<p>Marginal utility tells you how the utility changes as you alter x. That is the first derivative, which here is a function of x. This means it is increasing. The rate of that increase is constant as long as y is fixed (second derivative).</p>

CC BY-SA 4.0
2019-01-18T17:27:27.447Z
2019-01-18T23:52:23.410Z
2019-01-18T23:52:23.410Z
dimitriy
<p>Marginal utility (of <span class="math-container">$x$</span>) in your case is <span class="math-container">$U_x(x,y)=2xy^2$</span>. You use the sign of the derivative of MU, namely <span class="math-container">$U_{xx}$</span>, to tell whether MU is increasing, constant, or decreasing. </p>

<p>Specifically, you have </p>

<ul>
<li>increasing MU if <span class="math-container">$U_{xx}&gt;0$</span>,</li>
<li>constant MU if <span class="math-container">$U_{xx}=0$</span>, and </li>
<li>decreasing MU if <span class="math-container">$U_{xx}&lt;0$</span>.</li>
</ul>

<p>In your case, assuming that <span class="math-container">$y&gt;0$</span>, you'd have <span class="math-container">$U_{xx}&gt;0$</span>, hence increasing MU.</p>

CC BY-SA 4.0
2019-01-18T21:32:15.603Z
2019-01-18T21:32:15.603Z
Herr K.
<p>Does this utility function have increasing/decreasing or constant marginal utility?</p>

<p><span class="math-container">$ U(x,y) = x^2 y^2 $</span></p>

<p>Now, <span class="math-container">$ f_x = 2xy^2 $</span>, <span class="math-container">$ f_{xx} = 2y^2 $</span>, <span class="math-container">$ f_y = 2yx^2 $</span>, <span class="math-container">$f_{yy} = 2x^2 $</span></p>

<p><span class="math-container">$ f_{xx} $</span> has no <span class="math-container">$ x $</span> term in it -- so is the marginal utillity of <span class="math-container">$ x$</span> constant or increasing? It increases as <span class="math-container">$ y $</span> increases, of course, but it stays constant if we increase <span class="math-container">$ x $</span>. My textbook says that this is a case of increasing marginal utility, but I don't understand <strong>why</strong>. Similar problem for <span class="math-container">$ f_{yy} $</span> , which has no <span class="math-container">$ y $</span> term.</p>

CC BY-SA 4.0
2019-01-19T08:43:35.697Z
A note of caution: although $U_{xx}=0$ implies constant marginal utility, this observation is totally meaningless. We know that utility functions areinvariant to monotone transformations. So take the function $f(x)=x^2$ on domain $\mathbb{R}_+$. Then $f(U(x))$ preserves the order (because $f'(x)>0$) and is also a valid utility function for the same preferences. But $\partial^2[f(U(x))]/\partial x^2=(U_x)^2f''(U(x))+f'(U(x))U_{xx}=(U_x)^2f''(U(x))>0$. So the same preferences can be represented by both a constant MU function and an increasing MU function; it's entirely arbitrary.
Ubiquitous
<utility><marginal>
When can I say that a utility function has constant marginal utility?
WorldGov
# When can I say that a utility function has constant marginal utility?

(Asked by: username_0 on 2019-01-18)

Does this utility function have increasing/decreasing or constant marginal utility?

$ U(x,y) = x^2 y^2 $

Now, $ f\_x = 2xy^2 $, $ f\_{xx} = 2y^2 $, $ f\_y = 2yx^2 $, $f\_{yy} = 2x^2 $

$ f\_{xx} $ has no $ x $ term in it -- so is the marginal utillity of $ x$ constant or increasing? It increases as $ y $ increases, of course, but it stays constant if we increase $ x $. My textbook says that this is a case of increasing marginal utility, but I don't understand **why**. Similar problem for $ f\_{yy} $ , which has no $ y $ term.

**Tags:** utility, marginal

> **Comments:**
> **username_1**: A note of caution: although $U_{xx}=0$ implies constant marginal utility, this observation is totally meaningless. We know that utility functions areinvariant to monotone transformations. So take the function $f(x)=x^2$ on domain $\mathbb{R}_+$. Then $f(U(x))$ preserves the order (because $f'(x)>0$) and is also a valid utility function for the same preferences. But $\partial^2[f(U(x))]/\partial x^2=(U_x)^2f''(U(x))+f'(U(x))U_{xx}=(U_x)^2f''(U(x))>0$. So the same preferences can be represented by both a constant MU function and an increasing MU function; it's entirely arbitrary.  

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## 2 answers

### Answer by username_3 on 2019-01-18. Score: 5 | Accepted answer

Marginal utility (of $x$) in your case is $U\_x(x,y)=2xy^2$. You use the sign of the derivative of MU, namely $U\_{xx}$, to tell whether MU is increasing, constant, or decreasing.

Specifically, you have

* increasing MU if $U\_{xx}>0$,
* constant MU if $U\_{xx}=0$, and
* decreasing MU if $U\_{xx}<0$.

In your case, assuming that $y>0$, you'd have $U\_{xx}>0$, hence increasing MU.


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### Answer by username_2 on 2019-01-18. Score: 3

Marginal utility tells you how the utility changes as you alter x. That is the first derivative, which here is a function of x. This means it is increasing. The rate of that increase is constant as long as y is fixed (second derivative).


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