Utility function with max and min

I just want to make sure i got the intuition behind the crazy utility function: u = max(ax, ay) + min(x,y).

Because utility = max(ax, ay) + min(x,y), this means the consumer is able to substitute between two goods, and he is able to maximize one of them, consuming the less quantity possible of the other.  because this is a leontieff, you also know that  ax=ay, or x=y. For this homotetic function, we need to consider several cases.

One of them is px/py = a. If the consumer is indifferent among all choices such that x > y, then you need to consider the ranges such that y=0 (best scenario until the case y=x (worst scenario). when y=0 then he only consumes x, the maximum level of utility he can get is u=ax. To get the Hickisian demand, you know that at the optimum x=h. Then,  h=u/a . When x=y, he will consume the minimum amount of y, and the maximum of x. That is u= ax +y. But you know that y=x then u= x(a+1), rearranging a little bit, h= u/ (a+1).

Am I in the right track? thanks!

I think you’re on the right track. One way to think of the utility function is as follows:

u(x,y) =
ax + y if x >= y
x + ay if y >= x

(related to Leontieff, but not quite Leontieff)

we know that  0<a<1. I would recommend drawing what the indifference curves look like for this thing. If you look at the shape of the indifference curves, and think of where utility would be maximized for different budget sets, you will see that there are three cases (plus the two boundary cases between the three):

0 < p_x / p_y < a
a < p_x / p_y < 1/a
1/a < p_x / p_y

Overall, I would reccommend a graphical argument on this problem, rather than trying to solve Lagrangians or anything. You can check your answers against intuition by noting that a=0 corresponds to Leontieff [u(x,y) = min(x,y)] and a = 1 corresponds to perfect substitutes [u(x,y) = x+y]

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