﻿ The Envelope Theorem and Its Proof
San José State University
Department of Economics

applet-magic.com
Thayer Watkins
Silicon Valley
USA

 The Envelope Theorem and Its Proof

In analytic geometry there is an interesting and useful result that is called the Envelope Theorem. A special application of that theorem occurs in economics as a result of optimization. The theorem from analytic geometry needs to be covered first.

A curve in two dimensional space is best represented by parametric equations; i.e., x(t), y(t). For example, the parametric equations for a circle of radius r centered at the origin are:

#### x(t) = r cos(t), y(t) = r sin(t)

This circle can also be represented by the equation:

#### x2 + y2 = r2

This latter representation is of the form f(x,y) = 0.

A family of curves can be represented in the form:

#### g(s,y,c) = 0

where c is a parameter. There can be families of curves involving more than one parameter. The Envelope Theorem involves one-parameter families of curves.

The envelope of a family of curves g(x, y, c) = 0 is a curve P such that at each point of P, say (x,y), there is some member of the family that touches P tangentially. In other words, for each point of P, (x0, y0), there is a value of c, say c0, such that

#### g(x0, y0, c0) = 0

Since at each point of the envelope curve P there is a corresponding value of the parameter c, the envelope curve can be represented parametrically as (x(c), y(c)).

Since the equation defining the family of curves, g(x, y, c) = 0, is true for all values of c in some range, that equation can be differentiated with respect to c. The result is:

#### (∂g/∂x)(∂x/∂c) + (∂g/∂y)(∂y/∂c) + (∂g/∂c) = 0

For any particular curve of the family the parameter c is constant. Differentiating g(x, y, c) = 0 with respect to x with c held constant gives:

#### (∂g/∂x) + (∂g/∂y)(∂y/∂x) = 0

From the parametric equation for the envelope, (x(c), y(c)), it follows that

#### (∂y/∂x) = (∂y/∂c/ (∂x/∂c))

At the point of tangency the envelope curve and the corresponding curve of the family have the same slope. This means that

#### (∂y/∂x) = (∂y/∂c/ (∂x/∂c)) and (∂g/∂x) + (∂g/∂y)(∂y/∂x) = 0 imply(∂g/∂x)(∂x/∂c) + (∂g/∂y)(∂y/∂c) = 0

When this equation is compared with the equation obtained by differentiating the equation for the families of curves with respect to c; i.e.,

#### (∂g/∂x)(∂x/∂c) + (∂g/∂y)(∂y/∂c) + (∂g/∂c) = 0

the implication is that:

#### (∂g/∂c) = 0

Thus the way to find the envelope of a family of curves is to solve the two equations:

#### g(x,y,c) = 0 and (∂g/∂c) = 0

for x and y as functions of c.

### Examples

Consider the family of circles of radius 1 whose centers are on the x-axis

#### (x-c)2 + y2 -1 = 0

Differentiating this equation with respect to c and setting the result equal to 0 gives:

#### 2(x-c) = 0

This last equation implies x=c. Substituting this result into the equation for the family of circles gives y2 -1 = 0 or y = 1. Clearly this is the equation for the two envelope curves for the family.

Consider the family of circles given by

#### (x-6c)2 + y2 - 16c2 = 0

This is a family of circles centered on the x-axis such that as the center moves away from the origin the radius gets larger in the ratio of radius being two thirds the distance from the origin.

Differentiation of the equation for the family by c gives:

#### -12(x-6c) + 32c = 0 which implies that x = 104c/12 = 26c/3 hence that (64/9)c2 + y2 - 16c2 = 0 or y = (801/2/3)c

The equation for x in terms of c can be solved for c; i.e., c = 3x/26. This result when substituted into the parametric equation for y gives y in terms of x. Thus the envelopes for the family of circles are y = (801/2/26)x = (201/2/13)x, two straight lines through the origin with slope of (201/2/13).

## The Viner-Wong Envelope Theorem

In economics one could have a family of cost functions which give the cost of production of a plant as a function of output with the level of capital input as the parameter for the family. Each one of these would be called a short-run cost function because the level of capital is held fixed. The relevant cost function is the long run cost function which at each level of output chooses the level of capital that minimizes the cost. The first order condition for such a minimum cost is that the derivative of cost with respect to capital is equal to zero. This is a instance of the condition that was found above for the envelope of a family of curves. So the long run cost function is the envelope of the family of short run cost functions.

For an illustration of the above consider a product with a production function of

#### Q = AL2/3K1/3.

If K is given then the amount of labor, L, required to achieve different levels of Q is given by:

#### L = (Q/A)3/2/K1/2

The total cost of production is C = rK + wL, where r is the cost of capital and w is the wage rate. When the labor requirement function derived above is substituted for L in the cost function the result is:

#### C = rK + w(Q/A)3/2/K1/2

This can easily be put into the form appropriate for the application of the envelope theorem; i.e.,

#### f(C,Q,K) = C - rK - w(Q/A)3/2/K1/2 = 0

This the family of total cost functions with K as the parameter for the family. The condition from the envelope theorem for the value of the parameter K that corresponds to a point on the envelope curve is that ∂f/∂K = 0; i.e.,

#### ∂f/∂K = -r + w(1/2)(Q/A)3/2/K3/2 = 0

This implies that

Thus,

#### L = (Q/A)3/2/((w/2r)1/3(Q/A)1/2) = (w/2r)-1/3(Q/A) and C = r(w/2r)2/3(Q/A) + w (w/2r)-1/3(Q/A)

This is a straight line through the origin with a slope (average total cost) of

#### r(w/2r)2/3 + w (w/2r)-1/3

This example generalizes to the proposition that for constant returns to scale (homogeneous of degree one) production functions the envelope of the total cost functions is a straight line through the origin and the slope of the line is the average total cost, which is constant.