Sum of Homothetic Functions If two utility functions represent homothetic preferences, will their sum also be homothetic?
Defn: A function $h:\mathbb{R}^2\rightarrow \mathbb{R}$ is homogenous of degree $k$ if for every nonzero $\alpha$, $h(\alpha x, \alpha y)=\alpha^k h(x,y)$.

Defn: A function is homothetic if it is a monotonic transformation of a homogenous function.

Lemma: If $f$ is homothetic, i.e. $f=g\circ u$ for some strictly increasing $g$ and homogenous $u$ then 
$$
\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}} = \frac{g'(u(x,y))\frac{\partial u}{\partial x}}{g'(u(x,y))\frac{\partial u}{\partial y}}=\frac{\frac{\partial u}{\partial x}}{\frac{\partial u}{\partial y}}
$$
is homogenous of degree zero. 

Let

i) $u_1(x,y)=x+y$, 

ii) $u_2(x,y)=\log(2x+y)$

Then, $u_1$ and $u_2$ are homothetic functions since they are monotonic transformations of homogenous functions (of degree 1). Let

iii) $u_3(x,y)=x+y+log(2x+y)$. Then

$$
MRS_{u_3}=\frac{\frac{\partial u_3}{\partial x}}{\frac{\partial u_3}{\partial y}}=\frac{1+\frac{2}{2x+y}}{1+\frac{1}{2x+y}}=\frac{2x+y+2}{2x+y+1}
$$
which is not homogenous of degree 0. 

Hence, sum of homothetic functions is not necessarily homothetic.