<p>Let <span class="math-container">$\min\{x,z\}=\Omega$</span>, where <span class="math-container">$P_\Omega=P_x+P_z$</span>. Now the problem becomes <span class="math-container">$U(y,\Omega)=y\Omega$</span>, which is a standard Cobb-Douglas with degree 2 of homogeneity. Now in this case the choice for each good is:</p>
<p><span class="math-container">$y^*=\frac{\alpha_y100}{P_y(\alpha_y+\alpha_\Omega)}\implies y^*=\frac{100}{2P_y}\;\;\;\;\;\;\;$</span>in this case <span class="math-container">$\alpha_y=\alpha_\Omega=1$</span></p>
<p>For <span class="math-container">$\Omega$</span>: <span class="math-container">$\;\;\;\;\;\Omega^*=\frac{\alpha_\Omega100}{P_\Omega(\alpha_y+\alpha_\Omega)}\implies \Omega^*=\frac{100}{2P_\Omega} \implies \Omega^*=\frac{100}{2(P_x+P_z)}$</span></p>
<p>Now, since <span class="math-container">$P_x=P_y=P_z$</span>, let <span class="math-container">$P_x=P_y=P_z=P$</span> a general price, therefore substituting in our optimums:</p>
<p><span class="math-container">$y^*=\frac{100}{2P}\;\;\;\;\;\Omega^*=\frac{100}{4P}$</span></p>
<p>Now it's straightforward (since we already know <span class="math-container">$x^*=z^*$</span> and as this is the optimum for <span class="math-container">$\min\{x,z\}$</span> which is <span class="math-container">$x$</span> OR <span class="math-container">$z$</span>) that <span class="math-container">$y^*=\frac{100}{2P}&gt;\Omega^*=\frac{100}{4P}$</span>, so this implies that:</p>
<p><span class="math-container">$y^*&gt;x^*=z^*\;\;$</span>****</p>
<p>Also I found <a href="http://econdse.org/wp-content/uploads/2014/03/Answer-Key-Economics-MA-Entrance-2017.pdf" rel="nofollow noreferrer">this</a> document, where this question is number 13.</p>
<p>Hope this helps.</p>
<p>Disclaimer: It would be helpful if other people could assess this approximation, since I hadn't never seen this problem before.</p>

CC BY-SA 4.0
2020-07-12T22:37:51.793Z
Sorry, I made a little mistake with notation (I assumed it was =min{,}), but the conclusion is the same. I will correct those notation mistakes if it makes things more clear, please let me know. The answer should be $y>x=z$
manifold
CC BY-SA 4.0
2020-07-14T18:16:20.527Z
I think "$U(x,\Omega)=y\Omega$" should be "$U(y,\Omega)=y\Omega$."
MrAP
CC BY-SA 4.0
2020-07-14T18:17:35.727Z
@MrAP that's correct! I'll make the changes.
manifold
CC BY-SA 4.0
2020-07-12T20:50:46.823Z
2020-07-14T18:18:16.147Z
2020-07-14T18:18:16.147Z
manifold
<p>Hint:  Suppose the price of the goods is <span class="math-container">$P$</span> so that <span class="math-container">$N=100/P$</span> goods can be afforded in total.  Now consider which of the following yields more utility:</p>
<p>a) <span class="math-container">$x=y=z=N/3$</span>.</p>
<p>b) <span class="math-container">$x=z=N/4$</span> and <span class="math-container">$y=N/2$</span>.</p>

CC BY-SA 4.0
2020-07-12T21:29:13.930Z
2020-07-12T21:29:13.930Z
Adam Bailey
<p><strong>Question</strong>: Consider a consumer with utility function <span class="math-container">$U(x,y,z)=y\min\{x,z\}$</span>. The prices of all three goods are the same. The consumer has $100 to spend on these three goods.The demands will be such that:</p>
<p>(a) <span class="math-container">$y&lt;x=z$</span></p>
<p>(b) <span class="math-container">$y&gt;x=z$</span></p>
<p>(c) <span class="math-container">$x=y=z$</span></p>
<p>(d) None of the above</p>
<p><strong>My attempt</strong>: The consumer will consume equal amounts of <span class="math-container">$x$</span> and <span class="math-container">$z$</span> because otherwise the allocation would be inefficient, that is, he can obtain the same level of utility by spending less. So <span class="math-container">$x=z$</span>. I cannot figure out how is <span class="math-container">$y$</span> related to <span class="math-container">$x$</span> and <span class="math-container">$z$</span>. I think the answer would be (d) None of the above because it does not matter if <span class="math-container">$y$</span> is less than or greater than or equal to <span class="math-container">$x$</span> and <span class="math-container">$z$</span>.</p>

<microeconomics><utility><consumer-theory><consumption>
Relation between demands of $x, y$ and $z$
MrAP
# Relation between demands of $x, y$ and $z$

(Asked by: username_0 on 2020-07-12)

**Question**: Consider a consumer with utility function $U(x,y,z)=y\min\{x,z\}$. The prices of all three goods are the same. The consumer has $100 to spend on these three goods.The demands will be such that:

(a) $y<x=z$

(b) $y>x=z$

(c) $x=y=z$

(d) None of the above

**My attempt**: The consumer will consume equal amounts of $x$ and $z$ because otherwise the allocation would be inefficient, that is, he can obtain the same level of utility by spending less. So $x=z$. I cannot figure out how is $y$ related to $x$ and $z$. I think the answer would be (d) None of the above because it does not matter if $y$ is less than or greater than or equal to $x$ and $z$.

**Tags:** microeconomics, utility, consumer-theory, consumption


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## 2 answers

### Answer by username_1 on 2020-07-12. Score: 2 | Accepted answer

Let $\min\{x,z\}=\Omega$, where $P\_\Omega=P\_x+P\_z$. Now the problem becomes $U(y,\Omega)=y\Omega$, which is a standard Cobb-Douglas with degree 2 of homogeneity. Now in this case the choice for each good is:

$y^\*=\frac{\alpha\_y100}{P\_y(\alpha\_y+\alpha\_\Omega)}\implies y^\*=\frac{100}{2P\_y}\;\;\;\;\;\;\;$in this case $\alpha\_y=\alpha\_\Omega=1$

For $\Omega$: $\;\;\;\;\;\Omega^\*=\frac{\alpha\_\Omega100}{P\_\Omega(\alpha\_y+\alpha\_\Omega)}\implies \Omega^\*=\frac{100}{2P\_\Omega} \implies \Omega^\*=\frac{100}{2(P\_x+P\_z)}$

Now, since $P\_x=P\_y=P\_z$, let $P\_x=P\_y=P\_z=P$ a general price, therefore substituting in our optimums:

$y^\*=\frac{100}{2P}\;\;\;\;\;\Omega^\*=\frac{100}{4P}$

Now it's straightforward (since we already know $x^\*=z^\*$ and as this is the optimum for $\min\{x,z\}$ which is $x$ OR $z$) that $y^\*=\frac{100}{2P}>\Omega^\*=\frac{100}{4P}$, so this implies that:

$y^\*>x^\*=z^\*\;\;$\*\*\*\*

Also I found [this](http://econdse.org/wp-content/uploads/2014/03/Answer-Key-Economics-MA-Entrance-2017.pdf) document, where this question is number 13.

Hope this helps.

Disclaimer: It would be helpful if other people could assess this approximation, since I hadn't never seen this problem before.

> **Comments:**
> **username_1**: Sorry, I made a little mistake with notation (I assumed it was =min{,}), but the conclusion is the same. I will correct those notation mistakes if it makes things more clear, please let me know. The answer should be $y>x=z$  
> **username_0**: I think "$U(x,\Omega)=y\Omega$" should be "$U(y,\Omega)=y\Omega$."  
> **username_1**: @username_0 that's correct! I'll make the changes.  

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### Answer by username_2 on 2020-07-12. Score: 2

Hint: Suppose the price of the goods is $P$ so that $N=100/P$ goods can be afforded in total. Now consider which of the following yields more utility:

a) $x=y=z=N/3$.

b) $x=z=N/4$ and $y=N/2$.


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