<p>First of all, an utility function <span class="math-container">$u: X \rightarrow \mathbb{R}$</span> represents the preference relation <span class="math-container">$\succsim$</span> if: <span class="math-container">$$\forall a, b \in X, \; u(a) \geq u(b) \iff a \succsim b.$$</span></p>

<p>Well, if another function <span class="math-container">$v: X \rightarrow \mathbb{R}$</span> represents <span class="math-container">$\succsim$</span>, then: <span class="math-container">$$\forall a, b \in X, \; v(a) \geq v(b) \iff a \succsim b.$$</span></p>

<p>But the above implies the following: </p>

<p><span class="math-container">$$\forall a, b \in X, \; u(a) \geq u(b) \iff v(a) \geq v(b).$$</span></p>

<p>Which means that both functions preserves the order relation, just as Herr K. said it. Another important part of Wikipedia's article is <a href="https://en.wikipedia.org/wiki/Ordinal_utility#Uniqueness" rel="nofollow noreferrer">this one</a>. You see, any monotonically increasing (and it must be increasing, not nondecreasing) function preserves the preference relation. </p>

<p>So now I think you will be able to answer both questions. As a tip for the first one, think about the <span class="math-container">$\ln(x)$</span> function: is it a monotonic increasing function?</p>

CC BY-SA 4.0
2019-04-07T18:46:38.923Z
2019-04-07T18:51:41.513Z
2019-04-07T18:51:41.513Z
Marcelo Gelati
<p>I have this question for microecon that asks do the following utility functions represent the same preferences:</p>

<ol>
<li><p><span class="math-container">$u(x_1, x_2) = x1 \cdot x2, \; v(x_1, x_2) = \ln x_1 + \ln x_2$</span></p></li>
<li><p><span class="math-container">$u(x_1, x_2) = x1 \cdot x2, \; v(x_1, x_2) = x_1 + x_2$</span></p></li>
</ol>

<p>How do I go about approaching this problem since they are two different functions?</p>

CC BY-SA 4.0
2019-02-11T02:16:06.990Z
Did you learn that [utility is ordinal](https://en.wikipedia.org/wiki/Ordinal_utility)?
Herr K.
CC BY-SA 4.0
2019-02-11T04:50:05.017Z
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
CC BY-SA 4.0
2019-02-11T05:21:33.513Z
Read the article linked in my previous comment. The second sentence therein answers your question.
Herr K.
<utility><preferences>
How can two different utility functions represent the same preferences?
Abiola
# How can two different utility functions represent the same preferences?

(Asked by: username_0 on 2019-02-11)

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?

**Tags:** utility, preferences
> **Comments:**
> **username_1**: Did you learn that [utility is ordinal](https://en.wikipedia.org/wiki/Ordinal_utility)?  
> **username_0**: 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  
> **username_1**: Read the article linked in my previous comment. The second sentence therein answers your question.  

---

## 1 answers

### Answer by username_2 on 2019-04-07. Score: 2

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](https://en.wikipedia.org/wiki/Ordinal_utility#Uniqueness). 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?


---

