c. Graph the Lorenz curves \( L(x)=x^{p} \) corresponding to \( p=1.1,1.3,2,3,4 \). 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. A. \( p=1.1 \) is most equitable. \( p=4 \) is least equitable. \( y=x^{4} \) more closely resembles the line of perfect equality, whereas \( y=x^{1.1} \) least resembles the line. B. \( p=4 \) is most equitable. \( p=1.1 \) is least equitable. \( y=x^{4} \) more closely resembles the line of perfect equality, whereas the curve \( y=x^{1.1} \) least resembles the line. C. \( p=1.1 \) is most equitable. \( p=4 \) is least equitable. \( y=x^{1.1} \) more closely resembles the line of perfect equality, whereas the curve \( y=x^{4} \) least resembles the line. D. \( p=4 \) is most equitable. \( p=1.1 \) is least equitable. \( y=x \) more closely resembles the line of perfect equality, whereas the curve \( y=-x \) least resembles the line.

The correct answer is: **C. \( p=1.1 \) is most equitable. \( p=4 \) is least equitable. \( y=x^{1.1} \) more closely resembles the line of perfect equality, whereas the curve \( y=x^{4} \) least resembles the line.** ### Explanation: The **Lorenz curve** \( L(x) = x^p \) is used to represent the distribution of wealth or income within an economy. Here's how the parameter \( p \) affects the curve: 1. **Perfect Equality:** When \( p = 1 \), the Lorenz curve is \( L(x) = x \), which is a straight line representing perfect equality where each proportion of the population holds an equal proportion of the total wealth. 2. **Values of \( p > 1 \):** - **Higher \( p \):** As \( p \) increases above 1, the Lorenz curve \( L(x) = x^p \) becomes more convex (bowed downward) and lies further below the line of perfect equality \( y = x \). This indicates greater inequality, as a larger proportion of wealth is concentrated among the top percentiles of the population. - **Lower \( p \) (just above 1):** Values of \( p \) slightly above 1 (like \( p = 1.1 \)) result in Lorenz curves that are closer to the line of perfect equality, indicating a more equitable distribution of wealth. 3. **Interpretation of Specific Values:** - **\( p = 1.1 \):** The Lorenz curve \( L(x) = x^{1.1} \) is very close to the line \( y = x \), suggesting a relatively equitable distribution of wealth. - **\( p = 4 \):** The Lorenz curve \( L(x) = x^{4} \) is significantly bowed downward, indicating a high level of inequality with wealth concentrated among the top percentiles. Thus, **\( p=1.1 \)** represents the most equitable distribution (closest to perfect equality), and **\( p=4 \)** represents the least equitable distribution (farthest from perfect equality). **Answer:** C. \( p=1.1 \) is most equitable. \( p=4 \) is least equitable. \( y=x^{1.1} \) more closely resembles the line of perfect equality, whereas the curve \( y=x^{4} \) least resembles the line.