distinguish homogenous and homothetic I am not sure how to distinguish whether a function is homothetic. 
For example, 
Q1.
$$
U(x,y) = a\log(x) + b\log(y), 
$$
Q2.
$$
U(x,y) = \exp[(x^a+by^a)^3 +r]
$$
I can understand that these two functions are not homogenous.
But i don't know why these are homothetic. Can any body explain to me??
 A: A function is homogenous of order $k$ if
$$
f(tx, ty)=t^kf(x, y).
$$
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).
Consider now the function:
$$
f(x, y)=x^ay^b
$$
This is homogenous, since
$$
f(tx, ty)=(tx)^a(ty)^b=t^{a+b}x^ay^b=t^{a+b}f(x, y).
$$
Consider now the function
$$
g(z)=\log z
$$
which is monotone. We have
$$
g(f(x, y))=\log(f(x, y))=\log(x^ay^b)=a\log x+b\log y
$$
which is your first function. This is a monotone transformation of a homogenous function, so it is homothetic. 
Consider now
$$
f(x, y)=x^a+by^a
$$
which is homogenous since
$$
f(tx, ty)=(tx)^a+b(ty)^a=t^a(x^a+by^a)=t^af(x, y).
$$
Let
$$
g(z)=\exp(z^3+r)
$$
whose derivative is
$$
g^\prime (z)=3z^2 \exp(z^3+r)
$$
which is positive other than at the isolated point $z=0$, so the function $g$ is monotone. So it then follows that
$$
g(f(x, y))=\exp[(f(x, y))^3+r]=\exp[(x^a+by^a)^3+r].
$$
And hence, the function you provided is a monotonic transformation of a homogenous function, meaning that it is homothetic. 
