Minimum Utility Function

[Devika]

U (x,y) = min {2x+1, x+y,2y+1}

px=py=1 and m=2

Find the optimal bundles.

[Mauli]

hi devika:)
do u have the answer for this.?
i am getting something ,just want to confirm b4 sharing my approach.

[Devika]

Hi Mauli :)

The answer is x = [ 0.5, 1.5]
and y= [ 1.5, 0.5]

[Mauli]

this is what i am getting:
the consumer can consume anywhere on his budget line that is:
 p1x'+p2y'=m


  ie: x'+y'=2

[Mauli]

devika dont u think x,y =1,1
should also satisfy?

[Mauli]

or any other point on the budget line.

[Devika]

What was your approach mauli?

I plotted the graph, but no good came out of that.
What did you do?

[Mauli]

[Mauli]

i know its a very small figure.
sorry about that.
 see this.

[Mauli]

since px/py =1 it coincides with x+y.

[Mauli]

ps:i made the graph and got the required region by fixing utilities at different levels.
i)u>1
ii)0<u<1

[Devika]

Uff. I was drawing the graph incorrectly.
So here, the budget line coincides with the the x+y. So all points on that part are optimal.

And mauli, yes 1,1 is optimal. The answers given are intervals.

Thank you so much! :)

[MI]

If you plot the indifference curve

let say for 2 utils

2x +1 = 2 => x=0.5 and Y can take any values
2y +1 = 2 => y=0.5 and X can take any values

In between x+y =2

=> x and y = [0.5, 1.5]

[Mauli]

oh i thought you have written uthe optimal bundles in (x,y) form.

[Devika]

Thank you mauli and MI.

[Sumit]

Optimal bundle set={(x,y): x+y=2 where x belongs to [0.5,1.5] & y belongs to [0.5,1.5]}..

[riddhima]

Hey! I didn't draw any indifference curve to solve this question. I simply solved it the way we solve a min function.
2x+1= x+y = 2y + 1

2x+1=x+y gives us, x= y - 1
 putting in the budget equation i.e x + y =2
y-1 + y =2
solving which gives us y = 1.5 and x= .5

similarly, for x+y= 2y + 1
y = x-1
solving the budget equation for this gave , x= 1.5 and y= .5

Just correct me if I'm wrong in this approach!

Thanks!

[duck]

Hey Ridhimma.. :)

There are other bundles as well which are optimal apart from just these two bundles.
[As Sumit pointed out]
The approach that you used gives all optimal bundles when the utility function is of the following form: U(x,y)= min{ax,by} where a>0 , b>0.
In all other questions of "min" utility function, its good to plot IC and then check for optimum bundles.