2
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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).

$\endgroup$
  • $\begingroup$ 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? $\endgroup$ – BenBernke Jan 13 at 15:10
  • $\begingroup$ 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 $w^i$... $\endgroup$ – Bertrand Jan 13 at 15:31
  • $\begingroup$ Thanks a lot for the explanation and the Deaton paper. Very useful! $\endgroup$ – BenBernke Jan 13 at 21:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.