Envelope Theorem
Comparative statics via value functions, the shortcut behind Shephard's and Hotelling's lemmas.
Definition
Differentiating a value function with respect to a parameter: only the DIRECT effect counts, because the re-optimization terms vanish at the optimum.
Key equation
dV/dθ = ∂ℒ/∂θ evaluated at the optimum
The intuition
When a price nudges, the behavioral adjustment is second-order, you were already optimal. That single line generates Shephard's lemma, Hotelling's lemma, and Roy's identity: demand functions read straight off derivatives of value functions.
Exam tip
If asked to derive demand from an expenditure or profit function, the envelope theorem IS the answer, one derivative, no re-solving.
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