<p>I generalize the <a href="https://economics.stackexchange.com/a/54626/1601">previous answer</a> to a function <span class="math-container">$U(x), x\in \mathbb {R^n}$</span>, of several variables,  interpreting <strong><span class="math-container">$U'(x)$</span> as the gradient</strong> (the vector of partial derivatives) of <span class="math-container">$U(x)$</span>, as the question refers to a 'bundle' (but it can be an imprecision and it refers only to function of several variables, it is difficult to answer without reading the book).</p>
<p>The whole passage of the book should be read to see the context, but the statement in the question should mean that <strong>it is possible, but unlikely, that we have a local maximum of the utility function (or, more generally, a critical point) on the border of the feasible set.</strong></p>
<p>That is: we know that under the assumption of strict <strong>monotonicity</strong> (it is sufficient, actually, monotonicity or local non satiation) the budget constraint is <strong>fulfilled as equality</strong>, so that the optimal bundle <span class="math-container">$x^*$</span> is on the budget line..</p>
<p>This means that, in most cases,  the constraint is binding, and in general   the <strong>constrained optimum will be different from the unconstrained optimum.</strong></p>
<p>Therefore,  in general, in most cases,  we have <span class="math-container">$U'(x^*)\neq 0$</span>  (while <span class="math-container">$U'(x^*)= 0$</span> is instead the necessary condition for a maximum at an interior point for the unconstrained problem).</p>
<p>However,  we cannot exclude that, by chance, we can say,  we have at <span class="math-container">$x^*$</span> a critical point of <span class="math-container">$U(x)$</span> (a point where the derivatives are zero) on the line budget.</p>
<hr />
<p><span class="math-container">$^1$</span> <sup> Of course, reading the book is necessary to understand what’s the role of <strong>strict</strong> monotonicity, that is if the book considers it necessary for the statement. In my answer it is implicit that monotonicity or local non satiation is sufficient.
If this means that all the partial derivatives are strictly positive everywhere, actually, there isn't, by definition, any critical point.</sup></p>

CC BY-SA 4.0
2023-03-03T13:13:30.583Z
2023-03-06T08:20:07.117Z
2023-03-06T08:20:07.117Z
BakerStreet
<p>The statement in the question title makes no sense. If utility <span class="math-container">$U$</span> is a function of a (consumption) <em>bundle</em>, then it is defined on at least two variables, e.g. <span class="math-container">$U=U(x,y)$</span>. Thus it has partial derivatives, directional derivatives, a differential, and a gradient, but &quot;the derivative&quot; of <span class="math-container">$U$</span> is not well defined.</p>
<p>In case you have no bundle but a quantity <span class="math-container">$x$</span> of a single good, strict monotonicity of preferences means that the utility function <span class="math-container">$U(x)$</span> is strictly increasing. It could still have a zero derivative at the optimal quantity <span class="math-container">$x^*$</span>, however, if it happens to have a saddle point there, i.e. if <span class="math-container">$U'(x^*)=U''(x^*)=0$</span>.</p>
<p>As an example, if your utility is given by the (strictly increasing) function <span class="math-container">$U(x)=(x-1)^3+1$</span>, your income is <span class="math-container">$I=1$</span>, and the price of the good is <span class="math-container">$p=1$</span>, then your optimal choice is <span class="math-container">$x^*=1$</span>, while <span class="math-container">$U'(x^*)=0$</span>. That's &quot;unlikely&quot;, however, because you need both a utility function with a saddle point <em>and</em> this saddle point must happen to be the optimal choice given income and price.</p>

CC BY-SA 4.0
2023-03-03T11:14:06.157Z
Hi! I think this is an [inflection point](https://en.wikipedia.org/wiki/Inflection_point), not a [saddle point](https://en.wikipedia.org/wiki/Saddle_point)?
Giskard
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2023-03-03T12:39:00.960Z
@Giskard, it has to be both an inflection point ($U''=0$) and a critical point ($U'=0$). Such a thing is then called a saddle point, see last paragraph of the second section in your second link: https://en.wikipedia.org/wiki/Saddle_point#Mathematical_discussion
VARulle
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2023-03-03T10:37:34.657Z
2023-03-03T10:37:34.657Z
VARulle
<p>It states in my book that under strict monotonocity, the derivative of U(x*)=0 can be possible although it's unlikely to happen.</p>
<p>What does this exactly mean?</p>

<microeconomics><utility><preferences>
What does it mean if the derivative of the Utility function (at the optimal bundle) is 0?
aliosha karamazov
# What does it mean if the derivative of the Utility function (at the optimal bundle) is 0?

(Asked by: username_0 on 2023-03-02)

It states in my book that under strict monotonocity, the derivative of U(x\*)=0 can be possible although it's unlikely to happen.

What does this exactly mean?

**Tags:** microeconomics, utility, preferences

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## 2 answers

### Answer by username_1 on 2023-03-03. Score: 3

I generalize the [previous answer](https://economics.stackexchange.com/a/54626/1601) to a function $U(x), x\in \mathbb {R^n}$, of several variables, interpreting **$U'(x)$ as the gradient** (the vector of partial derivatives) of $U(x)$, as the question refers to a 'bundle' (but it can be an imprecision and it refers only to function of several variables, it is difficult to answer without reading the book).

The whole passage of the book should be read to see the context, but the statement in the question should mean that **it is possible, but unlikely, that we have a local maximum of the utility function (or, more generally, a critical point) on the border of the feasible set.**

That is: we know that under the assumption of strict **monotonicity** (it is sufficient, actually, monotonicity or local non satiation) the budget constraint is **fulfilled as equality**, so that the optimal bundle $x^\*$ is on the budget line..

This means that, in most cases, the constraint is binding, and in general the **constrained optimum will be different from the unconstrained optimum.**

Therefore, in general, in most cases, we have $U'(x^\*)\neq 0$ (while $U'(x^\*)= 0$ is instead the necessary condition for a maximum at an interior point for the unconstrained problem).

However, we cannot exclude that, by chance, we can say, we have at $x^\*$ a critical point of $U(x)$ (a point where the derivatives are zero) on the line budget.

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$^1$  Of course, reading the book is necessary to understand what’s the role of **strict** monotonicity, that is if the book considers it necessary for the statement. In my answer it is implicit that monotonicity or local non satiation is sufficient.
If this means that all the partial derivatives are strictly positive everywhere, actually, there isn't, by definition, any critical point.


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### Answer by username_2 on 2023-03-03. Score: 2

The statement in the question title makes no sense. If utility $U$ is a function of a (consumption) *bundle*, then it is defined on at least two variables, e.g. $U=U(x,y)$. Thus it has partial derivatives, directional derivatives, a differential, and a gradient, but "the derivative" of $U$ is not well defined.

In case you have no bundle but a quantity $x$ of a single good, strict monotonicity of preferences means that the utility function $U(x)$ is strictly increasing. It could still have a zero derivative at the optimal quantity $x^\*$, however, if it happens to have a saddle point there, i.e. if $U'(x^\*)=U''(x^\*)=0$.

As an example, if your utility is given by the (strictly increasing) function $U(x)=(x-1)^3+1$, your income is $I=1$, and the price of the good is $p=1$, then your optimal choice is $x^\*=1$, while $U'(x^\*)=0$. That's "unlikely", however, because you need both a utility function with a saddle point *and* this saddle point must happen to be the optimal choice given income and price.

> **Comments:**
> **username_3**: Hi! I think this is an [inflection point](https://en.wikipedia.org/wiki/Inflection_point), not a [saddle point](https://en.wikipedia.org/wiki/Saddle_point)?  
> **username_2**: @Giskard, it has to be both an inflection point ($U''=0$) and a critical point ($U'=0$). Such a thing is then called a saddle point, see last paragraph of the second section in your second link: https://en.wikipedia.org/wiki/Saddle_point#Mathematical_discussion  

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