Can anyone help with this utility function?

Question

Ok so preferences are captured in this utility function

U(x1,x2) = min(2x1,7x2)

From this an indifference curve must be drawn. Can anyone get me started? Or better yet show me how to do it?

Answer 1

Well, the min function means, when you look at a pair of x1/x2 (let's say x1 = 4, x2 = 8), then the min function gives a value of min(2*4,7*8).  The minimum of (8 or 56) is 8, so the utility function has a value of 8 at this point.  

An indifference curve tells you which amounts of x1 and x2 will make you as happy (that is, give you a value of 8 when you do the above calculation).  

As a hint, utility functions that have min functions in them are called Leontief preferences.  Another hint is that these indifference curves are strange: there will be a perfectly vertical part, and a perfectly horizontal part.

Let's look at the example again to see why.  What other values for x1 and x2 will give a result of 8?  Well, if x1 is 4, and x2 is 4, then the min function is still 8 (you can check that).  Actually, the min function gives 8 if x1 is 4 and x2 is anything above 8/7.  So, there will be a vertical line at x1 = 4 for all values of x2 above 8/7.  

If you do the min function for x1 = 4, and x2 = 8/7, then the result is min (2*4, 7*(8/7)) which gives min (8,8) which is 8.  But if x2 is lower than 8/7, then the min will be lower and this can no longer be on the same indifference curve.  

However, x2 can be 8/7 while x1 becomes bigger.  For example, if x2 is 8/7 and x1 is 8 [min(2*8, 7*(8/7)) = min (16, 8) = 8].  

Therefore, there will be a horizontal line at x2 = 8/7 for all values of x1 great than 4.  

You should probably draw a couple of these.  You can follow this example for a couple of different values besides 8, or you can just use the fact that the slope of the line going through the points on these Leontief indifference curves, where the min function contains the same numbers (like min (8,8), is 3.5 (you can get this from the 7 and 2 in the min function, because 7/2 is 3.5).