Does the utility function U(X, Y) = 12x^0.2 y^0.8 represent homothetic preferences?

I was given this question but I have no idea how to work it out. I understand what homothetic preferences are in theory, but I don't know how to calculate if a utility function represents homothetic preferences.

Any help would be hugely appreciated - I've been stuck on this for over an hour.


It depends on the definition of homotheticity. One definition states that consumer's preferences are homotethic if and only if his or her preferences can be represented by a utility function which is homogenous of degree 1.

For example, if we have two goods $x$ and $y$, then the utility function $u:\mathbb{R}^2\to\mathbb{R}$ represents homothetic preferences if and only if $$u(t\cdot x,t\cdot y)=t\cdot u(x,y)$$ for all $t>0$. Hence, to see if $u(x,y)=12x^{0.2}y^{0.8}$ is homothetic, check if $u(t\cdot x,t\cdot y)=t\cdot u(x,y)$.

Another definition states that a homothetic function is a function that can be written as a monotonic transformation of a homogenous function. According to this definition, $u(x,y)=x^3y^3+xy$ is homoethetic since $u(x,y)=g(v(x,y))$ with $g(z)=z^3+z$ and $v(x,y)=xy$.

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