Question

Draw a Lorenz curve showing the distribution of income for the five people in the following table. $$ \begin{array}{l|c} \text { Name } & \text { Annual Earnings } \\ \hline \text { Lena } & \$ 70,000 \\ \hline \text { David } & 60,000 \\ \hline \text { Steve } & 50,000 \\ \hline \text { Jerome } & 40,000 \\ \hline \text { Lori } & 30,000 \\ \hline \end{array} $$

Short answer

The Lorenz curve for the given table shows the distribution of income amongst the five individuals. It provides a visual representation of the economic inequality between these individuals. The closer the Lorenz curve is to the line of equality, the more evenly income is distributed; the further away, the greater the income inequality.

Step-by-step solution

Calculate Total Income

Add all the individual earnings together to find the total income: \(70,000 + 60,000 + 50,000 + 40,000 + 30,000 = \$250,000\)

Calculate Cumulative Percentage of Total Income

Arrange the individual incomes in increasing order and calculate the cumulative percentage of total income for each individual. For example, for Lori, who earns the least, her percentage of total income is \((30,000/250,000) * 100 = 12\%\). Consider these percentages to be Lorenz Y-values and the cumulative percentage of people as Lorenz X-values. Here, since there are 5 people, each individual represents 20% of the population.

Preparation for Plotting the Lorenz Curve

Combine the cumulative percentage of total income (Lorenz Y-values) for each individual with their corresponding Lorenz X-values, which represent the cumulative percentage of people. This will yield a set of coordinates. For example, Lori's coordinates are \((20, 12)\) as she makes up 20% of the people and 12% of the income.

Plot the Lorenz Curve

Starting from the origin (0,0), plot the coordinates on the graph: (20, 12), followed by the rest of the individual's coordinates. Connect these points to form the Lorenz curve. Also draw the line of equality which is the 45 degrees diagonal from the origin to the top right corner of the graph. This line represents a perfectly equal distribution of income. Comparing the Lorenz curve to the line of equality gives a visual representation of the income inequality among the individuals.

Key concepts

Income Distribution

Income distribution refers to how a nation’s total income is spread among its population. It's an indicator that helps us understand who gets what portion of the economic pie. When discussing income distribution, we're essentially talking about the frequency and amount of income individuals or households receive over a specific time period.

• It can highlight disparities or equalities within different socioeconomic groups.
• Income distribution is vital in shaping social and economic policies that aim to address wealth gaps.

Analyzing income distribution requires gathering data on people's earnings and understanding which portions of the population earn more or less. This understanding helps economists and policymakers work toward fairer economic systems.

Economic Inequality

Economic inequality is the unequal distribution of income and opportunities among a population. It's a broad concept that not only includes differences in income but also considers disparities in wealth, access to resources, and other opportunities. Greater inequality suggests a wider gap between the richest and the poorest in a society.

• Economic inequality can impact a country's health, education, and social cohesion.
• It often arises due to factors like education levels, globalization, technological changes, and government policies.

Understanding economic inequality is essential for developing measures to ensure more equitable growth and prosperity, reducing social tensions, and fostering a more inclusive economic environment.

Cumulative Income Shares

Cumulative income shares are a way to measure the share of total income held by cumulative segments of the population, typically arranged from the poorest to the richest. By looking at cumulative income shares, we can determine how income is distributed across different population segments.

• It helps in understanding how much of the total income is earned by specific percentages of the population.
• Cumulative income shares data form the basis for constructing a Lorenz curve.

To calculate cumulative income shares, arrange individual incomes in ascending order and calculate each segment's contribution to the total income cumulatively. This data represents the cumulative share of people compared to their cumulative share of income, forming foundations for further graphical analyses.

Graphical Analysis of Income

The graphical analysis of income, such as plotting a Lorenz curve, provides a visual representation of income distribution and inequality. The Lorenz curve is particularly useful in comparing actual income distribution to a perfectly equal distribution.

• It involves plotting cumulative income shares against cumulative population shares.
• The more the Lorenz curve sags below the line of equality (a 45-degree line representing equal distribution), the greater the level of inequality.

Creating a Lorenz curve begins by arranging all income earners from lowest to highest, calculating their cumulative income shares, and plotting these against their cumulative shares of the population. By analyzing these visual distinctions, observers can make informed assessments regarding the economic landscape.