Seriously geeky

Mas-Collel, Whinston and Green provide an elegant proof of the fact that lexicographic preferences cannot be represented by a utility function. Here's an alternative proof that is more intuitive to present. I use ^ to represent superscripts and _ to represent subscripts, as in Latex.

Lexicographic preferences in R^2 are defined as follows: x is weakly preferred to y if and only if either x_1>y_1 or (x_1=y_1 and x_2>=y_2). In words, even the smallest gain in the first dimension outweights even the largest gain in the second dimension.

For a contradiction, suppose there is a utility function u:R^2->R representing these preferences. Consider the line segment from (1,1) to (2,1). Along this line segment, u is increasing and hence continuous at all but a countable set of points (for a proof see Rudin Theorem 4.30). Find a point x where u is continuous. Now consider any point y=(x_1,x_2+e) with e>0. As y is strictly preferred to x, u(y)>u(x). But as we approach x from the right, i.e. for some point z=(x_1+h,x_2) with h>0 small enough, by continuity u(z) approaches u(x) and hence u(z) < u(y). However, by lexicographic preferences, z is strictly preferred to y as it is higher on the first dimension. This contradicts the premise that u represents preferences. QED.