I need to check whether the following function is homothetic or not: f(x,y)=x3y6+3x2y4+6xy2+9 for x,y ∈ R+
As it can be clearly expressed as a positive monotonic transformation of the homogeneous function xy2 on R+ therefore it must be a homothetic. But what's confusing me is that the ratio of its partial derivatives is not a homogeneous function of degree zero, which according to the textbook I'm using(Simon & Blume) is a condition that is satisfied by every homothetic function.