# How can two different utility functions represent the same preferences?

I have this question for microecon that asks do the following utility functions represent the same preferences:

1. $$u(x_1, x_2) = x1 \cdot x2, \; v(x_1, x_2) = \ln x_1 + \ln x_2$$

2. $$u(x_1, x_2) = x1 \cdot x2, \; v(x_1, x_2) = x_1 + x_2$$

How do I go about approaching this problem since they are two different functions?

• Did you learn that utility is ordinal? Feb 11, 2019 at 2:16
• yes, I know that the ordering of the numbers matter which is why there are many utility functions that represent the same preferences. I'm just getting stuck on this specific one Feb 11, 2019 at 4:50

First of all, an utility function $$u: X \rightarrow \mathbb{R}$$ represents the preference relation $$\succsim$$ if: $$\forall a, b \in X, \; u(a) \geq u(b) \iff a \succsim b.$$
Well, if another function $$v: X \rightarrow \mathbb{R}$$ represents $$\succsim$$, then: $$\forall a, b \in X, \; v(a) \geq v(b) \iff a \succsim b.$$
$$\forall a, b \in X, \; u(a) \geq u(b) \iff v(a) \geq v(b).$$
So now I think you will be able to answer both questions. As a tip for the first one, think about the $$\ln(x)$$ function: is it a monotonic increasing function?