequality.

• A Lorenz curve that lies completely on the equality line indicates perfect equality.
• A more bowed-out curve suggests greater inequality.
• The Gini coefficient may also be used here, derived from the Lorenz Curve, quantifying inequality on a scale from 0 (perfect equality) to 1 (perfect inequality).

By looking at values like \( f(0.4) \) and \( f(0.9) \), we can better understand how income is distributed among different portions of a population. This exercise highlights how the bottom percentages accumulate less wealth relative to their population size, spotlighting the need for economic policy interventions to address significant income gaps.