MHB Solving Homogeneous Function Confusion: ln(Y/X) in Numerator

[zkee]

Hey people!

I'm confused as to why the ln(Y/X) part of the numerator is not considered in the calculation of the degree of numerator.

View attachment 1983

Any help or websites to browse through for the answer would be appreciated!

 

[Fantini]

The definition of an homogeneous function means that if $t \in \mathbb{R}$ then $f(tx,ty) = t^n f(x,y)$, where $n$ is the degree of homogeneity. When you do this to $g(x,y) = \ln (y/x)$ what you get is

$$g(tx,ty) = \ln \left( \frac{ty}{tx} \right) = \ln \left( \frac{y}{x} \right) = g(x,y).$$

The $t$'s cancel, therefore it makes no contribution. This is a degree zero homogeneous function. :)

Best wishes,

Fantini.

 

[I like Serena]

Hey people!

I'm confused as to why the ln(Y/X) part of the numerator is not considered in the calculation of the degree of numerator.

Any help or websites to browse through for the answer would be appreciated!

Welcome to MHB, zkee! :)

The degree of y/x is 0.
Or in other words, $\ln(y/x)$ behaves like a constant.