Consumption Set and Preference Ordering

As discussed in the note for competitive equilibrium, the maximizing behavior of a consumer can only be modeled in relative ordinal measures, i.e. through preference orderings. A convenient way of storing this ordinal information would be to map it into a cardinal utility measure, although such a transforming function, if to be more efficient than remembering the orders directly, thus usually preferred to be continuous, can only serve a certain type of preference assigning scheme. An exception would be the lexicographic ordering, and the essence is that in such an ordering scheme, the economic agent assigns preferences at disjoint steps, and while the orders can still be stored in an arbitrary utility function, the irregular discontinuity points makes such a transformation saving no much in storing the information.

In a sense, a very strong sense, the preference ordering itself is a function.

Most of the specifications or assumptions about the preference orderings, such as local nonsatiation or convexity, really should be understood as normal assumptions about functions.