must have a non-zero width. Moreover, since U(x,1)<U(z,1) whenever x<z, these intervals must be disjoint for all x. This is not possible for an uncountable set of x-values. If there are a finite number of goods, and amounts can only be rational numbers, utility functions do exist, simply by taking 1/N to be the size of the infinitesimal, where N is sufficiently large, to approximate nonstandard numbers. In terms of real valued utility, one would say that the utility of Y and Z is infinitesimal compared with X, and the utility of Z is infinitesimal compared to Y. Thus, lexicographic preferences can be represented by utility functions returning nonstandard real numbers. Equilibrium in economies with lexicographic preferences If all agents have the same lexicographic preferences, then general equilibrium cannot exist because agents won't sell to each other (as long as price of the less preferred is more than zero). But if the price of the less wanted is zero, then all agents want an infinite amount of the good. Equilibrium cannot be attained with standard prices. The utilities are infinitesimal, but the prices are not. Allowing infinitesimal prices resolves this. Lexicographic preferences can still exist with general equilibrium. For example, Different people have different bundles of lexicographic preferences such that different individuals value items in different orders. Some, but not all people have lexicographic preferences. Lexicographic preferences extend only to a certain quantity of the good. The nonstandard (infinitesimal) equilibrium prices for exchange can be determined for lexicographic order using standard equilibrium methods, except using nonstandard reals as the range of both utilities and prices. All the theorems regarding existence of prices and equilibria extend to the case of nonstandard utilities, since the nonstandard reals form a conservative extension, meaning that any theorem which is true for reals can be extended to the nonstandard reals and remains true. See also Lexicographic optimization Notes External links Lexicographic preference relation cannot be represented by a utility function. In Economics.SE Recognizing linear orders embeddable in R2 ordered lexicographically. In Math.SE. Lexicographic Multi-Objective Linear Programming using Grossone Methodology: Theory and Algorithm . In Appl. Math. and Computation Consumer theory Lexicography Utility function types de:Präferenzrelation#Lexikographische Präferenzordnung