Neoclassical theory of the firm

The neoclassical theory of the firm portrays the marginal cost curve and the average cost curve as distinctly U-shaped. A typical drawing, taken from Hal Varian's Intermediate Microeconomics is shown below (Figure 20.2 in his book):

Unfortunately, these fabled "U-shaped cost curves" truly are fables. Even if we accept the underlying concepts of diminishing marginal productivity (debunked in Chapter 3), there is a mathematical relation between marginal cost and variable cost which ensures that only trivial levels of output can generate curves that look like this.

The short run output level

The neoclassical theory of the firm divides production time periods into 3 classes: the market period, when output cannot be altered; the short run, when all but one factor of production can be altered; and the long run, when all factors can be altered.

The supply curve in the short run is vertical: only the amount already produced can be offered. The supply curve in the short run: it is the sum of the marginal cost curves of all firms in a competitive market (a supply curve can't be drawn for any other market type). The marginal cost curve is presumed to fall at first due to increasing production efficiencies, then to rise as diminishing marginal productivity sets in. This results in both the marginal cost and the average cost curves being "U-shaped". Or so we are told.

The usual example given of a perfectly competitive firm is a wheat farm. There are roughly 750,000 wheat farms in the USA. The State of North Dakota is the second largest producer in the USA, and 40 per cent of farm revenue in comes from wheat, which is grown on 23,000 farms averaging 1,259 acres (versus the national average of 491 acres). Each acre produces on average 29 bushels, or 1.9 tons. If we measure in bushels, then if the wheat industry were in competitive equilibrium, the minimum point of the average total cost curve for each firm would be bushels. As we shall shortly see, a wheat farm producing this output couldn't possibly have a set of cost curves which look like the standard neoclassical drawing.

The Escher-drawing of marginal and average cost

As I comment in Chapter One of Debunking Economics, the drawings economists make often resemble Escher prints, where perspective is exploited to render scenes which are topologically impossible in the real world. But whereas Escher deliberately set out to draw the impossible, neoclassical economists normally believe that their drawings are accurate renditions of reality. We're about to show that this simply is not possible for their drawing of firm costs.

To illustrate this, we'll take a close look at the mathematical relations which exist between average and marginal cost, and output quantity. Average cost is defined as total cost divided by the quantity produced, where costs are a function of the quantity produced:

Marginal cost is defined as:

The formula for the slope of the average cost curve will therefore contain a term in marginal cost:

There is thus a simple mathematical relationship between the slope of the average cost curve, marginal cost, average cost, and output: the slope of the average cost curve is equal to the gap between marginal cost and average cost, divided by the output level.

That there is a mathematical relationship between marginal cost and the slope of the average cost curve should not be surprising, but it is something which most economists have missed. If we explore it in some detail, we find that a firm's marginal and average cost curves can't possibly look like this totemic drawing--even if marginal productivity does fall with output. We can illustrate this by asking "what level of output is consistent with the standard drawing of marginal cost and average cost?". We start to answer the question by reworking the formula above:

A trivial solution

This formula gives us a way of calculating what Q can be from the functions which are allegedly based on Q (except at the equilibrium price, where this formula is indeterminate {0/0}). The immediate observation is that, for Q to be large, either MC-AC must be big, or the slope of average cost must be small. But this isn't how economists draw it: returning to Varian's Figure 20.2, the gap between MC and AC is small, and the slope of AC is quite large--though the absence of any numerical values on the drawing makes it difficult to say how large.

However, a bit of scaling lets us put at least some proportional numbers on this graph (a bit of extrapolation made the calculations easier and probably more accurate). What do they translate to as actual numbers? We consider three output levels: the point where marginal cost equals average cost q1, and two higher output levels q2 and q3 where the quantity formula above applies. In what follows, I use average variable cost rather than average total cost; the mathematical relationship remains true, and the use of this curve simply makes it easier to read implicit values off Varian's drawing:

We start from

Rearrange:

Cancel pm and rearrange:

So what Varian has drawn, where q2=1.385 times qm, can't be on the marginal cost curve for an ac curve with this slope. Instead, the MC point for this AC curve is where

All this is unremarkable; it's highly unlikely that what Varian (or any other economist) sketches will precisely correspond to the actual marginal cost curve. However, what is worth remarking on is that the shape of actual marginal cost and average cost curves will be substantially different from the standard drawing.

Taking the example of a wheat farm producing 10,000 bushels per year, the average cost curve will be so flat that it will apparently be L shaped, rather than the U-shaped model of yore. If it is U-shaped, then the marginal cost curve will be almost vertical, rather than a curve which slopes upwards at only a slightly faster rate than the average cost curve. To get a pair of curve which slope like the archetypal drawing, only trivially small output levels can be involved.

What can valid MC/AC curves look like?

As noted above, the typical North Dakota wheat farm produces about 30,000 bushels of wheat a year. The average size of North Dakota farms is 3 times the national average, so it wouldn't be too wrong to argue that the average US wheat farm produces 10,000 bushels of wheat per year. It is probably valid enough to regard one year as the period of time a farmer would need to bring a new unit of fixed capital (land) into use, so this implies an average output per short period of about 10,000 bushels. The minimum of the average total cost curve must be somewhere in the range of 5,000 to 15,000 bushels. Just what can cost curves look like if the minimum point of the average total cost curve represents say 10,000 units?

We can't answer that in general, but we can see what kind of cost curves are consistent with these output levels and parts of the "data" extracted from Varian's chart. We can (a) see how big the gap between average and marginal cost has to be for the slope of the AVC curve to be the way Varian draws it, and (b) see what slope the AC curve could have, given the MC-AC gap assumed by Varian.

The general formula is

with the instances for q1 and q2 being:

where and If we assume that qm=10,000, this means that () and ().

The price of wheat varied between just below 2 to just above 3 dollars per bushel in 1999. If we assume that wheat farms profit maximise and set price equal to marginal cost, then we can treat $2 as the minimum of the short run average total cost curve, and $3 as the extremity of the marginal cost curve. With an output level of 10,000 bushels at the minimum of the average total cost curve, what shape MC and AC curves does this imply? Let's examine the extreme position where MC=3 and q=15000.

where

So the maximum slope of the average total cost curve is less than 1/15000. This is so close to horizontal that it's not funny. So the cost curve is not U-shaped, but L-shaped. To get an idea of this, a tangent from the average cost curve has the equation

where p(=ac) is between 2 and 3. If we take the lower value, we get

If we take the upper value, we get

So the y-intercept (a) lies between 1 and 2--and much closer to 1 than 2. This however is quite different to the figure drawn by Varian, where the tangent has a negative intercept with the price axis. The average cost curve is therefore much flatter than the standard neoclassical drawing implies, and the part of it near the relevant range of the marginal cost curve is, for all intents and purposes, horizontal. The "U-shaped cost curves" are therefore actually L shaped, with the average cost curve being an elongated L, while the marginal cost curve is more like an L rotated -90 degrees.