<p>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*} </p>

<p>When one plot the indifference curves, this is how they will look:</p>

<p><a href="https://i.sstatic.net/42Bz5.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/42Bz5.png" alt="Indifference Curves"></a></p>

<p>Here $\mu_1 &gt; \mu_2 &gt; \mu_3$.</p>

CC BY-SA 3.0
2017-01-23T09:32:02.007Z
2017-01-23T09:32:02.007Z
Amit
<p>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 $.</p>

<p>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)</p>

CC BY-SA 3.0
2016-03-23T04:01:57.397Z
Possible duplicate of [Finding demand function given a utility min(x,y) function](http://economics.stackexchange.com/questions/2969/finding-demand-function-given-a-utility-minx-y-function)
HRSE
CC BY-SA 3.0
2016-03-23T04:13:23.637Z
@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) .
earthboy
CC BY-SA 3.0
2016-03-23T10:27:37.737Z
@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.
Giskard
CC BY-SA 3.0
2016-03-23T14:24:05.810Z
@denesp $ U= ax+ y $when $x>y$ and $U=ay+x$ when $y>x$ . So i just maximize each cases individually wrt the budget?
earthboy
CC BY-SA 3.0
2017-04-15T07:11:38.070Z
You may benefit from watching this video: https://www.youtube.com/watch?v=C1JOiykokNM&index=8&list=PLUJGfL_499TJz9hUatBtUmz0vFMux2tTV
Amit
<microeconomics><mathematical-economics><utility><leontief>
maximizing leontief-type utility function
earthboy
# maximizing leontief-type utility function

(Asked by: username_0 on 2016-03-23)

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)

**Tags:** microeconomics, mathematical-economics, utility, leontief

> **Comments:**
> **username_1**: Possible duplicate of [Finding demand function given a utility min(x,y) function](http://economics.stackexchange.com/questions/2969/finding-demand-function-given-a-utility-minx-y-function)  
> **username_0**: @username_1 I think the given utility function is a little different from the usual leontief min(x,y) function (or I think so at least) .  
> **username_2**: @username_0 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.  
> **username_0**: @denesp $ U= ax+ y $when $x>y$ and $U=ay+x$ when $y>x$ . So i just maximize each cases individually wrt the budget?  
> **username_3**: You may benefit from watching this video: https://www.youtube.com/watch?v=C1JOiykokNM&index=8&list=PLUJGfL_499TJz9hUatBtUmz0vFMux2tTV  

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## 1 answers

### Answer by username_3 on 2017-01-23. Score: 4

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](https://i.sstatic.net/42Bz5.png)](https://i.sstatic.net/42Bz5.png)

Here $\mu\_1 > \mu\_2 > \mu\_3$.


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