<p>EDIT: Seems to that after your most recent edit the answer is still no. 
A relatively simple counterexample is:
$$
U(x,y,z) = \sqrt{x} + \sqrt{y} + \sqrt{z}.
$$
<br>This is clearly seperable and homothetic. But $D_z$ is is not independent of $p_x$ and $p_y$. I think this is clear just by looking at the utility function, as it is not of the Cobb-Douglas type. If you have doubts you can calculate $D_z$ from these equations:
\begin{eqnarray*}
MRS_{xz}(x,y,z) &amp; = &amp; \sqrt{\frac{z}{x}} = \frac{p_x}{p_z} \\
\\
MRS_{yz}(x,y,z) &amp; = &amp; \sqrt{\frac{z}{y}} = \frac{p_y}{p_z} \\
\\
m &amp; = &amp; p_x \cdot x + p_y \cdot y + p_z \cdot z.
\end{eqnarray*}
Then
\begin{eqnarray*}
m &amp; = &amp; p_x \cdot x + p_y \cdot y + p_z \cdot z \\
\\
m &amp; = &amp; p_x \cdot z \cdot \left(\frac{p_z}{p_x}\right)^2 + p_y \cdot z \cdot \left(\frac{p_z}{p_y}\right)^2 + p_z \cdot z \\
\\
\frac{m}{p_z}\frac{1}{\frac{p_z}{p_x} + \frac{p_z}{p_y} + \frac{p_z}{p_z}} &amp; = &amp; z.
\end{eqnarray*}
As you can see both $p_x$ and $p_y$ appear.</p>

CC BY-SA 3.0
2016-11-28T14:43:02.073Z
Thank you and  excuse me, I was misleading again. Utility is assumed to be monotonic and concave.
Ignasi Mata Pavia
CC BY-SA 3.0
2016-11-28T16:39:54.963Z
The utility function in my counterexample is monotonic and concave. I see in your question you mean strictly concave.
Giskard
CC BY-SA 3.0
2016-11-29T00:17:42.123Z
Thank you very much. Yes, of course I meant strictly concave. I was totally wrong, starting from a cobb douglas function was a huge mistake; without paying attention to other possibilities.
Ignasi Mata Pavia
CC BY-SA 3.0
2016-11-28T11:11:04.743Z
2016-11-28T17:03:27.383Z
2016-11-28T17:03:27.383Z
Giskard
<p>With 3 goods (x,y,z), linear Engel curves, where z is separable of x and y, and with all first derivatives are positives and second negatives.</p>

<p>Does the demand of z change with the price of x or y? So, is z independent of $p_{x}$ and $p_y$?</p>

<p>I think so, because I have made some examples where changing prices didn't change the optimal amount of z. However I cannot find a formal proof.</p>

CC BY-SA 3.0
2016-11-28T09:02:40.450Z
Maybe I'm wrong again, is it equivalent to ask if homothetic preferences imply independent goods?

Then, another question comes up, how can I compute algebraically a demand function without defining the utility function? Once I have the First Order Conditions and the Budget Constrain, I do not know how to proceed in order to get a general demand curve.
Ignasi Mata Pavia
CC BY-SA 3.0
2016-11-28T10:13:35.463Z
It is not clear to me where the details of your question end and where your own speculations begin. Does the problem state that $MRS_{xy}(x,y,z) = MRS_{xy}(x,y,z')$ for all $z, z'$ or is that something you suspect to be true?
Giskard
CC BY-SA 3.0
2016-11-28T10:59:46.197Z
Sorry for my misleading question. It is assumed that z is separable from x and y.
There are two things, weak separability and homothetic preferences. From such point, I would like to know if demand of z is independent of prices of x and y.
Ignasi Mata Pavia
CC BY-SA 3.0
2016-11-28T11:05:40.437Z
Please edit your question to show this information. (So other users do not have to read the comments.)
Giskard
<microeconomics><mathematical-economics>
Homothetic preferences and weak separability
Ignasi Mata Pavia
# Homothetic preferences and weak separability

(Asked by: username_0 on 2016-11-28)

With 3 goods (x,y,z), linear Engel curves, where z is separable of x and y, and with all first derivatives are positives and second negatives.

Does the demand of z change with the price of x or y? So, is z independent of $p\_{x}$ and $p\_y$?

I think so, because I have made some examples where changing prices didn't change the optimal amount of z. However I cannot find a formal proof.

**Tags:** microeconomics, mathematical-economics

> **Comments:**
> **username_0**: Maybe I'm wrong again, is it equivalent to ask if homothetic preferences imply independent goods?  Then, another question comes up, how can I compute algebraically a demand function without defining the utility function? Once I have the First Order Conditions and the Budget Constrain, I do not know how to proceed in order to get a general demand curve.  
> **username_1**: It is not clear to me where the details of your question end and where your own speculations begin. Does the problem state that $MRS_{xy}(x,y,z) = MRS_{xy}(x,y,z')$ for all $z, z'$ or is that something you suspect to be true?  
> **username_0**: Sorry for my misleading question. It is assumed that z is separable from x and y. There are two things, weak separability and homothetic preferences. From such point, I would like to know if demand of z is independent of prices of x and y.  
> **username_1**: Please edit your question to show this information. (So other users do not have to read the comments.)  

---

## 1 answers

### Answer by username_1 on 2016-11-28. Score: 5 | Accepted answer

EDIT: Seems to that after your most recent edit the answer is still no.
A relatively simple counterexample is:
$$
U(x,y,z) = \sqrt{x} + \sqrt{y} + \sqrt{z}.
$$
  
This is clearly seperable and homothetic. But $D\_z$ is is not independent of $p\_x$ and $p\_y$. I think this is clear just by looking at the utility function, as it is not of the Cobb-Douglas type. If you have doubts you can calculate $D\_z$ from these equations:
\begin{eqnarray\*}
MRS\_{xz}(x,y,z) & = & \sqrt{\frac{z}{x}} = \frac{p\_x}{p\_z} \\
\\
MRS\_{yz}(x,y,z) & = & \sqrt{\frac{z}{y}} = \frac{p\_y}{p\_z} \\
\\
m & = & p\_x \cdot x + p\_y \cdot y + p\_z \cdot z.
\end{eqnarray\*}
Then
\begin{eqnarray\*}
m & = & p\_x \cdot x + p\_y \cdot y + p\_z \cdot z \\
\\
m & = & p\_x \cdot z \cdot \left(\frac{p\_z}{p\_x}\right)^2 + p\_y \cdot z \cdot \left(\frac{p\_z}{p\_y}\right)^2 + p\_z \cdot z \\
\\
\frac{m}{p\_z}\frac{1}{\frac{p\_z}{p\_x} + \frac{p\_z}{p\_y} + \frac{p\_z}{p\_z}} & = & z.
\end{eqnarray\*}
As you can see both $p\_x$ and $p\_y$ appear.

> **Comments:**
> **username_0**: Thank you and  excuse me, I was misleading again. Utility is assumed to be monotonic and concave.  
> **username_1**: The utility function in my counterexample is monotonic and concave. I see in your question you mean strictly concave.  
> **username_0**: Thank you very much. Yes, of course I meant strictly concave. I was totally wrong, starting from a cobb douglas function was a huge mistake; without paying attention to other possibilities.  

---

