Problem 89 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

DISTRIBUTION OF INCOME A study suggests that the distribution of incomes for social workers and physical therapists may be represented by the Lorenz curves \(y=L_{1}(x)\) and \(y=L_{2}(x)\), respectively, where $$ L_{1}(x)=x^{1.6} \text { and } L_{2}(x)=0.65 x^{2}+0.35 x $$ For which profession is the distribution of income more equitable?

Step-by-Step Solution

Answer

The income distribution for social workers is more equitable.

Step 1 - Understand Lorenz Curves

Lorenz curves are used to represent the distribution of income or wealth within a population. The closer the Lorenz curve is to the line of equality (which is the diagonal line from (0,0) to (1,1)), the more equitable the distribution is.

Step 2 - Lorenz Curve for Social Workers

The Lorenz curve for social workers is given by the function: \(L_{1}(x)=x^{1.6}\).

Step 3 - Lorenz Curve for Physical Therapists

The Lorenz curve for physical therapists is given by the function: \(L_{2}(x)=0.65x^{2} + 0.35x\).

Step 4 - Compare Lorenz Curves

To determine which profession has a more equitable distribution of income, compare the functions graphically or algebraically. If one Lorenz curve is consistently closer to the line of equality than the other, then it represents a more equitable distribution.

Step 5 - Analyze Graphs

Graph \(y=L_{1}(x)=x^{1.6}\) and \(y=L_{2}(x)=0.65x^{2}+0.35x\). Notice that \(L_{2}(x)\) lies below \(L_{1}(x)\) for most parts. Therefore, \(L_{1}(x)\) is closer to the line of equality. A curve closer to the line of equality indicates a more equitable income distribution.

Step 6 - Conclusion

Since \(L_{1}(x)\) is generally closer to the line of equality compared to \(L_{2}(x)\), the income distribution for social workers (represented by \(L_{1}(x)\)) is more equitable than that for physical therapists (represented by \(L_{2}(x)\)).

Key Concepts

income distribution

Income distribution refers to how income or wealth is shared across the members of a population. This concept helps us understand economic equity and inequality within a society. When income is distributed equally, every member of the population has access to a comparable level of income.
One common method of visualizing income distribution is through the use of Lorenz curves, which graphically represent the distribution of income or wealth among a population. By analyzing these curves, we can draw conclusions about the equity of income distribution among different groups.

line of equality

The line of equality in the context of Lorenz curves is a fundamental reference point. This is a diagonal line that runs from the bottom-left corner (0,0) to the top-right corner (1,1) on the graph. This line represents perfect equality.
If the Lorenz curve overlaps or closely follows the line of equality, it means that the income distribution is very equitable. The further the Lorenz curve deviates from this diagonal line, the more unequal the distribution of income becomes. Hence, the line of equality serves as a benchmark against which we measure the equity of income distribution.

equity measurement

Equity measurement involves assessing how fairly income or wealth is distributed among members of a society. Lorenz curves are instrumental in this process. By comparing the area between the Lorenz curve and the line of equality, economists can quantify the degree of inequality.
In the context of the given exercise, comparing the Lorenz curves of social workers and physical therapists allows us to determine which profession has a more equitable distribution of income. A Lorenz curve closer to the line of equality indicates higher equity. Therefore, analyzing and comparing these curves is crucial for equity measurement.

graphical analysis

Graphical analysis is a powerful tool for understanding complex mathematical relationships. When it comes to income distribution, plotting Lorenz curves on a graph allows for a visual comparison of equity.
In the exercise, we compare the Lorenz curves of social workers (L_{1}(x) = x^{1.6}) and physical therapists (L_{2}(x) = 0.65x^{2} + 0.35x). By graphing these functions, we can clearly see the position of each curve relative to the line of equality.
Since L_{1}(x) = x^{1.6} lies closer to the line of equality than L_{2}(x) = 0.65x^{2} + 0.35x, it implies that social workers have a more equitable distribution of income compared to physical therapists. Graphical analysis thus provides a clear and intuitive way to compare and interpret the equity of income distribution.