2The method of Lagrange multipliers
IB Optimisation
2.2 Shadow prices
We have previously described how we can understand the requirement
∇f
=
λ∇h
.
But what does the multiplier λ represent?
Theorem. Consider the problem
minimize f(x) subject to h(x) = b.
Here we assume all functions are continuously differentiable. Suppose that for
each
b ∈ R
n
,
φ
(
b
) is the optimal value of
f
and
λ
∗
is the corresponding Lagrange
multiplier. Then
∂φ
∂b
i
= λ
∗
i
.
Proof is omitted, as it is just a tedious application of chain rule etc.
This can be interpreted as follows: suppose we are a factory which is capable
of producing
m
different kinds of goods. Since we have finitely many resources,
and producing stuff requires resources,
h
(
x
) =
b
limits the amount of goods we
can produce. Now of course, if we have more resources, i.e. we change the value
of
b
, we will be able to produce more/less stuff, and thus generate more profit.
The change in profit per change in b is given by
∂φ
∂b
i
, which is the value of λ.
The result also holds when the functional constraints are inequality con-
straints. If the
i
th constraint holds with equality at the optimal solution, then
the above reasoning holds. Otherwise, if it is not held with equality, then the
Lagrange multiplier is 0 by complementary slackness. Also, the partial derivative
of
φ
with respect to
b
i
also has to be 0, since changing the upper bound doesn’t
affect us if we are not at the limit. So they are equal.