# 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? – Herr K. Feb 11 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 – Abiola Feb 11 at 4:50
• Read the article linked in my previous comment. The second sentence therein answers your question. – Herr K. Feb 11 at 5:21

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?