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.