What does it mean if the derivative of the Utility function (at the optimal bundle) is 0?

Question

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?

Answer 1

I generalize the previous answer 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.}

Answer 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.