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

CC BY-SA 4.0
2019-01-13T15:10:58.863Z
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?
user11767
CC BY-SA 4.0
2019-01-13T15:31:50.307Z
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 $w^i$...
Bertrand
CC BY-SA 4.0
2019-01-13T21:14:21.357Z
Thanks a lot for the explanation and the Deaton paper. Very useful!
user11767
CC BY-SA 4.0
2019-01-13T14:04:27.780Z
2019-01-13T15:50:02.130Z
2019-01-13T15:50:02.130Z
Bertrand
<p>Consider an indirect utility function on the form </p>

<p><span class="math-container">$v^{i}(\textbf{p},w^{i}) = a^{i}(\textbf{p}) + b^{i}(\textbf{p})w^{i}$</span></p>

<p>Where <span class="math-container">$\textbf{p}$</span> is a vector of prices and <span class="math-container">$w$</span> denotes income of individual <span class="math-container">$i$</span>. I want to find the restrictions on <span class="math-container">$a^{i}(\textbf{p})$</span> and <span class="math-container">$b^{i}(\textbf{p})$</span> such that the preferences are homothetic. </p>

<p>I start by using Roy's identity to find the Marshallian demand for good <span class="math-container">$j$</span>:</p>

<p><span class="math-container">$x_{j}(\textbf{p},w^{i}) = -(\frac{\partial a^{i}}{\partial p_{j}} + \frac{\partial b^{i}(\textbf{p})}{\partial p_{j}} w^{i})/b^{i}(\textbf{p})$</span></p>

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

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

<microeconomics><preferences>
Homothetic preferences from indirect utility
user11767
# Homothetic preferences from indirect utility

(Asked by: username_0 on 2019-01-13)

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.

**Tags:** microeconomics, preferences

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## 1 answers

### Answer by username_1 on 2019-01-13. Score: 3 | Accepted answer

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).

> **Comments:**
> **username_0**: 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?  
> **username_1**: 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 $w^i$...  
> **username_0**: Thanks a lot for the explanation and the Deaton paper. Very useful!  

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