Greetings. In the previous video, we determined that for a simple monopolist which we call a one price monopolist, that monopolist will have marginal revenue less than price. We drew a very simple little diagram that said, "Look, suppose I had this situation where this is price and this is quantity and this was my demand curve, and I said at a $100 for price 50 people bought the product. But if I want to attract the 51st buyer into the market I'll have to drop it to 99, and that would in fact allow us to sell to 51 people." I'll draw an arrow to that spot. The reason that I found marginal revenue is less than price is because while I can get a price of $99 out of that person's pocket, I don't get to keep the full 99 as net revenue because by getting that price I had to lower the price to everybody else, I can only charge one price. Economists call this a simple monopoly otherwise known as a one price monopolist, and as a result of that marginal revenue is less than price. How much? Well, in this particular case marginal revenue was really only $49 because you got the 51st unit, your marginal revenue was equal to the $99 out of that consumer's pocket minus the $50 that you gave up by not staying at 50 units and therefore getting a $100. I instead lowered the price to 99 to sell those extra units, and so marginal revenue was in fact 49. What we want to do is make a simple rule about how this works, so I'm going to draw another graph. I'm going to start by saying let's consider formal construction of marginal revenue with a linear demand, so we're going to start by thinking this is a linear demand. So linear demand means that essentially price is equal to a minus bQ, that will be our linear form. The linear form says that essentially I'll do something like this and I know that my demand curve then, this is demand, it has a vertical intercept of a that comes from this. The vertical intercept is that first term, and the slope is equal to minus b, that comes from here, this is the minus b which is the slope, it's a simple straight line. Now I want to think about what the marginal revenue is, so to do this I'm going to do a little bit of calculus. Again, I'm not going to ever ask you to derive this calculus but I'm going to show you using calculus a neat graphical trick that you would have to use calculus to solve this but it will work perfectly, as long as you stay in the world of linear demand curves. The definition of marginal revenue is that marginal revenue is equal to the derivative of total revenue with respect to output. We know that, we know total revenue is just equal to price times quantity, fortunately for us we have an algebraic formula that tells us what price is, a minus bQ. So what we're going to do is we're going to take this a minus bQ and substitute this in and we'll write therefore the total revenue is equal to price times quantity, but we can rewrite that by understanding that price is just a minus bQ, so we'll put a minus bQ times Q. All I've done is take this straightforward algebraic form and substitute in for P, because we know from up here that P is just a minus bQ, but let's factor that through. That means the total revenue would be equal to aQ minus bQ squared, that's good, now we know marginal revenue from this formula. Marginal revenue would be equal to the derivative of total revenue which is the derivative of aQ minus bQ squared with respect to output. Again, those of you who know calculus, that's very straightforward, it would be equal to a minus 2bQ. Those of you who don't know calculus believe me, I didn't pull a rabbit out of my hat, talk to your friends in your group who do know calculus and say, "Is that really what happens there?" They'll say, "Yes, that's a simple derivative." The derivative which is the slope of the revenue function, if you have a linear demand curve the slope of the revenue function would be a minus 2bQ. Well, look at this, this is very interesting, it means that our marginal revenue curve has the same vertical intercept as the demand curve. Marginal revenue has the same vertical intercept as demand curve and that's good, we could think about drawing this curve by saying, "Well look, I know that marginal revenue curve has the same vertical intercept as demand, but what's the slope?" Well, its slope is twice the slope of the demand curve. The slope of the demand curve is minus 1b, the slope of the marginal revenue curve is minus 2b, which means that the marginal revenue curve goes down at twice the slope of the demand curve. The marginal revenue curve has a slope of MR equals minus 2b, so it goes down twice as fast as the demand curve. What that means is that this point, since it will go down twice as fast, if we want to call this point alpha, that makes this point to be 2-alpha because it's the midpoint. It's the midpoint because that line goes down twice as fast which means that essentially the marginal revenue curve starts going negative once you get past the midpoint of the demand curve, marginal revenue curve starts to go negative if you pass that point. Now, the important thing for you here is to think about what this mean to you. What it means to you as long as we use linear demand curves which I will, which is perfectly reasonable, as long as we have linear demand curves, the monopolist marginal revenue curve, you don't need to do the calculus. So monopolist marginal revenue curve will have the same vertical intercept as the demand curve, and it will go down at twice the slope. So if I ever give you for example just for the heck of it, let's draw another picture. Suppose I told you that the monopolist demand curve was 100 minus 6Q, that you would know the marginal revenue curve would be also 100 for the vertical intercept minus 12Q, and so it'd go down at twice the slope. All right, good. Thanks.