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 Supporting hyperplane > 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.