Thomas let's do an example of what you were talking about before. So let's do it for a particular segment. And we'll call that Segment 1 and this segment has particular demand conditions and cost conditions. And we want to analyze that both mathematically and graphically so that we can see, what's the best price to charge? What's the price we're going to make the most amount of money? So let me put a simple XY axis up here. And we've got Q, which is Quantity, that's the amount were selling on one axis. And P, which is Price, on the other axis. And Let's take the demand conditions as Q=10-2P, now that's a nice linear demand function. Right? It looks like what who you'd expect a demand function to look like. In that price and quantity are negatively related to each other. If price goes up, quantity is going to come down on that. And then let's specify some cost conditions as well. So let's say marginal cost equals 2. And that's constant marginal cost. Right? So how ever much we sell, each unit is going to cost us $2 more to produce. >> Why is it so important to emphasize marginal cost? Well, that's a good question. Because what we're going to see in a minute here is that what we have to solve for is the place where marginal cost equals marginal revenue in order to get the optimal price. And the reason that's the case I think is intuitive. You don't want to sell something if your marginal revenue is not at least above your marginal cost, right? >> Yes. >> That you will be loosing money. And so you'd be willing to sell all the way up to the point where your marginal revenue exactly equals your marginal cost ut not past that point, right? It doesn't make any sense. Now we can use this graphically too, right? Let's look at demand characteristics here, and let's get a graph of that. So. Hm. If I have P equals 0 then Q equals ten and then where do I sell nothing? What if P equals to five, two times five is ten and that look right so. It's kind of interesting graph because it's a little shallow, all right. So we've got five there and I've got a line that comes down just like that. >> Okay. >> Now the marginal cost is constant. Marginal cost equals two. So I can put two right here. >> It's the horizontal line. >> Yeah, that's right. So that's your marginal cost equal to two. And you can see that those two things do intersect each other. Now how do I find a place where marginal revenue equals marginal cost because that what we were talking? >> Yeah. >> Well it turns out that I need to do a couple of things to make that happen. One is I need to use Is the inverse demand function, and why I'm going to do this is just going to make it easier. I'm going to take this equation, and I'm going to solve it for P in terms of Q, right now. >> Mm-hm. >> It's Q as a function of P. I'm going to flip that around. And when I do that, and you all can check my math on this. But it's five minus one half Q, that's what the inverse demand function is going to look like. >> That looks right. >> Okay. Now we have to take that and get revenue right? so how many- >> It's P times Q. >> Right, okay so it's P times Q, so if I have revenue here and P let's just substitute that in right I guess that would be the easiest because then will get it all in terms of One variable which is Q so (5-1/2)Q and then to give revenue I need to multiply that by Q. But in revenue tha's not enough. I got my marginal cost I got get marginal revenue here not the total revenue. >> So we need to get rid of the Q. >> Exactly correct. So in this situation, what we need to do, now don't let Thomas scare you talking about derivatives, okay? If you don't remember how to take a derivative, we have a file in the resources section of this course that will remind you how to take a derivative in a situation like this. But he's exactly right. That's what we need to do. Let me do that now. I'll take that and that is with respect to Q. When I do that I am going to get a five. Then, I will get just a simple minus Q. That's the derivative of that function. Now again, why is this important? It's for what Thomas said before. We need to take the marginal revenue and equate it to the marginal cost and that's going to imply the best price. So if I go over here, equals marginal cost. Do we know what each of these one are? Yes we do. Marginal- >> Before you go there this is also linear function maybe we can plot without because >> Okay. >> I think you can even see this point on the graph. >> Okay that yeah that makes a whole lot of sense so if I take that is marginal revenue [CROSSTALK] >> 0, solve it five. >> Right, okay it's perfect. >> Price turns zero if it's >> If Q is five right? >> Right so we start at that same point but it looks it go down more steeply. So, what about five right here? That's where your marginal revenue. It is going to hit zero >> So on the graph we can already see which point we have to find >> Yeah, there it is, there is the point we are trying to find. And then once we find that point. I guess the question we are going to ask is what price should we charge that point, and that's going to be given by the demand function. Right up here >> Right. >> So I guess that's what we're, thank you for pointing that out, I think graphically that make a lot of sense and it helps the math here be more intuitive. >> All right so got marginal revenue equals 5- Q. And equal to marginal cost was that, that's two. So I think that's pretty very intuitive because if you take that Q we equals three, right? >> It's right. >> Q equals three. Okay now what is that Q, that's the amount that you're going to sell at the optimal price, the amount on the center. What's the optimal price going to be? Well we get that just by substituting that back into that inverse demand function. So if you take that three and put it back in there I think everybody can see that's thee halves right there and five minus three halves is 3.5, three and a half and so I'll put in here price equals 3.5 and that's this price right up here, that's 3.5 and then how much money are we going to make? Well profit which is just equal to your margin which is P minus MC multiplied by how much you're selling. And in this case we have 3.5 -2, and then where's the quantity coming? There's the three again. So what's this? That's 1.5 times three. So for this particular segment this particular domain conditions and cost conditions. What are we getting here, we're getting 4.5 in terms of the overall profitability. >> You know I'm a little bit disappointed. I thought we could do better. >> Yeah >> Maybe we can find a segment that has a higher willingness to pay? And is not as price sensitive so we have a different demand. >> Different demand function. >> Different demand function. Probably more costly to save those customers because if they want to pay more they want to get more value. >> Okay, so this construction might change as well? >> Yes. >> So, you want to take a look at that now? >> Let's do it. >> All right.