In this lesson, we're constructing cost curves. Previously, we looked at how to build a total cost curve. We got that information by understanding the production function, translating that production function and the inputs that it entails into dollars and figures of expenses, and we were able to build a total cost curve. In this video, we're going to think about building a new type of cost curve. So, we'll start with the definition, and we're going to define something called average cost. It's really average cost but will be a little bit more formal and say average total cost which we will start using the shorthand of ATC for average total cost. Average total cost is just basically per unit cost. That's the way to think about average total cost. It's defined as the ratio of total cost to output. So, if you think about our ongoing example of the mayonnaise at the Kraft plant, there's a total cost to run that plant, and Kraft's accountants can figure that all out. You add up everything that's involved, that's just the labor, the cost for the brick and mortar, that cost for the eggs, the cost for the glass, the cost for the design team, and the focus groups to look at the labeling to see what they want to put on the side of the jars, all of these things are all involved. If you take all of those costs and divide them by the total amount of jars of mayonnaise that went out, what you've got is a prorated share of cost that belongs to each one of those jars of mayonnaise. Each jar of mayonnaise say has, if you take the total cost and divide by the number of mayonnaise, maybe $3.28 of cost. Okay. That's the per unit cost. It's going to be important to us because as you can imagine, if I tell you that the per unit cost of $3.28 but that the price that the market will support is $3.18, it's probably not a good line of business to be in. At that point in time you begin to realize, well, I'm not making as much back as I actually put in each one of those jars of mayonnaise. So, we're going to have to know a lot about this average cost when we start thinking about why firms exist, what they're doing with their line of business. So, I'm going to expand this a bit talk about it a little bit more in terms of what's going on. We know that in fact I can rewrite this numerator as fixed cost plus variable cost over output. I can expand that out to the term of fixed cost over output plus variable cost over output. This term, we're going to relabel as average fixed cost, and on this term, we're going to relabel as average variable cost, otherwise known as AFC or AVC. I'll just rewrite these then as average total cost is equal to the sum of average fixed cost plus average variable cost. Just as we did when we built the total cost curve, we built the total cost curve by looking at the two components of it, the fixed cost component, we drew a graph for that, the variable cost component, we drew a graph for that, and then we added the two to get the total. We're going to do the same strategy here. I want to get to this, average total cost, I'm going to build this and add that after I build it. Okay, so let's start. What I'm interested in is figuring out what does the average fixed cost curve look like. Well, the average fixed cost is definitionally equal to fixed cost divided by output. You saw that on the previous slide. Average fixed cost is just fixed costs divided by output. So, in this slide we want to understand what that picture looks like. Well, immediately you can see that since the numerator is fixed, as I step along this axis to the right, increasing output, this denominator is going to grow and this numerator is constant which means this has to be a falling curve. In fact, it falls in a very specific way. It's not a straight line at all. It's a curve that looks like this. We'll call this curve, average fixed cost. Think about it, as output gets very large, let's go back to this picture, this ratio, as output gets very large, what's happening to this ratio? It's getting very small but does it ever get to zero? No, it asymptotically approaches the horizontal axis, but it's never really going to touch the horizontal axis which would be zero-level. Likewise, if output got very small and as you start reducing production a lot, this thing would go up very very fast in a hurry. In fact, if it went to zero, this thing would not be defined; it's infinite. Think about it, suppose I tell you that if you're going to sell cars you got to put advertisements on TV. Well, advertisements on TV cost a lot of money and you've got a $10 million ad campaign, but you sell 10 million vehicles like a big company like General Motors, well, it's about a buck a ca. Okay, but if you have to spend the same $10 million that General Motors spent to put your ads on TV and you're only going to sell a million cars, the per car share of that fixed cost is now ten bucks a car, still not a big deal. Suppose you're a boutique seller who only sells 250 thousand cars. They're expensive, and part of the reason they're expensive is because when you put the commercials on, now the per car share is pretty high. Okay. When you got to start making small numbers of cars, and you still try and advertise those, that average fixed cost is going to get pretty steep. That's what this curve reflects. If you showed this to somebody who was in a geometry class, they'd say that's a rectangular hyperbola. A rectangular hyperbola is exactly this. It's a hyperbola; it's curved but it's curved in a special way. If you were to take any point on this graph, the rectangle defined by that point would be the same area. That area, of course, as you can guess would be that constant fixed cost. No matter what point you pick on this graph, it will define a rectangle. The area of that rectangle will be a constant equal to that. All right, we're only halfway home. You remember, we have to do this, and we have to do this. The second one's a little bit harder than was before, but we got through it, and we'll get through this one. We draw another graph, another axis system. Let me put dollars and cents on the vertical and we put output on the horizontal. Now I want to think about the average variable cost which is equal to the variable cost over output. But when I did average fixed cost, the ball game was easy because as I increased my output, the denominator went up, but the numerator was constant. Not the case anymore. As I increase my output, the denominator is going to grow, but I also know the numerator is going to grow because more output is going to be more costly. There's no such thing as magic production. If I want to put more output out there, I know my variable costs are going to have to grow. So, what's really interesting to me then is, what's happening to the rate of change of these two? What's happening to the rate of change of these two? So, the way I'm going to think about this, is I'm going to ask you to take a simple abstraction with me. Let's go back and look at our variable cost curve. The way we designed it earlier. We had a variable cost curve that looked this, right? I know that what I'm looking for today is something called the average variable cost, which would be the ratio of the numerator, which is variable cost to the denominator, which is quantity. Well, suppose I've picked some amount here, right? Right here. We'll call this q sub zero. I know that at q sub zero, variable cost would be equal to this height. Let's call that vc at q zero. That would be the variable cost at output level q zero. But look what I'm asking. I've been asked to evaluate a ratio that is essentially variable cost, which is that height over output, which is that length across the horizontal axis. But, remember the old trick days, that's opposite over adjacent. If you take the opposite side of a triangle over the adjacent side of a triangle, that's going to give you essentially the slope of that line. As you can see, as we step out now, if I were to take a different output level, the base is the new higher output. The denominator is the new higher cost, and the triangle formed by those two, that slope, that slope of that third line, the hypotenuse is falling. Eventually, we get out to some point, like about right here, where that line will in fact just be tangent. Then, if we continue to increase output we're going to see our triangles are now going to have the third line. The hypotenuse is now going to start getting steeper and steeper. So, don't get too confused in a trigonometry. Understand that what's happening here is that, at low levels of output, as I increase my output, that slope of that line, that is the third side of the triangle is getting smaller and smaller. It's falling. What that means is that the average variable cost is falling. Now, it's not negative. I'm not saying it's going negative because there's no such thing as negative variable cost, but it's going down. The ratio itself is collapsing. That's the magic we needed because when we were over here looking at the picture we said, what I need to know is somehow as I increase output, what's happening to the relative size of the numerator and the denominator? Now I know that the numerator and the denominator, that ratio is falling over a certain range and then rising. An economist will draw that like this way, I will call this average variable cost, and we call this u-shaped average cost curves. Where's that shape come from? It comes again from that idea that on the production function, there's some low levels of output where as you increase output, your costs don't increase as much as your output does. So, the average of that variable cost function is actually falling. That would be this region. But after some point, that famous inflection point trying to get more output gets increasingly expensive, and we can see the average cost starts to go up. So, we've got the two curved up. Let's put them together. We want to build an average cost. We now know that we have an average fixed cost curve that looks like this. We have an average variable cost curve that's got some general u-shape to it. I'll just put it here, and we'll talk more about what that general u-shape might be as we go forward. Average variable cost. Now, I just want to add the two. If I add the two it's going to look something this. This is our family of average cost curves. We had three different total costs we were looking at. We were looking at a fixed cost curve, a variable cost curve, and the sum of those was the total cost. Now we have an average fixed cost curve. We have an average variable cost curve, and the sum of those is this average total cost. Now, it's important that we talk a bit about something that you should see here, is the fact that Larry's just a sloppy draftsman or is there something going on here, and in fact, there is something going on here. So, we're going to say there's a couple regularities here. First of all, you'll note the way I've drawn it, the output that minimizes average variable cost is less than the output that minimizes average total cost. That is in fact a regularity, it's not just Larry sloppy drafting. That's the way it's going to be, and has to do with all properties you could prove. If I gave you a cost function and you knew something about calculus, you could take that cost function, construct an average cost function, which is dividing it by q, take the derivative of that. Once you have your average cost curve you'll know exactly how to minimize that by taking the first order condition. All things you could do with an average variable cost, and you could prove what I'm writing down right now, that the output that minimizes average variable cost is less than the output, it's actually less than or equal, it might be equal to one then, the output that minimizes average total cost. The other thing that's happening here is that for high output, average variable cost and average total cost converge. They don't quite touch. But you can see that for high output, because average fixed cost is vanishing, for high output levels, average fix cost is asymptotically approaching the axis. That means, those two curves, average total cost and average variable costs are going to getting closer and closer as you go out, farther output. So, this is our family of average cost curves. We have one more cost curve we need to introduce, and actually it's the most important one, but it builds upon these.