8. Consider the C̈hotikapanichm̈odel for a Lorenz curve,...

8. Consider the \"Chotikapanich\" model for a Lorenz curve, where \begin{equation*} L(x) = \frac{1}{e^k - 1}(e^{kx} - 1), \end{equation*} where k is an (as yet) unknown constant. (a) Check that, regardless of the value of k, for this formula we get L(0) = 0 and L(1) = 1. (b) Write one or two sentences explaining why it is important, in the context of distribution of income, that any Lorenz curve should satisfy the equations L(0) = 0 and L(1) = 1. (c) According to the Gini Index, which model shows greater income inequality for a country: a model with Lorenz curve given by the Chotikapanich model with k = 2.4, or by the Lorenz curve $L(x) = \frac{3}{5}x^3 + \frac{2}{5}x^2$?

Transcript

00:01 Okay, all right.

00:02 So, sorry.

00:07 So this question, as long as you understand what lorenz curve is, then it should make sense.

00:13 But it seems like maybe don't understand what that means.

00:16 So the lorenz curve is just a percent of households by income.

00:23 So, like, as you increase percent of households, we'll be increasing, like, these people will have have, as we increase the percentage, these people on the right will have more and more money.

00:40 So we're going from poor, poorest to riches, essentially.

00:47 That's how we're drawing this graph.

00:52 And this is just amount of money or, sorry, i say income, income earned by that, you know, prospective percentage.

01:12 Right so if we're here just to make some sense of this as we go from here to here this is the money that we're adding on but if we increase by the same percentage like let's say this was you know 5 % 5 % and these are increases in a percent of household we see that one gives us more um give us more area so this just you can think of it as well, you can think it as infinitely small rectangles, but i don't want to confuse you.

01:50 So, infinite small rectangles.

01:51 The point is, you know what, i should draw just to be clear.

01:57 I brought it out as we go from here to here and here to here.

02:09 So say this is the value.

02:15 You can see that as we increase the amount of households that the income earned will increase, increase more than it did for the last 5 percentage.

02:28 And that's here, again, to show you that we're going from poorest to riches, right? so, of course, the higher, more percentage we add on, the more money proportionally to the last percentage that was being add on will be made because of how this graph is made.

02:49 Anyways, let's just get to the question.

02:51 Why is l of 0 equal to 0? what that just means, think about it, is that zero percent of the households...

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