# For any small perturbation dx, utility cannot change, or else, x* would not be optimal

I am having trouble understanding something that Varian says in "Microeconomic Analysis: Third Edition." For those of you who have the book handy, the question that I have regards something said on Chapter 7 (Utility Max), page 101.

The setting. Suppose $$\mathbf{x}$$* is the optimal consumption bundle. Now suppose we have a small perturbation $$\mathbf{dx}$$ of $$\mathbf{x}$$*, such that we are still on the budget constraint (so we increase consumption of one good, and decrease consumption of another). So we have $$\mathbf{p}(\mathbf{x}^{*} \pm \mathbf{dx})=m$$ where $$\mathbf{p}$$ is the price vector and $$m$$ is income. Since $$\mathbf{px}=m$$, we can deduce that $$\mathbf{pdx}$$=0. That is $$\mathbf{dx}$$ is orthogonal to $$\mathbf{p}$$.

So far everything makes sense. What doesn't make sense is the next part that says for any perturbation of $$\mathbf{x}$$*, utility can't change, otherwise this consumption bundle wouldn't be optimal.

The problem could be that when I visualize the problem, I imagine $$\mathbf{dx}$$ being some small vector along the budget line, starting at $$\mathbf{x}$$* and ending just a little ways from $$\mathbf{x}^{*}$$. So even if we were at $$\mathbf{x}^{*}$$, wouldn't any deviation along the budget constraint put us on a slightly smaller indifference curve?

• Let $dx$ be a small increase in consumption of the first good. You get some small decrease in utility, $du$. Now imagine $dx$ becoming infinitesimally small. What happens to the fraction $\frac{du}{dx}$? Commented Dec 20, 2021 at 7:51
If you are at the point of maximization, any deviation should either be of no benefit to you or violate some constraints. By continuity, it means that, unless you are constrained, at the optimal point the marginal benefit of deviation should be $$0$$. Since we have picked the subspace where $$p\mathbf{x} = m$$, deviating inside it can't violate any constraints, hence it implies that the marginal benefit of deviation is $$0$$.
We can describe the optimal bundle as the solution to the optimization problem: \begin{align*} \mathbf{x^*} &= \arg\max_{\mathbf{x} \in \mathbb{X}} u(\mathbf{x})\\ &\text { s.t. } \mathbf{p}\mathbf{x} = m \end{align*} From that we can easily derive the FOC: $$\mathbf{D} u(\mathbf{x}^*) = \lambda \mathbf{p}$$ Assuming $$\mathbf{p}(\mathbf{x^*}+d\mathbf{x})=m$$, we get that $$\mathbf{p}d\mathbf{x} = 0$$. Therefore, $$\mathbf{D} u(\mathbf{x}^*)d\mathbf{x} = \lambda \mathbf{p}d\mathbf{x} = 0$$