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How do I maximize the utility function: $ U(x,y)= max(ax,ay)+min(x,y) $ , where $ 0<a<1 $ with respect to prices $ p_{x}, p_{y} $ respectively and income $ m $.

I know leontief-type utility functions are solved by graphing and not by using Lagrangians, buy how do I graph this function?. ( I know there would be kinks on the graph, but how do I find those kinks)

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    $\begingroup$ Possible duplicate of Finding demand function given a utility min(x,y) function $\endgroup$
    – HRSE
    Commented Mar 23, 2016 at 4:01
  • $\begingroup$ @HRSE I think the given utility function is a little different from the usual leontief min(x,y) function (or I think so at least) . $\endgroup$
    – earthboy
    Commented Mar 23, 2016 at 4:13
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    $\begingroup$ @earthboy It is a trick question. All that the $\min$ and $\max$ operators hide is that the functions have two branches, one where one argument is bigger, the other where the other argument is bigger. (You can slap the even case to either branch.) So in this case try to see what happens if $x > y$, if $x <y$, etc. If you do it right your utility function should become much much more simple. $\endgroup$
    – Giskard
    Commented Mar 23, 2016 at 10:27
  • $\begingroup$ @denesp $ U= ax+ y $when $x>y$ and $U=ay+x$ when $y>x$ . So i just maximize each cases individually wrt the budget? $\endgroup$
    – earthboy
    Commented Mar 23, 2016 at 14:24
  • $\begingroup$ You may benefit from watching this video: youtube.com/… $\endgroup$
    – Amit
    Commented Apr 15, 2017 at 7:11

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You can re write $u(x, y)$ as \begin{eqnarray*} u(x, y) = \begin{cases} ax + y & \text{ if } x > y\\ ay + x & \text{ if } x \leq y\end{cases} \end{eqnarray*}

When one plot the indifference curves, this is how they will look:

Indifference Curves

Here $\mu_1 > \mu_2 > \mu_3$.

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