Economics Student Seeks Answer: Derivatives Question

[Jamp]

Hello, I am an economics student but this question is purely mathematical.

Imagine the function U(x,y)=x^0.5y^0.5 ( sqrt(x) times sqrt(y) ) in space. If we then imagine the same function equal to a constant we get the 2d functions corresponding to the intersections of horizontal planes with the original 2 variable function, right? (if anyone here is taking economics, these would be the indifference curves). So that for every constant we get a function that is convex to the origin.
Now imagine the function V=U^4=(x^2)(y^2) in space. If we do the same as above for this one, then for every constant we have a function that is concave to the origin, correct?
The problem arises now. My professor says that, the derivative of the one variable functions (the convex and concave ones, the indifference curves) is the ratio of the partial derivatives of the original 2 variable function, which is always the same if the second is a monotonous increasing transformation of the first. Evidently, for the first one: (0.5x^-0.5y^0.5)/(0.5x^0.5y^-0.5) = y/x and for the second one 2xy^2/2x^2y = y/x. The problem is, if the slopes of the functions are the same for every point, then how come one is convex and the other concave? Can this be true? By drawing the curves I can see that it can, but only if we draw the curves that way. Why is it always true? I need to understand what the ratio of the partial derivatives of the original function means geometrically and logically, what does it represent?

Thanks!

 

[AKG]

What do you mean by "concave to the origin" and "convex to the origin"? Note the curve defined by V(x,y) = C for some (positive) constant C is symmetrical in the x-z and y-z planes. The restriction of this curve to the x > 0, y > 0 quadrant of space is THE EXACT SAME THING as the curve defined by U(x,y) = C^1/4. So they're basically the same types of curves, and so I would assume their either both "concave to (0,0)" or both "convex to (0,0)".

 

[Jamp]

What I mean is, take the first function:
U(x,y) = (x^0.5)(y^0.5)
For any constant, for example 10: 10=(x^0.5)(y^0.5) <=> y=100/x is convex to (0,0).
However, 10=(x^2)(y^2) <=> y=sqrt(10-x^2) is concave.
These two different functions can not, as far as I can see, have the same slope for all x. However, both have the same ratio of partial derivatives in x over in y (y/x), which is supposed to be the slope of the curves of the original function for some constant. So how can this be true?

 

[AKG]

What I mean is, take the first function:
U(x,y) = (x^0.5)(y^0.5)
For any constant, for example 10: 10=(x^0.5)(y^0.5) <=> y=100/x is convex to (0,0).
However, 10=(x^2)(y^2) <=> y=sqrt(10-x^2) is concave.

This is wrong.

$$10 = x^2y^2 \Leftrightarrow y = \pm \frac{\sqrt{10}}{x}$$

I think you confused x²y² with x²+y².

 

[Jamp]

Thanks... I was thinking about it again on my way to school today and realized that, don't know how I did that... always a good idea to restart from the beginning.. :shy: