Okay, now let's move to the land of price optimization using some

demand information, right?

We've looked at those elasticities, but now we want to try to find the price that

really is the best given the conditions that we're in.

Now, in front of you is our good old demand function.

You've seen this before, right?

Q = 10- 2P and I'll just graft that below.

And there's a marginal cost equal to 1.

And we're going to take this demand and formation and this marginal cost

information and come up with the price that would be best in this situation.

Now, how do we do it?

Well here's what we do.

First we need to write down a profit function.

And when I write down pi, I mean profit.

And that's true in just about every textbook you'll ever look at,

if it has price optimization they'll use pi for profit, it's just a convention.

But what is profit in this setting?

Well, it's the amount of money you make on each unit,

kind of your marginal profit per unit, or your margin,

which is just (P-MC), price minus marginal cost, and then you have to

multiply that by the number that you sell, right, there's that Q right there.

The number you sell multiplied how much you make on each unit,

that'll give you the profit.

Now, what do we do with this profit function?

Well the first thing we do is we substitute in the demand information.

And in our case that is 10- 2P, I'm getting

that straight from the previous slide, the description of that demand function.

So I substitute that in, and if you notice at this point because

we know what MC is, this is a function, the profit function,

all in terms of one unknown, and that one unknown is price.

That's what we're trying to find.

So all I've done here is multiply this through and simplify it a bit.

Now, what do we do?

Now we gotta do some of that calculus, nobody freak out.

We're going to use that simplified profit function and

a little bit of calculus to come up with the best price.

So here's that simplified profit function again.

What are we going to do?

Well, we're going to maximize it related to price.

So what does that mean?

We're going to take the derivative, set it equal to zero, and that, when it's solved,

should give us the best price in this particular situation.