Overheads Chpt.7

Cost Structure of the Firm

We can look at the business firm from at least two points of view: productivity, inputs, and outputs (as we have just done) or inputs and costs. In advanced microeconomics, these two points of view are called "duals." They are equally valid, but they point up different things. By looking at the firm from the point of view of costs, we shift our perspective somewhat, and gain a much more direct understanding of supply.

We also look more directly at the difference between the long and short run. In the short run, we have two major categories of costs:

Fixed and Variable Costs

*Variable costs are costs that can be varied flexibly as conditions change. In the John Bates Clark model of the firm that we are studying, labor costs are the variable costs. Fixed costs are the costs of the investment goods used by the firm, on the idea that these reflect a long-term commitment that can be recovered only by wearing them out in the production of goods and services for sale. Of course, productivity and costs are inversely related, so the variable costs will change as the productivity of labor changes.

Cost Tables

Let's return to the numerical example in the last chapter. In that example we considered only the variable costs -- the labor costs in the John Bates Clark model of the firm. So we don't know what the fixed costs are. Let's assume that they are $80,000 per week, and the wages and all other costs are on a weekly basis too. Here is a table with the output, fixed, variable, and total (fixed plus variable) costs in the example.

Q FC VC TC
0 80000
80000
945 80000 50000 130000
1780 80000 100000 180000
2505 80000 150000 230000
3120 80000 200000 280000
3625 80000 250000 330000
4020 80000 300000 380000
4305 80000 350000 430000
4480 80000 400000 480000
4545 80000 450000 530000
4500 80000 500000 580000

Fixed and Variable Costs 2

Here is a picture of the fixed costs (FC), variable costs (VC) and the total of both kinds of costs (TC) for the productivity example in the last unit:

Figure 1

Opportunity Cost, Again

What is the connection between the distinction we have just made -- fixed vs variable costs -- and opportunity cost, the key concept in some earlier units?

In economics, all costs are included -- whether or not they correspond to money payments. If we have opportunity costs with no corresponding money payments, they are called implicit costs. The implicit costs (as well as the money costs) are included in the cost analysis we have just given.

There is some correlation between implicit costs and fixed or variable costs, but this correlation will be different in such different kinds of firms as

Unit Cost

Costs may be more meaningful if they are expressed on a per-unit basis, as averages per unit of output. In this way, we again distinguish

Unit Cost Example

Here are the average, average variable, and average fixed costs for our example firm.

Table 1

Q AC AFC AVC
945 138 85 53
1780 101 45 56
2505 92 32 60
3120 90 26 64
3625 91 22 69
4020 95 20 75
4305 100 19 81
4480 107 18 89
4545 117 18 99

A Picture of Unit Costs

Here are the average cost (AC), average variable cost (AVC) and average fixed cost (AFC) in a diagram. This is a good representative of the way that economists believe firm costs vary in the short run.

Figure 2

Marginal Cost

As before, we want to focus particularly on the marginal variation. In this case, of course, it is marginal cost. Marginal cost is defined as

As usual, Q stands for (quantity of) output and C for cost, so Q stands for the change in output, while C stands for the change in cost. As usual, marginal cost can be interpreted as the additional cost of producing just one more ("marginal") unit of output.

Marginal Cost Example

Let's have a numerical example of the Marginal Cost definition to help make it clear. In the John Bates Clark style example we have been using, total cost is 280000 for an output of 3120, and it is 33000 for an output of 3625. So we have

and

so that

for a marginal cost of $99.01 for the next unit produced. As usual, this is an approximation, and the smaller the change in output we use, the better the approximation is.

A Picture of Marginal Cost

Here is a picture of marginal cost for our example firm, together with average cost as output varies.

Figure 1

Notice how the marginal cost rises to cross average cost at its lowest point.

Maximization of Profits, Again

We can now give another rule for the maximization of profits. The new rule is really just the same rule as we saw before, only now we state it in terms of price and costs. It is the equimarginal principle in yet another form.

The question is: "I want to maximize profits. How much output should I sell, at the given price?"

The answer is: increase output until

p=MC

The point is illustrated by the following table, which extends the marginal cost table in an earlier page to show the price and the profits for the example firm.

Marginal Cost Example with Profits

Table 3

Output Average
Cost		 Marginal
Cost		 price profit
0 0 100 0
9.45
945 137.57 100 -35503.65
52.91
1780 101.12 100 -1993.60
59.88
2505 91.82 100 20490.90
68.97
3120 89.74 100 32011.20
81.30
3625 91.03 100 32516.25
99.01
4020 94.53 100 21989.40
126.58
4305 99.88 100 516.60
175.44
4480 107.14 100 -31987.20
285.71
4545 116.61 100 -75492.45
769.23

Marginal Cost and Firm Supply

We have discovered the principle of supply for the individual firm.

Remember: what is supply? It is the relation between the price and the quantity that people want to sell. For an individual firm, that is: the relation between the price and the quantity the firm wants to sell.

So we ask: at a given price, how much will a (profit- maximizing) firm want to sell? The answer: enough so that the price is equal to marginal cost. In other words, the marginal cost curve is the supply curve for the individual firm.

Shutting Down and Bankruptcy 1

As long as the firm produces something, it will maximize its profits by producing "on the marginal cost curve." But it might produce nothing at all. When will the firm shut down?

The answer goes a bit against common sense. The firm will shut down if it cannot cover its variable costs. So long as it can cover the variable costs, it will continue to produce.

This is an application of the opportunity cost principle. Just because fixed costs are fixed, they are not opportunity costs in the short run -- so they are not relevant to the decision to shut down. Even if the company shuts down, it must pay the fixed costs anyway. But the variable costs are avoidable -- they are opportunity costs! So the firm will shut down it it cannot meet the variable (short run opportunity) costs. But as long as it can pay the variable costs and still have something to apply toward the fixed costs, it is better off continuing to produce.

Shutting Down and Bankruptcy 2

It is important not to confuse shut-down with bankruptcy. They are two different things. If a company cannot pay its interest and debt payments (usually fixed costs), then it is bankrupt. But that doesn't mean it will shut down. Bankrupt firms are often reorganized under new ownership, and continue to produce -- just because they can cover their variable costs, and so the new owners do better to continue producing than to shut down.

More on Cost

Before we leave the cost curves behind, let's look at one more example. It will be a little more complicated. The real world is pretty complicated and many economists would say that this example is complicated enough to be help understand some real-world problems. We will again focus on the breakdown of cost into fixed and variable costs, in this example.

Let's start with the total cost curve. Here it is:

Figure 4

In Figure 4, we have the total cost on the vertical axis, with output varying from zero to 1000 units on the horizontal axis. As usual, both are measured per unit of time -- per week, perhaps. Notice how the cost first levels off, and then rises even more steeply as output increases. It is this non-linear complexity that many economists think is common in real production processes, in the short run.

Averages

As usual, we will want to look at these costs as averages per unit, and break them down by fixed and variable. As before, the average fixed cost curve is pretty simple. Here it is:

Figure 5

It's the average variable cost curve that will be a little more complicated. Here it is:

Figure 6

In our earlier examples, the AVC was just a straight line, and that's possible. In this case, we see that it starts high, first declines, and then increases as output increases still further. The idea is that, in the short run, costs can be increased by operating below the design capacity of the firm's plant and equipment, just as they can be increased by operating above capacity. That's why the variable cost is high at both ends of the scale, and lowest in the middle -- in the range of outputs for which the plant and equipment was designed.

Now let's put these together:

Synthesis

Now let's put the average fixed and variable costs together on the same diagram:

Figure 7

Now that we have them both on the same axes, we can get the average cost easily by just adding them vertically. This is sometimes called the average total cost -- meaning it is total (both fixed and variable) but average (per unit of output) at the same time. Here is the diagram with all three curves.

Figure 8

Notice how the AFC -- Average Fixed Cost -- forms a sort of a wedge between the Average Variable Cost -- AVC -- and the Average Cost -- AC -- or average total cost, so that the AC is tilted a bit to the right.

Complicated as this is, we still need to bring in one more curve into the picture. Of course, it is the marginal cost curve.

Marginal Cost

The next step is to add the Marginal Cost curve to the other three. Here is the diagram with all four curves:

Figure 9

The marginal cost curve is shown here in orange. Trace it out and see how it interacts with the average cost curves. Notice how it crosses the average cost curve at its lowest point. We have already talked about the reasons for that. Notice also that, a bit to the left, it crosses the AVC curve also at its lowest point. That can happen because the AC and AVC are not parallel -- they get closer together.

OK, its a little complicated, and not very artistic. But we are getting to the point. We can use these curves to visualize the individual firm's short-run supply curve.

Supply

Remember the two rules:

To maximize profits, at each price the firm should sell just enough so that the marginal cost is equal to the price -- that is, the quantity supplied will be on the marginal cost curve.

But the firm will shut down, and produce nothing, if it cannot cover the variable costs.

So the firm's supply curve is the segment of the marginal cost curve above the average variable cost curve. In Figure 10, below, it is shown by the stippled orange curve "S" for supply:

Figure 10

These curves may be a little complicated, but it's worthwhile learning how to look at them and sort them out, because they contain a lot of good information about the way a firm can maximize its profit in the short run. But what about the long run?

Long Run Cost

Thus far we have not considered the long run in the theory of the firm. We will now think a bit about the long run, using the concept of average cost.

We have defined "the long run" as "a period long enough so that all inputs are variable." This includes, in particular, capital, plant, equipment, and other investments that represent long-term commitments. Thus, here is another way to think of "the long run:" it is the perspective of investment planning.

So let's approach it this way: Suppose you were planning to build a new plant -- perhaps to set up a whole new company -- and you know about how much output you will be producing. Then you want to build your plant so as to produce that amount at the lowest possible average cost.

To make it a little simpler we will suppose that you have to pick just one of three plant sizes: small, medium, and large. Here's the way they look in a picture:

Envelope Curve 1

Here are the average cost curves for the small (AC1), medium (AC2) and large(AC3) plant sizes:

Envelope Curve 2

If you produce 1000 units, the small plant size gives the lowest cost.

If you produce 3000 units, the medium plant size gives the lowest cost.

If you produce 4000 units, the large plant size gives you the lowest cost.

Therefore, the long run average cost (LRAC) -- the lowest average cost for each output range -- is described by the "lower envelope curve," shown by the thick, shaded curve that follows the lowest of the three short run curves in each range.

Long Run Average Cost in General

More realistically, an investment planner will have to choose between many different plant sizes or firm scales of operation, and so the long run average cost curve will be smooth, something like this:

As shown, each point on the LRAC corresponds to a point on the SRAC for the plant size or scale of operation that gives the lowest average cost for that scale of operation.

Returns to Scale 1

In our pictures of long run average cost, we see that the cost per unit changes as the scale of operation or output size changes. Here is some terminology to describe the changes:

increasing returns to scale = decreasing cost
average cost decreases as output increases in the long run
constant returns to scale = constant costs
average cost is unchanged as output varies in the long run
decreasing returns to scale = increasing costs
average cost increases as output increases in the long run

Returns to Scale 2

Here are pictures of the average cost curves for the three cases:

increasing returns to scale = decreasing cost
constant returns to scale = constant costs
decreasing returns to scale = increasing costs

Increasing Returns to Scale

Economists usually explain "increasing returns to scale" by indivisibility. That is, some methods of production can only work on a large scale -- either because they require large-scale machinery, or because (getting back to Adam Smith, here) they require a great deal of division of labor. Since these large-scale methods cannot be divided up to produce small amounts of output, it is necessary to use less productive methods to produce the smaller amounts. Thus, costs increase less than in proportion to output -- and average costs decline as output increases.

Constant Returns to Scale

We would expect to observe constant returns where the typical firm (or industry) consists of a large number of units doing pretty much the same thing, so that output can be expanded or contracted by increasing or decreasing the number of units. In the days before computer controls, machinery was a good example. Essentially, one machinist used one machine tool to do a series of operations to produce one item of a specific kind -- and to double the output you had to double the number of machinists and machine tools.

Decreasing Returns to Scale

Decreasing returns to scale are associated with problems of management of large, multi-unit firms. Again with think of a firm in which production takes place by a large number of units doing pretty much the same thing -- but the different units need to be coordinated by a central management. The management faces a trade-off. If they don't spend much on management, the coordination will be poor, leading to waste of resources, and higher cost. If they do spend a lot on management, that will raise costs in itself. The idea is that the bigger the output is, the more units there are, and the worse this trade-off becomes -- so the costs rise either way.

The LRAC Curve 1

In our examples, the LRAC is (more or less roughly) u-shaped, like this:

The LRAC Curve 2

The idea is that:

That's reasonable -- but we should recall that it is pretty much a guess, and may or may not apply in a particular case!

Summary about Cost

By thinking in terms of cost, rather than productivity, we gain several points of understanding of supply: