Homothetic preferences from indirect utility

Question

Consider an indirect utility function on the form

$v^{i}(\textbf{p},w^{i}) = a^{i}(\textbf{p}) + b^{i}(\textbf{p})w^{i}$

Where $\textbf{p}$ is a vector of prices and $w$ denotes income of individual $i$. I want to find the restrictions on $a^{i}(\textbf{p})$ and $b^{i}(\textbf{p})$ such that the preferences are homothetic.

I start by using Roy's identity to find the Marshallian demand for good $j$:

$x_{j}(\textbf{p},w^{i}) = -(\frac{\partial a^{i}}{\partial p_{j}} + \frac{\partial b^{i}(\textbf{p})}{\partial p_{j}} w^{i})/b^{i}(\textbf{p})$

To check for homothetic preferences I find an expression for $\frac{x_{j}}{x_{k}}$ such that it is independent of income. I find that $\frac{x_{j}}{x_{k}} = \frac{(\frac{\partial a^{i}(\textbf{p})}{\partial p_{j}}+\frac{\partial b^{i}(\textbf{p})}{\partial p_{j}}w^{i})}{(\frac{\partial a^{i}(\textbf{p})}{\partial p_{k}}+\frac{\partial b^{i}(\textbf{p})}{\partial p_{k}}w^{i})}$

So (assuming I have not made an error differentiating), the expression is independent of income as long as $\frac{\partial b^{i}(\textbf{p})}{\partial p_{k}} = \frac{\partial b^{i}(\textbf{p})}{\partial p_{j}} = 0$ and there is no restriction needed on $a^{i}(\textbf{p})$ other than it needs to be differentiable.

Answer 1

Your conclusion is a bit hasty:
(i) why did you implicitly exclude the case $a^i(\textbf{p})=0$?
(ii) if your individual is rational then $v^i$ is homogeneous of degree zero in $(\textbf{p},w^i)$ which has implications on $a^i$ and $b^i$... Your demand system should also satisfy the budget constraint (adding-up).