Question

Exercise 4. Let ≥ be a rational, monotone and continuous preference on R2 +. Suppose that ≥ is such that there is at least one strict preference statement, i.e. there exists two bundles x and y in R2 + such that x＞y.

(a) Is there a discontinuous utility representation of ≥ ? Justify your answer.

(b) Would your answer change if if we didn’t have at least one strict preference statement? Justify your answer.

I don't know where my professor brought this question, but our class is based on the 'intermediate microeconomics with calculus by Varian'.

I already checked out the solutions of this textbook posted on this website, and i figured out that my professor does not directly bring exercises from this textbook.

Answer 1

(a) yes, there exist one utility function representation for this kind of preferences. for existence of discontinuous utility representation, we can show that for these two bundles x anf y, can be assigned utility x1 and y1. here x1 > y1. any x1 that are weakly prefered over x will be assigned utility value greater than or equals to x1 accordingly. for all those bundles that y weakly prefer, we can assigne utility value less than or equal to y1 accordingly. anybundle that are strongly predered over y but weakly prefered by x can be assigned utility value inbetween x1 and y1. but we just have to rnsure that x1 must be sufficiently large to ensure discontunity.  

(b) yes, then we will not be able to construct any utility function that represents this preference and is discontinious. because any utility function that represents this preference has to take a single value but then it is not discountinious.