Monotonic Transformation

def. Monotonic Transformation. A transformation of a Utility Function is one that preserves the order of preferences of the original function. i.e., $f$ is a monotomic transformation of utility function $u(\vec{x})$ iff:

$$\forall (\overrightarrow{x^A},\overrightarrow{x^B})\text{, if }u(\overrightarrow{x^A})>u(\overrightarrow{x^B})⟹f\circ u(\overrightarrow{x^A})>f\circ u(\overrightarrow{x^B})$$

Simple monotonic transformations:

• Adding or subtracting a constant
• Multiplying or dividing by a positive constant
• For always positive functions, exponentiation to a constant
• For always positive functions, logarithm to a constant base

thm. monotonic transformation equivalence. If one utility function is a monotonic transformation of another, you may treat it for all practical purposes as the same utility function.