Homothetic preferences from indirect utility

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.

(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).
• I think I see your points. So for (i), the appropriate restriction should be $a^{i}(\textbf{p}) \neq 0$ along with the differentiability restriction. Point (ii) is less clear to me. Is the restriction on $b^{i}(\textbf{p})$ reasonable? Jan 13 '19 at 15:10
• Your initial indirect utility function is actually quasi-homothetic (see Deaton and Muellbauer, 1980, Section 5.4), and is homothetic if $a^i(p)=0$. Assuming that $b^i$ is price independent is very strong as it implies that all your demands are independent of income $w^i$... Jan 13 '19 at 15:31