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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.

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  • $\begingroup$ To get some intuition, try plotting the level set for $u(x_1, x_2)=1$ and $u(x_1, x_2) = 2$. $\endgroup$
    – Matthew Gunn
    Mar 9, 2019 at 23:40
  • $\begingroup$ Answer is posted here : qr.ae/pvDf9H $\endgroup$
    – Amit
    Dec 4, 2022 at 7:26

1 Answer 1

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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.

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  • $\begingroup$ 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. $\endgroup$
    – Aude
    Mar 9, 2019 at 15:39
  • $\begingroup$ @Aude Yes that's how the compact utility function expression works. $\endgroup$
    – Alecos Papadopoulos
    Mar 9, 2019 at 15:43

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