<p>One needs to go case-by-case and arrive at a utility function with branches. To get you started, if <span class="math-container">$x_1 &lt; x_2/2 \implies \min(2x_1,x_2) = 2x_1$</span>, but then also <span class="math-container">$\min(x_1,2x_2) = x_1$</span>. Therefore in this case, <span class="math-container">$u(x_1,x_2) = 2x_1$</span>. etc </p>

<p>There are two other intervals to consider as regards the relation between <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span>, so in all you will obtain a utility function with three branches. The middle one will be a <span class="math-container">$\max$</span> operator, and it can usefully be split in two, resulting in four branches.</p>

CC BY-SA 4.0
2019-03-09T15:39:37.770Z
Thanks for the response. So in the first case, u(x1,x2) = 2x1 because that's the larger of the 2 min values? I'm sorry if that's an obvious question, but this problem is a little beyond my current level of understanding.
Aude
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2019-03-09T15:43:00.100Z
@Aude Yes that's how the compact utility function expression works.
Alecos Papadopoulos
CC BY-SA 4.0
2019-03-09T15:23:53.947Z
2019-03-09T15:23:53.947Z
Alecos Papadopoulos
<p>Compute the optimal choice for a consumer with the following utility function:</p>

<p><span class="math-container">$$u(x_1, x_2) =\max \{\min(2x_1, x_2), \min(x_1, 2x_2)\}$$</span></p>

<p>I'm familiar with computing optimal choice for perfect complements but this has 2 expressions and I'm stumped with how to go about doing this. I'd also appreciate if anyone could explain what exactly is happening in a function like this.</p>

CC BY-SA 4.0
2019-03-09T23:40:12.087Z
To get some intuition, try plotting the level set for $u(x_1, x_2)=1$ and $u(x_1, x_2) = 2$.
Matthew Gunn
CC BY-SA 4.0
2022-12-04T07:26:18.633Z
Answer is posted here : https://qr.ae/pvDf9H
Amit
<microeconomics><utility><consumer-theory><consumption><leontief>
Optimal choice for a weird leontief function
Aude
# Optimal choice for a weird leontief function

(Asked by: username_0 on 2019-03-09)

Compute the optimal choice for a consumer with the following utility function:

$$u(x\_1, x\_2) =\max \{\min(2x\_1, x\_2), \min(x\_1, 2x\_2)\}$$

I'm familiar with computing optimal choice for perfect complements but this has 2 expressions and I'm stumped with how to go about doing this. I'd also appreciate if anyone could explain what exactly is happening in a function like this.

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

> **Comments:**
> **username_1**: To get some intuition, try plotting the level set for $u(x_1, x_2)=1$ and $u(x_1, x_2) = 2$.  
> **username_2**: Answer is posted here : https://qr.ae/pvDf9H  

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## 1 answers

### Answer by username_3 on 2019-03-09. Score: 2

One needs to go case-by-case and arrive at a utility function with branches. To get you started, if $x\_1 < x\_2/2 \implies \min(2x\_1,x\_2) = 2x\_1$, but then also $\min(x\_1,2x\_2) = x\_1$. Therefore in this case, $u(x\_1,x\_2) = 2x\_1$. etc

There are two other intervals to consider as regards the relation between $x\_1$ and $x\_2$, so in all you will obtain a utility function with three branches. The middle one will be a $\max$ operator, and it can usefully be split in two, resulting in four branches.

> **Comments:**
> **username_0**: Thanks for the response. So in the first case, u(x1,x2) = 2x1 because that's the larger of the 2 min values? I'm sorry if that's an obvious question, but this problem is a little beyond my current level of understanding.  
> **username_3**: @username_0 Yes that's how the compact utility function expression works.  

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