One type of function often used to model Lorenz curves is f(x) = ax+(1−a)xp. Suppose that and that the Gini index for the distribution of wealth in a country is known to be 4/9, and assume that a = 1/3.
Determine the correct value of p and use that to find how much of the wealth is owned by the wealthiest 5% of the population?
Ah, I figured out where I went wrong. I didn't distribute one of the negatives for f(x). Just redid it and got p=5. Not sure you did your integral right.
My integral was set up as 2∫[0,1] [x - x/3 - (2/3)x^p]dx since the Gini Index is G=2∫[0,1] (x - f(x))dx
Since our function is now f(x)=(1/3)x + (2/3)x^5, this means that the wealthiest 5% of the population own 16.6% of total income
And I meant 16.75% of total income my bad
we've already done two of these for you. Why don't you take a stab at this one?
I'm having trouble determining this one. Seems like p comes out negative, but I don't know what I might've done wrong.
I keep getting p=-7, but that might be wrong. @oobleck tell me what you get if you try the problem.
∫[0,1] (x - (x+2px)/3) dx = 1 - p/3
1 - p/3 = 4/9
p/3 = 13/9
p = 13/3
How did you get -7?
Here's a graph of the Lorenz Curve I did: .desmos.com/calculator/zyrkdoo05p
To find the correct value of p, we need to use the given information that the Gini index is 4/9 and a = 1/3.
The Gini index is a measure of income or wealth inequality, and it ranges from 0 (perfect equality) to 1 (maximum inequality). In this case, the Gini index is 4/9.
The Lorenz curve is a graphical representation of wealth inequality, with the x-axis representing the cumulative proportion of the population sorted by wealth and the y-axis representing the cumulative proportion of total wealth owned by that proportion of the population.
The equation for the Lorenz curve in this case is f(x) = ax + (1-a)x^p, where a = 1/3. This equation represents a general form of the Lorenz curve that depends on the parameter p.
To find the correct value of p, we can use the fact that the Gini index is equal to twice the area between the Lorenz curve and the line of perfect equality (the line y = x).
The area between the Lorenz curve and the line y = x can be determined by integrating the difference between f(x) and x from 0 to 1.
Integrating f(x) - x from 0 to 1, we have:
∫ [(1/3)x + (2/3)x^p - x] dx from 0 to 1
Simplifying the expression, we have:
∫ [(1/3)x + (2/3)x^p - x] dx = (1/3)(1/2)x^2 + (2/3)(1/(p+1))x^(p+1) - (1/2)x^2 from 0 to 1
Plugging in the values and simplifying, we have:
(1/6) + (2/3)(1/(p+1)) - (1/2) = 4/9
Simplifying further, we have:
(1/6) - (1/2) + (2/3)(1/(p+1)) = 4/9
(-2/3) + (2/3)(1/(p+1)) = 4/9
Now, we can solve this equation for p.
(-2/3)(p+1) + 2/3 = 4/9
Multiplying through by 9 and simplifying, we have:
(-6/9)(p+1) + 6/9 = 4/9
(-6/9)(p+1) = -2/9
Multiplying through by -9/6 and simplifying, we have:
p + 1 = 2/3
Subtracting 1 from both sides, we have:
p = 2/3 - 1
p = -1/3
Therefore, the correct value of p is -1/3.
Now, to find how much wealth is owned by the wealthiest 5% of the population, we need to determine the x-coordinate on the Lorenz curve that corresponds to the 95th percentile. We can then plug that value into the equation f(x) = (1/3)x + (2/3)x^p to find the corresponding y-coordinate.
To find the x-coordinate, we need to solve the equation:
∫ [(1/3)x + (2/3)x^p] dx from 0 to x = 0.95
Unfortunately, there is no analytical solution for this equation. We need to use numerical methods such as numerical integration or Newton's method to approximate the value of x.
Once we have the approximated x-value for the 95th percentile, we can plug it into the equation f(x) = (1/3)x + (2/3)x^p to find the corresponding y-coordinate or the proportion of wealth owned by the wealthiest 5% of the population.