# Optimal choice for a weird leontief function

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.

• 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 Mar 9 '19 at 23:40

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.

• 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 Mar 9 '19 at 15:39
• @Aude Yes that's how the compact utility function expression works. – Alecos Papadopoulos Mar 9 '19 at 15:43