# Lorenz curve extension to 3d?

While thinking about Lorenz curves and economic inequality I wondered if you can extend Lorenz curves to Lorenz surfaces by revolving a Lorenz curve about the line of perfect equality.

Would such a Lorenz surface have any economical interpretation?

sub question:

Can Lorenz curves be analytic functions, such as $$y=x^2?$$

Here is a picture of the Lorenz surface I'm imagining inside a cube. The line of perfect equality would now be the longest diagonal of the cube.

• I think the answer would depend on what do you want to measure on the third axis? Given that the curve is used to measure inequality measuring the cumulative shares what would you even put as the 3rd dimension? Also I do not understand the second question the curve is by definition given by the PDF's and CDF's that you empirically find in your data. Sure once you find what the curve is in particular case it might happen to be approximately $y=x^2$ or some other analytic function but you should not just assume its shape the Lorenz curve is very much empirical concept
– 1muflon1
Oct 13 '20 at 19:35
• Any ideas for the 3rd dimension? Oct 13 '20 at 22:56