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.

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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.
The most equitable distribution is when $$ p=1.1 $$, and the least equitable is when $$ p=4 $$.

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