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?
9 answers
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?
1 - p/3 = 4/9
p/3 = 13/9
p = 13/3
How did you get -7?
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
Here's a graph of the Lorenz Curve I did: .desmos.com/calculator/zyrkdoo05p
And I meant 16.75% of total income my bad