How can I tell if 2 different utility functions represent the same preferences?

Question

I need to verify that $u(x,y)=x^{1/3}y^{1/3}$ represents the same preferences as $v(x,y)=x^3y^3$. Obviously these are completely different functions with different derivatives, so what am I comparing? What makes 2 utility functions represent the same preferences?

Answer 1

Recall the definition:

The function $u: X \rightarrow \mathbb{R}$ represents $\succeq$ on $X$ if for any $x,y \in X$, then $x \succeq y \iff u(x) \geq u(y)$

We can show that if $u: X \rightarrow \mathbb{R}$ represents $\succeq$ on $X$, then for any strictly increasing function, $f: \mathbb{R} \rightarrow \mathbb{R}$, the function $v(x) = f(u(x))$ also represents $\succeq$

(Short) Proof: For any $x,y \in X$, if $u$ represents $\succeq$, then $x \succeq y \iff u(x) \geq u(y)$ by definition.

Because $v$ is strictly increasing, $f(u(x)) \geq f(u(y)), \implies v(x) \geq v(y)$.

Thus $x \succeq y \iff v(x) \geq v(y), \implies v$ represents $\succeq$

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In your case, you have $u(x,y) = x^\frac{1}{3}y^\frac{1}{3}$, and can set $v(x, y) = (u(x,y))^9 = x^3y^3$, your desired result.

Answer 2

Another trick is to compare their marginal rate of substitution. If for a similar marginal rate of technical substitution (i.e. relative prices) they yield the same relative inputs, then they represent the same preferences (basically one is a monotonic transformation of the other; see below).

For the first preferences you give, this is:

$$ \frac{\dfrac{\partial u}{\partial x}}{\dfrac{\partial u}{\partial y}} = \frac{y}{x} $$

It is trivial to show that the second function yields the same MRS. Hence, they represent the same preferences.

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Another way to look at this is to notice that optimising a function $f(x)$ is the same than optimising a monotonically transformation of such function. Thus, apply the natural logarithm to the two functions and you get, respectively:

$$ \ln u = \frac{1}{3}\left(\ln x + \ln y \right) $$

$$ \ln v = 3\left(\ln x + \ln y \right) $$

Notice that $\ln v = 9\ln u$. Hence, they are indeed the same.