Let X = R 2 +. The lexicographic preference relation L is defined as x L y if either x1 > y1 or x1 = y1 and x2 y2. The name derives from the way a dictionary is organized; that is commodity 1 has the highest priority in determining the preference ordering, just as the first letter of a word does in the ordering of a dictionary. When the level of commodity is the same in two bundles, the amount of commodity 2 comes into play. (1) Verify that L is complete, transitive, and monotone. (2) For each x R 2 +, determine the indifference set of x, I(x) = {y R 2 + : y x}. (3) Claim: If a utility representation for L exists, then we can associate a number nx R (a utility value) to each indifference set I(x) such that nx 6= ny if I(x) 6= I(y) (two distinct indifference sets must give different utility values). Do you think that lexicographic preferences are representable by a utility function? Why? (4) Consider the statement from 5.2. Do you think the converse is true? In other words, are all complete, transitive, and reflexive preferences representable by a utility function? What if they are also monotone?