Homogeneous of Degree Two Utility Functions and Homothetic Preferences. The understanding that I am not clear is in when do homothetic preferences represent a utility function and vice-versa. My solution to the problem is posted below the problem:
A consumer’s preferences are described by a utility function that is homogeneous
of degree two: For all $\alpha &gt; 0$ and $x \in R^{L}_{+} $ ,
$u(\alpha x) = \alpha^2 u(x)$
The problem that I am not getting clear is:
Q) &quot;Are this consumer’s preferences homothetic? Show that they are or give a counterexample.&quot;
My solution:
According to Mas Colell et al. &quot;Microeconomic Theory&quot; (chapter 3, page 50)

Therefore, this given consumer's preferences are not homothetic as it doesn't generate a utility function that is homogeneous of degree 1 (HOD(1)). A counter example would be a utility function that is HOD(1) like the Cobb Douglas Utility Function
$ U(x_1, x_2) = x_{1}^{\alpha} x_{2}^{1-\alpha} $
To conclude, this consumer's preferences are not homothetic as it represents a utility function of HOD(2). While , according to Mas Colell et al. preference $\pmb{\succsim}$  is homothetic $\textbf{if and only if}$ it admits a utility function that is HOD(1).
Could you please help me in understanding where I am going wrong with what Mas-Colell mentioned above &quot;necessary and sufficient condition&quot; and how a utility function that is HOD(2) implies that $\pmb{\succsim}$ is homothetic.
Thanks.
First of all, in order to provide a counterexample, you need to construct a utility function that is homogeneous of degree two, but is not homothetic. Therefore, the counterexample you gave in your solution doesn't work.

To prove the statement directly, let $u(x)$ be a utility representation that is homogeneous of degree two. That is, $u(\alpha x)=\alpha^2 u(x)$. Therefore, if $x\sim y$, which means $u(x)=u(y)$, we have 
$$u(\alpha x)= \alpha^2 u(x)=\alpha^2 u(y)=u(\alpha y).$$
This means $\alpha x\sim \alpha y$, and hence the preferences are homothetic.

We can also use the proposition in MWG: A continuous $\succeq$ is homothetic if and only if it admits a utility function $u(x)$ that is homogeneous of degree one. One caveat is that the utility representation is unique up to monotone transformations, so even if one representation $u(x)$ is not homogeneous of degree one, the preferences could still be homothetic if a monotone transformation of the representation, $\phi(u(x))$, is.

In this question, if we consider a monotone transformation $\hat{u} (x)=(u(x))^\frac{1}{2}$, this $\hat{u}(x)$ still represents the preferences $\succeq$. Notice that 
$$\hat{u} (\alpha x)=(u(\alpha x))^\frac{1}{2}=(\alpha^2 u(x))^\frac{1}{2}=\alpha (u(x))^\frac{1}{2}=\alpha\hat{u} (x),$$
meaning that this new representation is homogeneous of degree one. Therefore, by the proposition above, the preferences are homothetic.