So what if you know more than just a demand schedule, kind of a list of numbers? But you actually know kind of a statistical relationship between quantity and price? Well, in order to do that you have to work with something called a functional form. Now what is a functional form? A functional form is a relationship between price and quantity that you get from somewhere. Now, where do you get it? We're going to talk at length in this class about where we get, there's lots of different ways you can get this. Consumer surveys, experiments, data that you get from stores to estimate statistical relationship, we're going to go through all of that. But right now I just want to get down in a very simple way what you can understand from this demand relationship. So, right over here we have Q=f(P), very generic. Lets assume for a minute that Q=10-2P. Now again, where did that come from? Well I just wrote it on the board and assumed it. Once we get farther into the course we'll actually come up with this kind of thing on our own. Now okay, what can we do with Q=10-2P? Well one of the things that we can do is we can graph it, kind of like we've graphed that demand relationship before. I can come down here and, I can put quantity on one axis. I can put price on the other axis. And, this is a line, right? That's the equation of a line. Now, how do I actually draw it here? Well I think it's pretty intuitive if you look at this. That if you set P equal to 5, and you've got 2 times 5 here, which is 10, that implies that Q equals 0. So we can go down here and find what's called the x intercept or the quantity intercept, and that's going to be out here at P equals 5 and quantity equals 0, so up here we have P equals 5. And at that point you don't sell any of this kind of stuff. Now what if I set price equal to 0? Well, that's the point which I should sell the most, right? And in fact if I put a 0 in here for P the 2 times 0 is just 0. So that implies that quantity is going to equal 10. So what does that mean? Well it means that this is actually way out here at 10, and in between, I'm going to draw the best line I know how to draw up here on the board, okay, that is what's called the demand function. If you've had a microeconomics class, you've seen some things that look like this, you've seen graphs like this, and they were called demand functions. If you haven't had a microeconomics class, that's basically what it is. It's just a graph that depicts all the price and quantity combinations. Now, of course, we're in a pricing course, so what we want is not just the whole demand function. We want to find the price. On that demand function so that we make the most amount of money, right? And we're going to call that P* where do we find P* from if we have that kind of demand relationship? Well where that comes from is we go back to that profit function we talked about before. But this time the profit function is a little bit different. Remember that profit, and I'm going to write it here as Pi, Pi equals profit, is just the quantity you sell multiplied by however much you make. And remember that's P, the price, minus however much it costs you to make. Well in this situation, we know what Q is. Q=10-2P. Costs are known, so what I can do is I can write the profit function one more time. And that is q, which is 10-2P, and I can multiply that by P-C, and what's nice about that is that it's a profit function written only in one variable. There's only one thing that we really don't know here. Because we know costs, that thing that we do not know is the price, which we get to set. And we want to find the profit function at P*. The maximum amount that we can make. Now, how do we find that? Well, I hate to break it to you. This is going to involve just a little bit of calculus. [SOUND] Nobody freak out about that. It's very basic. If you don't remember calculus, if you've never had calculus, I can teach you enough calculus in order for you to be able to solve this problem. And we are going to get to that calculus material but we're going to do it right before the price optimization that happens later in the course because that's where it will prove to be useful. Right now we're going to move from one economic concept, demand, to another economic concept that you almost certainly saw in your microeconomics course if you ever took one of those.