Economist perspective

Suppose we make products out of materials.
We make of the th product,
and to make products
we need of raw material .
We have raw materials.
We earn .

So we want to maximise s.t. .

We are now offered a small amount of material .
How much should we pay for this?

Let the value function be
For small enough we have

Now the amount we should pay shouldn’t exceed .
Which is also .
We saw this happen in Theorem (Gradient)

Note that if ,
then we are not willing to pay anything for that material.
This corresponds to the Complimentary Slackness.

When someone is selling us the materials at price ,
and wants to buy the products later,
they are maximising ,
where we can say that ,
because we want to optimise our profits for .
So they are actually solving the dual problem to ours.
It turns out that if is the supporting plane,
none of us has an incentive to change our strategies.