Utility Maximization with a transformed min function

Question

I was just wondering what the steps one would take to maximize the utility of a function of the form U(X,Y) = min{X,Y} + X subject to income I = $p_x$X + $p_y$Y where $p_x$ is the price of X and $p_y$ is the price of Y. I understand that normally this would be solved using the Lagrangian method but obviously you cannot differentiate this particular function.

Thanks.

Answer 1

I would make a case decisions. Find out, what happen in all these cases.

1) $\text{max} \ \ U(X,Y)=Y+X$

s.t. $Y \leq X$ and $I=p_xX+p_yY$

a) $p_x \leq p_y$

b) $p_x > p_y$

2) $\text{max}\ \ U(X,Y)=2X$

s.t. $Y \geq X$ and $I=p_xX+p_yY$

a) $p_x \leq p_y$

b) $p_x > p_y$