The envelope theorem
When you are already optimising, small changes to the thing you chose cost you nothing to first order.
The claim
Suppose you have chosen $x$ to maximise $f(x, a)$, where $a$ is some parameter you do not control — a price, a tax rate, a wage. Call the best value you can achieve $V(a) = \max_x f(x, a)$.
The envelope theorem says that to work out how $V$ responds to a small change in $a$, you can ignore the fact that your optimal $x$ will also shift:
\[\frac{dV}{da} = \frac{\partial f}{\partial a}\bigg|_{x = x^*(a)}\]Only the direct effect matters. The indirect effect — through the adjustment of $x$ itself — vanishes.
Why it vanishes
Because you were already at an optimum. At $x^*$, the derivative $\partial f/\partial x$ is zero. So when $x$ shifts a little, $f$ changes by (zero) × (the shift), which is zero to first order. You were standing at the top of the hill; stepping slightly sideways costs nothing measurable.
What it quietly assumes
That $x^*$ really is an interior optimum where the first-order condition holds with equality, and that $f$ is differentiable there. Both fail more often than textbooks suggest.
Where it breaks
At a corner solution the first-order condition does not hold — you are at the edge of the feasible set, pressed against a constraint, and the derivative is not zero. The indirect effect no longer disappears.
It also gives you only a first-order result. For a large change in $a$, the adjustment of $x$ matters a great deal, and the envelope approximation drifts away from the truth.
Why it is worth knowing
It is the reason Shephard’s lemma, Hotelling’s lemma and Roy’s identity all work, and the reason the derivative of a value function in dynamic programming is tractable at all. Once you see it, a dozen separate results collapse into one idea.