<p><span class="math-container">$u(x_1,x_2)=\min(x_1,x_2)+5\max(x_1,x_2)=\max(x_1+5x_2,5x_1+x_2)$</span>
which is not quasi-concave, but is an increasing quasi-convex function. So, demand will be at one of the two end-points of the budget line. Comparing the utilities at the two end-points, we get the demand as follows:</p>
<p><span class="math-container">\begin{eqnarray*}(x_1^d,x_2^d)(p_1,p_2,w)\in\begin{cases}\left\{\left(\frac{w}{p_1},0\right),\left(0,\frac{w}{p_2}\right)\right\} &amp; \text{if } p_1=p_2 \\ \left\{\left(\frac{w}{p_1},0\right)\right\} &amp; \text{if } p_1&lt;p_2 \\ \left\{\left(0,\frac{w}{p_2}\right)\right\} &amp; \text{if } p_1&gt;p_2 \end{cases}\end{eqnarray*}</span></p>
<p><a href="https://i.sstatic.net/oT3iuWdA.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/oT3iuWdA.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2024-07-16T10:59:09.597Z
2024-07-17T04:34:09.610Z
2024-07-17T04:34:09.610Z
Amit
<p>By simplifying the utility function as @walrasian_auctioneer indicated, we have <span class="math-container">$u(x_1, x_2) = x_1 + x_2 + 4 \max \{x_1, x_2\}$</span>. Easy to find that both <span class="math-container">$(x_1 +x_2)$</span> and <span class="math-container">$\max \{x_1, x_2\}$</span> has the same condition for maximization, that is, depends on <span class="math-container">$p_1 &gt; p_2$</span> or <span class="math-container">$p_1 \neq p_2$</span> or <span class="math-container">$p_1 &lt; p_2$</span>.</p>

CC BY-SA 4.0
2024-11-13T02:20:48.410Z
Now a new problem is: what is the Walrasian demand for utility $u(x_1, x_2) = 5 \min\{x_1, x_2\} + \max \{x_1, x_2\}$.
Xuhang
CC BY-SA 4.0
2024-11-13T02:14:35.277Z
2024-11-13T02:17:07.563Z
2024-11-13T02:17:07.563Z
Xuhang
<p>Why don't you just plug in some values for <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span>?
Start with something like</p>
<p><span class="math-container">$x_1=1, x_2=1$</span> and find utility <span class="math-container">$u(1,1)=1+5*1$</span>.</p>
<p>Then increase <span class="math-container">$x_1$</span> or <span class="math-container">$x_2$</span> and let</p>
<p><span class="math-container">$x_1=2, x_2=1$</span> and find utility <span class="math-container">$u(2,1)=1+5*2=11$</span>, and</p>
<p><span class="math-container">$x_1=1, x_2=2$</span> and find utility <span class="math-container">$u(1,2)=1+5*2=11$</span>.</p>
<p>The utilities of both bundles are identical, and the goods seem to be substitutable.
In fact for any <span class="math-container">$x_1&lt;x_2$</span> you will find
<span class="math-container">$u(x_1,x_2) = x_1 + 5 x_2$</span>, and, similarly,
for any <span class="math-container">$x_1&gt;x_2$</span> you will find
<span class="math-container">$u(x_1,x_2) = x_2 + 5 x_1$</span>.</p>
<p>Now, just for now, assume both goods have the same price <span class="math-container">$p_1=p_2=1$</span> and you have a budget of <span class="math-container">$w=3$</span> such that you can buy both bundles giving utility 11 up there.
You can see that it would be better to put all your money in one of the goods and get
<span class="math-container">$u(3,0)=u(0,3) = 0 + 5*3=15$</span>. Now think about different prices and a general income <span class="math-container">$w$</span>. Does it again make sense to buy only one of the goods? Is it the cheaper one?</p>

CC BY-SA 4.0
2020-12-01T12:32:04.683Z
2020-12-01T12:32:04.683Z
Bayesian
<p>A consumer has the utility function <span class="math-container">$u(x_1; x_2) = \min(x_1; x_2) + 5 \max(x_1; x_2)$</span>.
Find its Walrasian demand <span class="math-container">$x^*(p; w)$</span>.</p>
<p>I've tried searching it up when we have two Leontief functions summed together but to no luck. If this was a simple case we'd just have <span class="math-container">$x_1=x_2$</span> and substitute it to the budget constraint to find the demand for both. Should we take each function separately or as a whole together? Kinda lost here.</p>

CC BY-SA 4.0
2020-11-30T11:41:59.960Z
I understand that the indifference curves together would sort of form rectangle shapes but honestly even with the hints I'm kinda lost. It's my first time seeing something like this and don't know where to start. If you could give one more hint I feel like I'd be able to start working on it.
Ana Ellis
CC BY-SA 4.0
2020-11-30T12:24:21.753Z
Can you draw a single indifference curve?
Michael Greinecker
CC BY-SA 4.0
2020-11-30T14:01:28.393Z
Is the solution supposed to be something like this? https://drive.google.com/file/d/1jQD67U8orP0oFmvqRL511vbSNv9TTRPn/view?usp=sharing
Ana Ellis
CC BY-SA 4.0
2020-11-30T20:36:18.960Z
@AnaEllis No. If you understand why indifferences for Leontief-utility look the way they do, it shouldn't be hard to do the same for $\max$.
Michael Greinecker
CC BY-SA 4.0
2020-11-29T20:47:52.667Z
You could start by drawing indifference curves for the utility function given by $v(x_1,x_2)=\max\{x_1,x_2\}$. This might give you the crucial insight to solve your problem.
Michael Greinecker
CC BY-SA 4.0
2020-11-29T21:02:38.307Z
Hint: $\min \{x,y\} + \max \{x,y\} = x + y$.
Walrasian Auctioneer
<microeconomics><utility><demand><walrasian><leontief>
Walrasian demand with a twist of Leontief function
Ana Ellis
# Walrasian demand with a twist of Leontief function

(Asked by: username_0 on 2020-11-29)

A consumer has the utility function $u(x\_1; x\_2) = \min(x\_1; x\_2) + 5 \max(x\_1; x\_2)$.
Find its username_2 demand $x^\*(p; w)$.

I've tried searching it up when we have two Leontief functions summed together but to no luck. If this was a simple case we'd just have $x\_1=x\_2$ and substitute it to the budget constraint to find the demand for both. Should we take each function separately or as a whole together? Kinda lost here.

**Tags:** microeconomics, utility, demand, walrasian, leontief

> **Comments:**
> **username_1**: You could start by drawing indifference curves for the utility function given by $v(x_1,x_2)=\max\{x_1,x_2\}$. This might give you the crucial insight to solve your problem.  
> **username_2**: Hint: $\min \{x,y\} + \max \{x,y\} = x + y$.  
> **username_0**: I understand that the indifference curves together would sort of form rectangle shapes but honestly even with the hints I'm kinda lost. It's my first time seeing something like this and don't know where to start. If you could give one more hint I feel like I'd be able to start working on it.  
> **username_1**: Can you draw a single indifference curve?  
> **username_0**: Is the solution supposed to be something like this? https://drive.google.com/file/d/1jQD67U8orP0oFmvqRL511vbSNv9TTRPn/view?usp=sharing  
> **username_1**: @username_0 No. If you understand why indifferences for Leontief-utility look the way they do, it shouldn't be hard to do the same for $\max$.  

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## 3 answers

### Answer by username_5 on 2020-12-01. Score: 4 | Accepted answer

Why don't you just plug in some values for $x\_1$ and $x\_2$?
Start with something like

$x\_1=1, x\_2=1$ and find utility $u(1,1)=1+5\*1$.

Then increase $x\_1$ or $x\_2$ and let

$x\_1=2, x\_2=1$ and find utility $u(2,1)=1+5\*2=11$, and

$x\_1=1, x\_2=2$ and find utility $u(1,2)=1+5\*2=11$.

The utilities of both bundles are identical, and the goods seem to be substitutable.
In fact for any $x\_1<x\_2$ you will find
$u(x\_1,x\_2) = x\_1 + 5 x\_2$, and, similarly,
for any $x\_1>x\_2$ you will find
$u(x\_1,x\_2) = x\_2 + 5 x\_1$.

Now, just for now, assume both goods have the same price $p\_1=p\_2=1$ and you have a budget of $w=3$ such that you can buy both bundles giving utility 11 up there.
You can see that it would be better to put all your money in one of the goods and get
$u(3,0)=u(0,3) = 0 + 5\*3=15$. Now think about different prices and a general income $w$. Does it again make sense to buy only one of the goods? Is it the cheaper one?


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### Answer by username_3 on 2024-07-16. Score: 1

$u(x\_1,x\_2)=\min(x\_1,x\_2)+5\max(x\_1,x\_2)=\max(x\_1+5x\_2,5x\_1+x\_2)$
which is not quasi-concave, but is an increasing quasi-convex function. So, demand will be at one of the two end-points of the budget line. Comparing the utilities at the two end-points, we get the demand as follows:

\begin{eqnarray\*}(x\_1^d,x\_2^d)(p\_1,p\_2,w)\in\begin{cases}\left\{\left(\frac{w}{p\_1},0\right),\left(0,\frac{w}{p\_2}\right)\right\} & \text{if } p\_1=p\_2 \\ \left\{\left(\frac{w}{p\_1},0\right)\right\} & \text{if } p\_1<p\_2 \\ \left\{\left(0,\frac{w}{p\_2}\right)\right\} & \text{if } p\_1>p\_2 \end{cases}\end{eqnarray\*}

[![enter image description here](https://i.sstatic.net/oT3iuWdA.png)](https://i.sstatic.net/oT3iuWdA.png)


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### Answer by username_4 on 2024-11-13. Score: 0

By simplifying the utility function as @username_2\_auctioneer indicated, we have $u(x\_1, x\_2) = x\_1 + x\_2 + 4 \max \{x\_1, x\_2\}$. Easy to find that both $(x\_1 +x\_2)$ and $\max \{x\_1, x\_2\}$ has the same condition for maximization, that is, depends on $p\_1 > p\_2$ or $p\_1 \neq p\_2$ or $p\_1 < p\_2$.

> **Comments:**
> **username_4**: Now a new problem is: what is the username_2 demand for utility $u(x_1, x_2) = 5 \min\{x_1, x_2\} + \max \{x_1, x_2\}$.  

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