Envelope theorem

The envelope theorem gives the derivative of an optimized value with respect to a parameter from the objective’s direct parameter effect, without differentiating the optimizing choice.

Value function: A function that assigns each parameter value the best attainable objective value under specified constraints. The theorem computes how this optimized objective changes as its parameter varies.
Constrained optimization: The problem of maximizing or minimizing an objective subject to restrictions on feasible choices. The envelope theorem applies to optimized values, including values defined by constraints.
Profit function: A function giving a firm's maximum profit as a function of prices, costs, or other parameters. The theorem derives marginal effects on maximum profit without differentiating chosen inputs.
Comparative statics of the optimizer: The analysis of how an optimal choice itself changes when parameters change. The envelope theorem gives value changes, not the optimizer’s sensitivity.
First-order condition: An equation requiring a differentiable objective’s gradient to vanish at an interior optimum. Stationarity makes the optimizer’s derivative disappear from the value derivative.
Comparative statics: The study of how an equilibrium or optimum changes when an underlying parameter changes. The theorem is a tool for comparative statics of optimized objective values.
Indirect utility function: A consumer’s maximum utility as a function of income and market prices. Its derivatives connect optimized welfare to prices and income.
Sensitivity analysis: The study of how outputs or solutions respond to changes in model inputs. Envelope results provide a focused sensitivity measure for optimized objective values.
Lagrange multiplier: A variable used to encode a constraint in an optimization problem’s Lagrangian. With constraints, multipliers enter the envelope formula through the Lagrangian.
Global maximum: A feasible point whose objective value is at least as large as at every other feasible point. The value function is defined by taking the objective’s maximum over feasible choices.

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Cost curveRelated: The long-run average-cost curve can be understood as an envelope of short-run curves.