Envelope theorem

The envelope theorem is a theorem about optimization problems (max

Utility maximization problem

In microeconomics, the utility maximization problem is the problem consumers face: "how should I spend my money in order to maximize my utility?" It is a type of optimal decision problem.-Basic setup:...

& min) in microeconomics

Microeconomics

Microeconomics is a branch of economics that studies the behavior of how the individual modern household and firms make decisions to allocate limited resources. Typically, it applies to markets where goods or services are being bought and sold...

. It may be used to prove Hotelling's lemma

Hotelling's lemma

Hotelling's lemma is a result in microeconomics that relates the supply of a good to the profit of the good's producer. It was first shown by Harold Hotelling, and is widely used in the theory of the firm. The lemma is very simple, and can be stated:...

, Shephard's lemma

Shephard's lemma

Shephard's lemma is a major result in microeconomics having applications in the theory of the firm and in consumer choice. The lemma states that if indifference curves of the expenditure or cost function are convex, then the cost minimizing point of a given good with price p_i is unique...

, and Roy's identity

Roy's identity

Roy's identity is a major result in microeconomics having applications in consumer choice and the theory of the firm. The lemma relates the ordinary demand function to the derivatives of the indirect utility function...

. It also allows for easier computation of comparative statics

Comparative statics

In economics, comparative statics is the comparison of two different economic outcomes, before and after a change in some underlying exogenous parameter....

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

).

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

Cost curve

In economics, a cost curve is a graph of the costs of production as a function of total quantity produced. In a free market economy, productively efficient firms use these curves to find the optimal point of production , and profit maximizing firms can use them to decide output quantities to...

, 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

Well-behaved

Mathematicians very frequently speak of whether a mathematical object — a function, a set, a space of one sort or another — is "well-behaved" or not. The term has no fixed formal definition, and is dependent on mathematical interests, fashion, and taste...

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.

A curve in a two dimensional space is best represented by the parametric equations like x(t) and y(t).
The family of curves can be represented in the form ,
g(x,y,c) = 0 where c is the parameter

Generally, the envelope theorem involves one parameter but there can be more than one parameters involved as well.

The envelope of a family of curves g(x,y,c) = 0 is a curve such that at each point on the curve there is some member of the family that touches that particular point tangentially.
This forms a curve or surface that is tangential to every curve in the family of curves forming an envelope.

Consider an arbitrary maximization (or minimization) problem where the objective function

depends on some parameters

The function

is the problem's optimal-value function — it gives the maximized (or minimized) value of the objective function

as a function of its parameters

.

Let

be the (arg max) value of

, expressed in terms of the parameters, that solves the optimization problem, so that

. The envelope theorem tells us how

changes as a parameter changes, namely:

That is, the derivative of

with respect to

is given by the partial
derivative of

with respect to

, holding

fixed, and then evaluating at the optimal choice

General envelope theorem

There also exists a version of the theorem, called the general envelope theorem, used in constrained optimization problems which relates the partial derivatives of the optimal-value function to the partial derivatives of the Lagrangian

Lagrangian

The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...

function.

We are considering the following optimization problem in formulating the theorem (max may be replaced by min, and all results still hold):

Which gives the Lagrangian function:

Where:

is the dot product

Dot product

In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

Then the general envelope theorem is:

Note that the Lagrange multipliers

are treated as constants during differentiation of the Lagrangian function, then their values as functions of the parameters are substituted in afterwards.

Envelope theorem in generalized calculus

In the calculus of variations

Calculus of variations

Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...

, the envelope theorem relates evolute

Evolute

In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. Equivalently, it is the envelope of the normals to a curve....

s to single paths

Path (topology)

In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f. One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path...

. This was first proved by Jean Gaston Darboux

Jean Gaston Darboux

Jean-Gaston Darboux was a French mathematician.-Life:Darboux made several important contributions to geometry and mathematical analysis . He was a biographer of Henri Poincaré and he edited the Selected Works of Joseph Fourier.Darboux received his Ph.D...

and Ernst Zermelo

Ernst Zermelo

Ernst Friedrich Ferdinand Zermelo was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem.-Life:He graduated...

(1894) and Adolf Kneser

Adolf Kneser

Adolf Kneser was a German mathematician.He was born in Grüssow, Mecklenburg, Germany and died in Breslau, Germany ....

(1898). The theorem can be stated as follows:

"When a single-parameter family of external paths from a fixed point O has an envelope
Envelope (mathematics)
In geometry, an envelope of a family of curves in the plane is a curve that is tangent to each member of the family at some point. Classically, a point on the envelope can be thought of as the intersection of two "adjacent" curves, meaning the limit of intersections of nearby curves...

, the integral from the fixed point to any point A on the envelope equals the integral from the fixed point to any second point B on the envelope plus the integral along the envelope to the first point on the envelope, J_OA = J_OB + J_BA."

Envelope theorem

General envelope theorem

Envelope theorem in generalized calculus

See also