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I have successfully done the UMP for perfect complements. I got

$$x^* = y^* = \frac{m}{p_x +p_y}$$ This makes intuitive sense because for whichever good I have the least of, I don't want to buy more units of the other. So the total amount I should spend should be the optimum quantity times $p_x + p_y$.

However, I am completely lost about how to do the EMP version of this. I keep going in circles.

I have done the Lagrangian and gotten for $x > y$, $e = p_y \bar{U}$ and for $x <y$ that $e = p_x \bar{U}$.

Here is my work:

https://math.stackexchange.com/questions/1252173/can-one-optimize-a-function-with-minx-y-as-a-constraint/1252341#1252341

But this doesn't make sense to me. don't see how this relates to the solution I got for the UMP which did make sense.

What I am misunderstanding about the EMP that is leading to these nonsensical results that don't match what I did for the UMP?

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You already got your answer in the math forum, from the economics point of view maybe it could help that you can use the relation between the expenditure function and the indirect utility to get your solution. Your indirect utility of the UMP is $v(p,w)=\frac{m}{p_x+p_y}$, this provides the expenditure function for the EMP $e(p,\overline{u})=\overline{u}\ (p_x+p_y)$. Now the hicksian demand is $h(p,u)=(u,u)$. This is the case of course of the utility $u(x,y)=min(x,y)$.

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