maximizing leontief-type utility function How do I maximize the utility function:
$ U(x,y)= max(ax,ay)+min(x,y) $ , where $ 0&lt;a&lt;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)
You can re write $u(x, y)$ as
\begin{eqnarray*} u(x, y) = \begin{cases} ax + y &amp; \text{ if } x &gt; y\\ ay + x &amp; \text{ if } x \leq y\end{cases} \end{eqnarray*} 

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



Here $\mu_1 &gt; \mu_2 &gt; \mu_3$.