In a study conducted by a certain country's Economic Development Board, it was found that the Lorentz curve for the distribution of income of college teachers was described by the function
f(x) = 13/14x^2 + 1/14x
and that of lawyers by the function
g(x) = 5\7x^4 + 2/7x.
(a) Compute the coefficient of inequality for each Lorentz curve. (Round your answers to four decimal places.)
college teachers
lawyers
To compute the coefficient of inequality for each Lorentz curve, we need to integrate the provided functions, f(x) and g(x), and then calculate the area between the Lorenz curves and the line of perfect equality. Here's how to do it step by step:
(a) For the Lorentz curve of college teachers:
1. Integrate the function f(x) = 13/14x^2 + 1/14x by finding its definite integral over the range [0, 1]. The definite integral of f(x) can be expressed as:
∫[0,1] (13/14x^2 + 1/14x) dx
Evaluating this integral will give us the area under the curve of the Lorentz curve for college teachers.
2. Calculate the area between the Lorentz curve and the line of perfect equality, which is represented by the function y = x.
3. Compute the coefficient of inequality by dividing the area between the Lorentz curve and the line of perfect equality by the total area under the line of perfect equality (which is 0.5).
Now let's compute the coefficient of inequality for college teachers:
1. Integrating the function f(x):
∫[0,1] (13/14x^2 + 1/14x) dx = (13/14) * ∫[0,1] x^2 dx + (1/14) * ∫[0,1] x dx
= (13/14) * [x^3/3] from 0 to 1 + (1/14) * [x^2/2] from 0 to 1
= (13/14) * (1/3) + (1/14) * (1/2)
= 13/42 + 1/28
= 91/294
2. The area between the Lorentz curve and the line of perfect equality can be expressed as:
Area = 0.5 - ∫[0,1] f(x) dx
= 0.5 - (91/294)
= 0.5 - 0.30952
= 0.19048
3. Compute the coefficient of inequality:
Coefficient of Inequality = Area / 0.5
= 0.19048 / 0.5
≈ 0.38095
Rounded to four decimal places, the coefficient of inequality for the Lorentz curve of college teachers is approximately 0.3809.
Now you can follow the same steps to calculate the coefficient of inequality for lawyers using the given function g(x) = 5/7x^4 + 2/7x.