Today, we want to construct the last of these cost curves and that's going to be called the marginal cost. So, let's put down here definition. The definition is something called marginal cost or MC. That's what we'll use going forward. The definition of marginal cost is the change in total cost for any change in output. Again, as I said to you before, economists always use the word marginal to talk about a change. So, let's just work out the details here by actually putting the formula down. Marginal cost is equal to, formally, what we would say is the derivative of total costs with respect to output, that's if we had a calculus-based course. We would draw that and will be the slope of the cost function, the change in total cost for a change in output. If you don't like those derivative terms, we can just use Delta, the change in total cost or the change in output, or we can also just write it out, the change in TC divided by the change in q. This is really just the slope of the total cost curve. Again, economists use marginal anytime they're thinking about the slope. So, in this case, the marginal cost curve is the slope of the total cost curve. The marginal cost curve is the change in cost for change in output or the rise over the run if you think about your old definition for slope along that cost curve. Well, that shouldn't be too hard. Let's just take a look at what a total cost curve looks like to refresh our memory. We'll start with an axes system. We got dollars and cents on the vertical axis. We got output on the horizontal axis.You recall that our total cost curve was the sum of two different costs, a fixed cost, let's have F_0, and variable cost. The sum of those two was our total cost curve. Now, we know the detail we're trying to find here is that I want to know marginal cost which is equal to the slope of that. Formally, we would say the derivative of total costs with respect to output, or we could just say it's equal to, we'll do a. k. a., the slope. Well, you can see on this graph that the slope is positive. This curve is everywhere increasing. Again, that intuitively just has to be the case. You can imagine that the extra cost for producing an extra unit would actually be negative. That would be a great world to be in. If I want to make more jars of mayonnaise, suddenly, my costs go down. That's not going to happen. In order to make more jars of mayonnaise, I've got to actually spend extra money. But this curve tells us how much the extra money is. If we just think about the slope here, slope is flattening here. That means it's positive but it's going down, and then after some inflection point, the slope starts to go up. If you were to take this in the days of thinking about trying figure the slope, the slope at any point is the slope of the line tangent to the curve at that. So, you can see these curves are getting flatter and then after some point as we continue to step up our output production, these curves gets steeper and steeper. It gets increasingly expensive on the margin to make more of this product. So, our marginal cost curve is going down and then going up. As we increase our output along this horizontal axis, it's going down and then this is going up. Well, let's see what that might look like. That's just a general representation of the fact that total cost function had any region where the slope was flattening out, that is, the marginal cost was going down. Then the slope started to scream up and get very expensive to produce that product. That's where this up marginal cost is going up. I've got to figure out where this curve goes. I know, for example, I had another axes system that we just did in a previous video, that showed, lo and behold, that the average cost was also U-shaped. Where does the marginal cost curve fit relative to that U-shaped average cost curve? So, it's a little bit of a harder problem but not one we can't find. I want to give you an example. Suppose I told you that on any given semester, you went into a semester with a grade point average of 3.5 on your grade point. As a result of your semester's work, you actually had for that semester a 3.0 for that semester. What would happen to your grade point average? Well, you all know that. Right away, you can calculate that. If your grade point average was 3.5 but this particular semester, you had 3.0 is you're going to pull that grade point average down. On the other point, if you went into this semester with a 3.5 grade point average and then had a four-point straight A, what's going to happen to your overall average? It's going to go up. So, the key finding here to keep in the back of your head is that any time the average is going down, it must mean the marginal is below it pulling it down. Anytime the average is going up, it must be the marginal's above it pulling it up. If your average is going up, it must mean that you're doing better than you have in the past, so now, your average is going up. If your average is falling, it means you're doing worse than you used to and you're pulling down the average. You can think about that for your bowling average. You can think about it for a grade point average, all sorts of things will make that fit. Well, what that means is that if I were to construct, say, just some arbitrary axis system here and where I put on the horizontal axis, q, and the vertical axis, s, and if I had this to be my U-shaped average cost curve, the question is, where do I put that U-shaped marginal cost curve? Well, I know now that for this particular region, we say, average cost is increasing over the range highlighted by yellow. But on this range, which I'll try a different highlight color, for this range, average total cost is decreasing over the red range. Well, I understand from the previous example that if my average is going up, it must mean the marginal is above it. If my average is going down, it must be the marginal is below it. Well, the only way I can make that explained with my U-shaped, remember I got to fit a U-shaped marginal cost on here, is that that U-shaped marginal cost must be going down and then come up and go right through the minimum of that average. At the minimum point, this would be q, minimize average total cost, marginal cost equals average cost at the minimum because we know everything to the right of that point, will higher output, marginal cost has got to be above average cost because average cost is climbing. But to the left of that point, for lower output points, since average cost is falling throughout that red range, marginal cost has to be below it. Well, the only way you can make a U-shaped curve be below and above on those two different segments is if it goes right through the minimum of the average total cost curve. So, that's going to help us out a lot. Now, we know how to draw this curve. I'm going to put that on a larger axes system here so we can look at the entire collection. We know that we have an average fixed cost curve which is asymptotically approaching both axes. We have an average variable cost curve which is U-shaped. We have an average fixed cost curve which is also U-shaped. It has its minimum to the right of the minimum of the average variable cost. So, we see that the the minimum here is the minimum here. That's our regularity condition. The output that minimizes average total cost is going to be at a greater output than the one that minimized average fixed cost. So now, I want my marginal cost curve on here and it's going to look like it's also U-shaped. It goes through the minimum of the average variable cost and the minimum of the average total cost. This set of curves, we'll refer to as the family of short-run cost curves. These are going to be the workhorses that we're going to use when we're thinking about firms' decision making and optimization behavior, okay? But I need to talk a bit about this before we go on. Marginal cost hits average total cost and average variable cost at their minimums because of that relationship between marginal and average. Remember, when we went back here, I'll go back a couple slides and remind you, I found the marginal cost by taking the derivative of the total cost. But the marginal cost is also the derivative of the variable cost, right? Because we talked about earlier that the variable cost curve and the total cost curve have the same slopes at any given output. They are just vertically displaced. So, I could have also found the marginal cost by taking the derivative of the variable cost. That's why we know that the marginal cost curve and the marginal cost is going to hit the minimum average total cost and it's also going to hit the minimum of the average variable cost curve. But I didn't talk to you about, where does the marginal cost curve hit the average fixed cost curve? In order to think about that, let's draw a new picture here. Say, I know that total cost is equal to fixed cost plus variable cost. I know that marginal cost is equal to the derivative of that with respect to output or the rate of change, which would be the derivative, the change in fixed costs with respect to output plus the derivative of variable costs with respect to output. But what do we know about this term right here? Well, that term doesn't exist. There is no such thing as a change in fixed cost. By definition in the short run, fixed cost can't change. So, this term doesn't exist in the definition of marginal cost. Therefore, marginal cost and fixed cost, and marginal cost and average fixed cost are wholly independent. So, I have no idea where those two intersect. I just happen to draw it there. But, since they're completely independent, you don't know where that is. In fact, you don't even know where average fixed cost is versus average variable cost. If you were to take this picture right now and suppose this represented your firm. Suppose, you have a restaurant in downtown Manhattan. These look like your cost curves. Then all of a sudden, the city comes along and says, "Hey, we have to have a new restaurant tax. It's a once a year fixed fee. Doesn't matter how many people you serve, you're going to have to pay us $10,000 to be able to have a restaurant here in the heart of Manhattan." What that's going to do is it's going to jack up your fixed costs but not change your variable cost. That means, the average fixed cost curve on this will be shifting up. The average variable cost curve won't move. The average total cost will move up because it's the sum of the two. So, if your average fixed cost is getting shifted up in, say, some general direction like this, it's going to push up the average total cost but it's not going to move the average variable cost. In fact, this curve could push like that. It could have regions above the average variable cost. There's nothing saying that can't happen. Again, variable cost and fixed cost are apples and oranges. Marginal cost and fixed cost are apples and oranges. We don't know how those relate. Average variable cost, total cost, and marginal cost are all completely interrelated.