0

I am asked to prove that the following function is homothetic:

$u(x,y)=\alpha \frac {x^\delta}{\delta}+\beta \frac{y^\delta}{\delta}$

With $\delta \le 1, \delta \ne 0$

In the study book, a homothetic function is defined as a function whos level curves' slopes depend only on the ratio of $y$ to $x$. That was easy for me to prove: $\frac {dy}{dx}|_{u(x,y)=k}=-\frac{u_{x}}{u_{y}}=-\frac{\alpha}{\beta}(\frac{x}{y})^{\delta -1}$ for all values of $\delta$ except $\delta =1$ which gives us the constant $-\frac{\alpha}{\beta}$. Is there another definition for homothetic functions? If not how do I show that the specific case $\delta =1$ still allows for $u(x,y)$ to be homothetic?

Omrane
  • 1,384
  • 2
    Possible duplicate of distinguish homogenous and homothetic – mlc Apr 11 '17 at 04:44
  • As explained in the answer to http://math.stackexchange.com/questions/573646/distinguish-homogenous-and-homothetic, there is an alternative definition: a function is homothetic if it is a monotonic transformation of a homogenous function (note that this second function does not need to be homogenous itself). – mlc Apr 11 '17 at 04:45

0 Answers0