Deriving the Marginal Cost Curve Page 1 of 2 - Cengage

Production and Costs Marginal Costs Deriving the Marginal Cost Curve

Page 1 of 2

In this lecture we're going to draw a marginal cost curve and I'm going to show you first the easy way to draw the marginal cost curve and then I'll show you a relationship between cost and productivity that you can see by looking at the curve.

First, let's go back to the numbers that we derived before--the marginal cost of producing another unit of the good. And I'll put these numbers over there so that you can work with them. And what we'll do is plot the points in a space where we have output on the horizontal axis and cost on the vertical axis. So the first point will be two television sets and the marginal cost of producing an extra television set when you're making two is $500. So we'll put a dot right here at two televisions and $500. One, two, three, four, $500, two televisions. Ten television sets takes the marginal cost down to $125. So I'll put a dot there. If we're up to 30 television sets, the marginal cost is down to $50 a set, and if we are at 40 television sets we have $100 for our marginal cost. Forty-five gives us a marginal cost of $200, 48 gives us a marginal cost of $333, so I'll put a dot here. And finally, at 49 television sets we have a marginal cost of $1,000.

Now, of course, these are not the only quantities of televisions we could be producing. These are not the only cost numbers we could have. So to represent the points in between, we will connect the dots with a line and that will become the marginal cost curve. So let me do that now. I connect my dots with a green line and the green line becomes the marginal cost curve. There we go--marginal cost, the cost of producing an extra unit of output given the number of units that you're already producing. When you're only producing two units, the marginal cost of an extra unit is high. But if you're producing 30 televisions, the marginal cost of an extra unit is low.

Now, you might ask yourself, "Why is that?" Why does the marginal cost of production vary as you change the quantity that you are producing? What do you think? The answer is this. Cost and productivity are reciprocally related. If it takes more workers to produce a television set, you're going to have to pay more for that television set. And if the additional worker, if the next worker that you hire produces only a small fraction of a television set, then you have to hire a lot of workers to produce a whole television set. Again, it's the reciprocal relationship between costs and productivity. If the next worker produces only one-quarter of a television set, it takes four workers to produce a whole one. But if the next worker produces half of a television set, it only takes two workers to produce a whole one. The inverse relationship between cost and productivity.

You can see that inverse relationship in the graph. Does this graph look like something that we've seen before? It certainly does. Let's take the graph now and flip it over and look at it from the other direction. What does this look like? Well, let me first of all fill in the little chunk that I lost due to the tape. And you can see you've got this upside down, U-shaped curve, which is exactly the shape of the marginal product curve. I need to re-label my axes. If I put output on this axis like we did before, and if I put labor down here on this axis, then this curve becomes the marginal product curve. I can just put an MP right here. See, the marginal cost curve and the marginal product curve are just mirror images of each other. That's because cost and productivity are reciprocally related. When productivity is increasing, when the marginal product is increasing due to teamwork and specialization, then the marginal cost of an extra unit of output is falling. When the marginal product is falling, the marginal cost of producing an extra unit is rising, because you're having to hire more and more workers at the margin to produce an extra television set.

So the point that I'd like to make here is that marginal cost and marginal product are mirror images of each other. They are reciprocals of each other. Now, I can show this same point using some simple mathematics, and I'm going to do that in just a minute. But first, I want to show you another way of deriving the marginal cost curve based on the definition of marginal cost. Remember, the definition of marginal cost is the change in variable cost that results from producing an extra unit of output. So what we can do is look at the variable cost curve and show how to derive the marginal cost curve from it, because the two concepts are related by definition. Let's look now at the variable cost curve. The variable cost curve shows us how much we have to pay to hire the variable input labor that's necessary to produce any given quantity of television sets.

Now, look at the variable cost curve and notice that the variable cost curve has a very steep slope close to the origin. And the slope of the variable cost curve gets flatter and flatter until we reach this point of inflection. At that point the slope stops getting flatter and begins to get steeper again. The curve gets steeper and steeper and steeper, and finally becomes completely vertical when we reach the capacity of our plant. Remember, in the short run you reach a point where you simply cannot produce any more television sets, no matter how much labor you hire. At that point,

Production and Costs Marginal Costs Deriving the Marginal Cost Curve

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you can hire all the extra labor you want, but you don't get any extra TVs. At that point the variable cost curve becomes vertical, it goes straight up.

Well, think now about what this slope tells you. The slope of the variable cost curve is the rise over the run. The rise in this diagram is a change in variable cost. That is, an increase in the vertical coordinates mean that the variable cost is increasing, you're spending more money on labor. The run in this case is an increase in the production of television sets that is, producing more television sets as you hire more labor. So the slope of the curve, the rise over the run, is the change in variable cost that results from a change in output.

Well, that's the definition of the marginal cost. So if you want to graph the marginal cost curve and you want to do it in a kind of intuitive way that relates it to the definition of marginal cost, you can work from the variable cost curve itself. The marginal cost curve is a graph of the slope of the variable cost curve at every point.

Well, if I want to draw a marginal cost curve down here in the diagram below the variable cost curve, I've got to do this kind of carefully. First thing I have to do is make sure that I've got the same variable on the horizontal axes. That is, if you're going to stack one graph on top of another, you've got to make sure that the axis that they share are measuring the same things. So I'm measuring output on both of these horizontal axes. Upstairs I'm measuring variable cost on the vertical, and downstairs I'm measuring marginal cost on the vertical.

So next thing I have to do is I've got to draw carefully a line down here below that shows the changing slope of the variable cost curve as we move along it. The most important point in this graph is going to be this point right here. That's the point of inflection, the point at which the variable cost curve stops being concave and starts being convex. This is the point at which the slope stops diminishing and starts increasing. So if I'm real careful about this, I can indicate the quantity of television sets at which that inflection occurs and drop that on downstairs into the marginal cost diagram and make that the minimum point on my marginal cost curve. Marginal cost is decreasing, that is, the slope is decreasing up to that point, and then beyond that point it's going to be increasing again. So I've identified the minimum point on the marginal cost curve.

Now if I wanted to be totally mathematically precise, I'd have to figure out exactly what the slope is at that point and put my dot down here at the number that represents that slope. But I'm not going to be that careful. I'm just trying to represent the main idea here. We know that this is the inflection point. This is the point of minimum marginal cost. Well, that's also going to let us be able to draw the curve now, because over here, on this side, as output is increasing, marginal cost is diminishing, so I can draw marginal cost that is falling down to this point. So the marginal cost will be decreasing as this slope is decreasing, until I reach the inflection point.

Now, beyond the inflection point, marginal cost starts to increase again. As you see, the slope gets steeper and steeper. So beyond that point I draw an upward sloping marginal cost curve that looks like this. And this curve heads on up to infinity as the variable cost curve becomes vertical, because you know, the slope of a vertical line is infinity, so the marginal cost curve is shooting on up so that it has a reading of infinity by the time the variable cost curve goes vertical. Well, I can label this curve now "marginal cost."

It's the same curve that I draw before when I was carefully plotting these points from over there. But now I'm doing it intuitively. I'm using the definition of the marginal cost curve, the slope of the variable cost curve, and I'm using that now to derive the marginal cost curve graphically. I like this derivation because it makes clear the connection between marginal cost and variable cost. This curve represents the slope of this one.

What I'm going to do next is show you a mathematical relationship between cost and productivity that I think will give you more insight into what marginal cost is all about.

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