Envelope theorem
Encyclopedia
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 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
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.

Envelope theorem

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, JOA = JOB + JBA."

See also

  • Arg max
  • Optimization problem
    Optimization problem
    In mathematics and computer science, an optimization problem is the problem of finding the best solution from all feasible solutions. Optimization problems can be divided into two categories depending on whether the variables are continuous or discrete. An optimization problem with discrete...

  • Random optimization
    Random optimization
    Random optimization is a family of numerical optimization methods that do not require the gradient of the problem to be optimized and RO can hence be used on functions that are not continuous or differentiable...

  • Simplex algorithm
    Simplex algorithm
    In mathematical optimization, Dantzig's simplex algorithm is a popular algorithm for linear programming. The journal Computing in Science and Engineering listed it as one of the top 10 algorithms of the twentieth century....

  • Topkis's Theorem
    Topkis's Theorem
    In mathematical economics, Topkis's theorem is a result that is useful for establishing comparative statics. The theorem allows researchers to understand how the optimal value for a choice variable changes when a feature of the environment changes...

  • Variational calculus
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