Because we use the envelope theorem in constrained optimization problems often
in the text, proving this theorem in a simple case may help develop some
intuition. Thus, suppose we wish to maximize a function of two variables and
that the value of this function also depends on a parameter, \(a: f\left(x_{1},
x_{2}, a\right) .\) This maximization problem is subject to a constraint that
can be written
as: \(g\left(x_{1}, x_{2}, a\right)=0\)
a. Write out the Lagrangian expression and the first-order conditions for this
problem.
b. Sum the two first-order conditions involving the \(x^{\prime}\) s.
c. Now differentiate the above sum with respect to \(a\) - this shows how the
\(x\) 's must change as \(a\) changes while requiring that the first-order
conditions continue to hold.
A. As we showed in the chapter, both the objective function and the constraint
in this problem can be stated as functions of
\(a: f\left(x_{1}(a), x_{2}(a), a\right), g\left(x_{1}(a), x_{2}(a), a\right)=0
.\) Differentiate the first of these with respect to \(a\). This shows how the
value of the objective changes as \(a\) changes while keeping the \(x^{\prime}\) s
at their optimal values. You should have terms that involve the \(x^{\prime}\) s
and a single term in \(\partial f / \partial a\)
e. Now differentiate the constraint as formulated in part (d) with respect to
\(a\). You should have terms in the \(x\) 's and a single term in \(\partial g /
\partial a\)
f. Multiply the results from part (e) by \(\lambda\) (the Lagrange multiplier),
and use this together with the first-order conditions from part (c) to
substitute into the derivative from part (d). You should be able to show that
\\[
\frac{d f\left(x_{1}(a), x_{2}(a), a\right)}{d a}=\frac{\partial f}{\partial
a}+\lambda \frac{\partial g}{\partial a}
\\]
which is just the partial derivative of the Lagrangian expression when all the
\(x^{\prime}\) 's are at their optimal values. This proves the envelope theorem.
Explain intuitively how the various parts of this proof impose the condition
that the \(x\) 's are constantly being adjusted to be at their optimal values.
g. Return to Example 2.8 and explain how the envelope theorem can be applied
to changes in the fence perimeter \(P\) -that is, how do changes in \(P\) affect
the size of the area that can be fenced? Show that in this case the envelope
theorem illustrates how the Lagrange multiplier puts a value on the
constraint.
