The following section from DiscussEconomics on microeconomics and preferences discusses the mathematical representation of preference using utility functions.

`Using `

*utility function *: U(x) = U (x1, x2, x3.......xn)

(Where U is in fact mu.)

This assigns a number (utility number to every consumption bundle in a person’s preference ordering. 1. Now if someone is indifferent between two bundles, the U function assigns the **same **number to both bundles:

2. Now if you prefer 1 bundle to another, the utility function assigns a **larger** number to the preferred bundle:

Many (any number) of utility functions can be constructed to represent the same preference ordering. They are not unique except with respect to rank.

*some U (x1, x2) exists for any preferences that satisfy the following: 1 completeness, 2 continuity, 3 transitivity, 4 non-satiation.

Also remember that utility numbers are ordinal, no cardinal in that only relative rankings matter.

**Here is an example I’ve written, hopefully you don’t have trouble following:**

With any transformation which preserves the ranking of a utility function, the new utility function is legitimate.

**Monotomic Transformation**

If function ‘f’ has the property that the larger the number you enter, the larger the number that comes out (monotomicaly increases) then the new functions V(x) is also a utility function.

**Therefore:** (Where B1 is bundle 1)

Utility numbers can be negative, however, it still must rank correctly and prefer. So end the introduction to utility functions (coupled with indifference curves).