<p>You are right. Decreasing or increasing marginal utility is a property of the utility function and not of the underlying preference relation. Take for example:</p>
<p><span class="math-container">$u_1(x, y) = x^{\frac{1}{2}}y^{\frac{1}{2}}$</span> and
<span class="math-container">$u_2(x, y) = x^2y^2$</span>.</p>
<p><span class="math-container">$u_2$</span> is a monotonic transformation of <span class="math-container">$u_1$</span> so they represent the same preference relation.</p>
<p>Using <span class="math-container">$u_1$</span> the marginal utility of x is <span class="math-container">$\frac{1}{2}\sqrt{\frac{y}{x}}$</span>, which is decreasing in x.</p>
<p>Using <span class="math-container">$u_2$</span> the marginal utility of x is <span class="math-container">$2xy^2$</span>, which is increasing in x.</p>

CC BY-SA 4.0
2021-01-30T07:40:03.477Z
2021-01-30T07:40:03.477Z
Grada Gukovic
<p>When we differentiate the utility function with respect to some input <span class="math-container">$x_i$</span>, we get a number that tells us how "fast" the utility function is changing at some point with respect to <span class="math-container">$x_i$</span>. Doesn't that mean that when we compare marginal utilities, we are comparing with something that is based on the structure of the numeric output of the utility function, which shouldn't be the case since it's ordinal?</p>

<p>I know I'm missing something here intuitively, but I can't seem to figure out what it is. </p>

<utility>
Doesn't the concept of marginal utility speak to a cardinal utility function?
Vasting
# Doesn't the concept of marginal utility speak to a cardinal utility function?

(Asked by: username_0 on 2019-08-31)

When we differentiate the utility function with respect to some input $x\_i$, we get a number that tells us how "fast" the utility function is changing at some point with respect to $x\_i$. Doesn't that mean that when we compare marginal utilities, we are comparing with something that is based on the structure of the numeric output of the utility function, which shouldn't be the case since it's ordinal?

I know I'm missing something here intuitively, but I can't seem to figure out what it is.

**Tags:** utility

---

## 1 answers

### Answer by username_1 on 2021-01-30. Score: 0

You are right. Decreasing or increasing marginal utility is a property of the utility function and not of the underlying preference relation. Take for example:

$u\_1(x, y) = x^{\frac{1}{2}}y^{\frac{1}{2}}$ and
$u\_2(x, y) = x^2y^2$.

$u\_2$ is a monotonic transformation of $u\_1$ so they represent the same preference relation.

Using $u\_1$ the marginal utility of x is $\frac{1}{2}\sqrt{\frac{y}{x}}$, which is decreasing in x.

Using $u\_2$ the marginal utility of x is $2xy^2$, which is increasing in x.


---

