Envelope Theorem

The envelope theorem is a theorem about optimization problems (max & min) in microeconomics. It may be used to prove Hotelling's lemma, Shephard's lemma, and Roy's identity. It also allows for easier computation of comparative statics in generalized economic models.

The theorem exists in two versions, a regular version (unconstrained optimization) and a generalized version (constrained optimization). The regular version can be obtained from the general version because unconstrained optimization is just the special case of constrained optimization with no constraints (or constraints that are always satisfied, i.e. constraints that are identities such as or .

The theorem gets its name from the fact that it shows that a less constrained maximization (or minimization) problem (where some parameters are turned into variables) is the upper (or lower for min) envelope of the original problem. For example, see cost minimization, and compare the long-run (less constrained) and short-run (more constrained – some factors of production are fixed) minimization problems.

For the theorem to hold, the functions being dealt with must have certain well-behaved properties. Specifically, the correspondence mapping parameter values to optimal choices must be differentiable, with it being single-valued (and hence a function) a necessary but not sufficient condition.

The theorem is described below. Note that bold face represents a vector.