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?

  • 1
    $\begingroup$ Did you learn that utility is ordinal? $\endgroup$
    – Herr K.
    Feb 11 '19 at 2:16
  • $\begingroup$ 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 $\endgroup$
    – Abiola
    Feb 11 '19 at 4:50
  • $\begingroup$ Read the article linked in my previous comment. The second sentence therein answers your question. $\endgroup$
    – Herr K.
    Feb 11 '19 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: $$\forall a, b \in X, \; v(a) \geq v(b) \iff a \succsim b.$$

But the above implies the following:

$$\forall a, b \in X, \; u(a) \geq u(b) \iff v(a) \geq v(b).$$

Which means that both functions preserves the order relation, just as Herr K. said it. Another important part of Wikipedia's article is this one. You see, any monotonically increasing (and it must be increasing, not nondecreasing) function preserves the preference relation.

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?


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