Question

Consider the utility function u = [min fx1; x1 + x2g] 2 . Does this utility function represent the same preferences as the utility function u = x1?

Answer 1

Answer: Yes the utility function represents the same preferences as the utility function U=X1.

Explanation: Here since this is a min function, that value of Xi will be chosen which is minimum out of the two given. Now there can be three cases here:

Case 1: If X1=X1+X2. Here in this case the final utility would become 0. So we don't consider this case.

Case 2: If X1>X1+X2. Here if this is the case then even if X2 is zero the function will be min(X1,X1) so the utility that we get here will be X1.

Case 3: If X1<X1+X2. Here in this case since X1 is already less than X1+X2, the utility will be X1.

So this utility function is actually a monotonic transformation of the utility function U= X1.