So last video, we've derived that the marginal cost curve was a U-shaped function. I want to draw just a brief return to that. We have here dollars on the vertical axis and quantity on the horizontal axis. We discovered that the marginal cost curve was some U-shaped function that looks like this. Again, how did they know where to draw that? I don't. I just the general function, I know it's U-shaped. Now, should have been farther to the right, should have been higher, should have been lower, I don't know because I didn't give you the explicit functional form of the cost curve. If we had that, we could know exactly where it goes. Something is making my head think about something. You know what? Another one of these little thought bubbles. I recall that in fact, in one of our previous lectures, we discovered that the average cost curve was itself U-shaped. So since both of these curves have the same horizontal axis, output, both these curves measure the same vertical axis, dollars and cents, and they're both U-shaped. I bet you guess what we have to do next. We have to figure out where do they fit relatively. How do these curves come to play against each other relatively? This is actually a very important little piece of understanding. Let me take a small sidetrack here. If you actually knew calculus, you're not required to know calculus, but if you knew calculus, you could take a cost curve and you could take the derivative of that cost curve, and that would be your marginal cost and you could plot that. But you could also take the cost curve divided by output, and then you could map out that, and that would be the average cost function. You could, through calculus, figure out exactly what the relative size each one of those is on any given cost function I give you. It wouldn't be a hard exercise at all if you understood calculus. Right now, we are not do it in calculus, we're going to do it in graphs. But we're going to do in graphs that we know we could actually prove if we decided to set down or personally use and actually do a proof by calculus. We're not going to do that but we could. So I want to start by making you think about something here. I'm going to give you a story. Suppose you're in some program in college. At the moment in this program, you actually have a grade point average of 3.5, and that's your average grade point. Then at the current course, you actually end up getting a B, so you get a three-point for that course. So if you get a B for the marginal course, what's happening to your average? If your current average is 3.5 and then your marginal performance is three, it's going to pull that average down. If on the other hand, we have a 3.5 grade point average and you get a four-point, an A, what's going to happen to your average? It's going to go up. So when the marginal is greater than the average, it pulls the average up. When the marginal is below the average, it pulls the average down. If you think about sports, think if you ever go bowling, if your bowling average is 150, if you're averaging 150 to bowling lane and enrolled 180, your average is going to go up. If your average is 150 and then you have a bad day enroll a 120, your average is going to go down. So again, the average curve is going to go down if marginal is below. The average curve is going to go up if marginal is above it. So now, we are equipped with the logic to figure out what these two curves look like. So I'm going to draw my axes system. I'm going to put on there some arbitrary average total cost. It's U-shaped, nothing wrong with this picture, I didn't do any tricks here. Again, you could say, what did you put there, Larry? I got the pen. I know it's U shaped. I don't what to identify anything wrong here, just the general position of it. Now, I want to ask the question. This particular region of the average cost function, the average cost function is going up, correct? If the average cost function is going up, hang it up, I'm going to label this point as the output that minimizes average total cost. That's that output point right there. Everything to the right of that, average total cost is going up. If average total cost is going up, what do we know about marginal cost? Marginal cost has to be bigger than average cost. The only way average cost can go up is if marginal cost is above it. On the other hand, what about this region of the cost curve, is going down. Now, if average total cost is going down, what do we know about marginal cost? Well, it's got to be below it. If your average is falling, the marginals got to be below it. You turn three-point for 3.5 average. If you have a 3.5 grade point average and you turn out three-point, you're pulling it down. If your average is 150 at the bowling lane and you enroll 120, it's going down. Well, the only way that works is if the marginal cost curve which is also a U-shaped goes right through the minimum of the average total cost curve. The only way that works is that U-shaped marginal cost curve has to go right through that minimum. Now, if you decide to go on get yourself a PhD in economics, they'll make you prove that through calculus, and it's a very straightforward proof, you could do it. You don't even have to worry about PhD. If you understood calculus, you could prove it. If I gave you a cost function and you understood calculus, you could easily prove this. But we're not interested in that. My proof to you, I hope, was intuitively quite powerful. The only way you can pull your average up is if your marginal performance is stronger, getting your average up. If your marginal performance is weaker than your average, what's it doing? It's pulling your average down. Well, the only way you can make that work is if that U-shaped marginal cost curve goes screaming right through the minimum of the average total cost curve, okay? Thanks.