<p>According to the logical positivist view of decision theory, utility functions are just descriptions of <strong>observable behavior</strong> and have no intrinsic meaning absent this vantage. In other words, $u(x) &gt; u(y)$ indicates that $x$ would be chosen in favor of $y$, but the magnitude of the difference carries no addition information. If $u(x) = 1$ and $u(y) = 0$ the decision maker prefers $x$; if $u(x) = 10000$ and $u(y)= 0$ the decision maker prefers $x$. In either case, we can draw the same conclusions. </p>

<p>What does this mean with regard to your question? It tells us what we mean by <em>equality</em> of utility functions: we mean that these two different (so no definitionally equal) functions represent the same observable data. Specifically, exactly when $u_1(x) &gt; u_1(y)$ do we also have $u_2(x) &gt; u_2(y)$. As mentioned in the comments, this is precisely when we can compose $u_1$ with a strictly increasing function and pop out $u_2$. (This is because all we care about is the ordering between objects, and strictly increasing functions <em>preserve</em> order). </p>

<p>So, what might our strictly increasing function look like? Well, we can take the log, so that we get $\frac23 ln(x_1) + \frac13 ln(x_2)$ then multiplying by 6 and adding 3 does the trick. Each of these was strictly increasing so too is the composition:
$$ (x,y) \mapsto 6 (ln(x) + ln(y)) + 3$$
is our desired function.</p>

CC BY-SA 4.0
2018-05-28T22:06:17.770Z
2018-05-28T22:06:17.770Z
201p
<p>I recently joined an econ class. I am so lost on how to prove their equality. As a math standpoint, these are completely different equations. Please help! </p>

<pre><code>u1 (x1, x2) = x1^(2/3) x2^(1/3)
u2 (x1, x2) = 4ln (x1) + 2 ln(x2) +3
</code></pre>

CC BY-SA 4.0
2018-05-24T06:30:17.940Z
Related: https://economics.stackexchange.com/a/2993/42
Herr K.
<utility><preferences>
Compare utility functions
Vraj Shroff
# Compare utility functions

(Asked by: username_0 on 2018-05-24)

I recently joined an econ class. I am so lost on how to prove their equality. As a math standpoint, these are completely different equations. Please help!

```
u1 (x1, x2) = x1^(2/3) x2^(1/3)
u2 (x1, x2) = 4ln (x1) + 2 ln(x2) +3

```

**Tags:** utility, preferences

> **Comments:**
> **username_1**: Related: https://economics.stackexchange.com/a/2993/42  

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## 1 answers

### Answer by username_2 on 2018-05-28. Score: 3 | Accepted answer

According to the logical positivist view of decision theory, utility functions are just descriptions of **observable behavior** and have no intrinsic meaning absent this vantage. In other words, $u(x) > u(y)$ indicates that $x$ would be chosen in favor of $y$, but the magnitude of the difference carries no addition information. If $u(x) = 1$ and $u(y) = 0$ the decision maker prefers $x$; if $u(x) = 10000$ and $u(y)= 0$ the decision maker prefers $x$. In either case, we can draw the same conclusions.

What does this mean with regard to your question? It tells us what we mean by *equality* of utility functions: we mean that these two different (so no definitionally equal) functions represent the same observable data. Specifically, exactly when $u\_1(x) > u\_1(y)$ do we also have $u\_2(x) > u\_2(y)$. As mentioned in the comments, this is precisely when we can compose $u\_1$ with a strictly increasing function and pop out $u\_2$. (This is because all we care about is the ordering between objects, and strictly increasing functions *preserve* order).

So, what might our strictly increasing function look like? Well, we can take the log, so that we get $\frac23 ln(x\_1) + \frac13 ln(x\_2)$ then multiplying by 6 and adding 3 does the trick. Each of these was strictly increasing so too is the composition:
$$ (x,y) \mapsto 6 (ln(x) + ln(y)) + 3$$
is our desired function.


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