All books involve a compromise between content and length. The web makes it possible to overcome that traditional limitation of the printed page, by providing an accessible home for material which had to be sacrificed during the editorial process.
The topics which are explored in the additional material are:
Capacity utilisation
Actual corporate pricing
Monopoly/perfect competition comparison invalid
Marginal cost equals price does not maximise profits
To take an example, therefore, from a very trifling manufacture; but one in which the division of labour has been very often taken notice of, the trade of the pin-maker; a workman not educated to this business (which the division of labour has rendered a distinct trade), nor acquainted with the use of the machinery employed in it (to the invention of which the same division of labour has probably given occasion), could scarce, perhaps, with his utmost industry, make one pin in a day, and certainly could not make twenty. But in the way in which this business is now carried on, not only the whole work is a peculiar trade, but it is divided into a number of branches, of which the greater part are likewise peculiar trades. One man draws out the wire, another straights it, a third cuts it, a fourth points it, a fifth grinds it at the top for receiving, the head; to make the head requires two or three distinct operations; to put it on is a peculiar business, to whiten the pins is another; it is even a trade by itself to put them into the paper; and the important business of making a pin is, in this manner, divided into about eighteen distinct operations, which, in some manufactories, are all performed by distinct hands, though in others the same man will sometimes perform two or three of them. I have seen a small manufactory of this kind where ten men only were employed, and where some of them consequently performed two or three distinct operations. But though they were very poor, and therefore but indifferently accommodated with the necessary machinery, they could, when they exerted themselves, make among them about twelve pounds of pins in a day. There are in a pound upwards of four thousand pins of a middling size. Those ten persons, therefore, could make among them upwards of forty-eight thousand pins in a day. Each person, therefore, making a tenth part of forty-eight thousand pins, might be considered as making four thousand eight hundred pins in a day. But if they had all wrought separately and independently, and without any of them having been educated to this peculiar business, they certainly could not each of them have made twenty, perhaps not one pin in a day; that is, certainly, not the two hundred and fortieth, perhaps not the four thousand eight hundredth part of what they are at present capable of performing, in consequence of a proper division and combination of their different operations (Smith 1776)
In the book, I note that a number of
‘conditions’ which economists impose upon their models are actually ‘proofs
by contradiction’ that their theories contain errors. The most blatant example of
this are what are known as the ‘Sonnenshein-Mantel-Debreu’ conditions. These say
that, in order to be able to aggregate the individual utility which different consumers
derive from consuming different commodities:
Ÿ (a) all consumers indifference curves must have the same slope; and
Ÿ (b) the slope must be such that spending doesn’t change with
income—therefore all Engels curves must be straight lines.
But condition (a) means that all consumers must be
identical—so that there is really only one consumer. Condition (b) means that all
commodities must be identical—so that there is really only one commodity. This means
that utility can’t be aggregated across different consumers and different
commodities—it is a ‘proof by contradiction’ that what economists are
trying to do is impossible.
This discovery should have caused economists to
abandon the notion that they can model society simply by adding up all the individuals in
it, but they were so wedded to this idea from the time of Bentham that they refused to see
a contradiction when it stared them in the face.
In the book, I promised to give an example of how
mathematicians used ‘proof by contradiction’ properly to transcend an old belief
that all numbers were rational, and ushered in one of many great leaps in our
understanding of mathematical logic. So here goes:
This proof, though mathematical, is very easy to
understand. Start by assuming that the square root of 2 is rational: that it can be
accurately expressed as the ratio of two integers. Therefore “the square root of 2
equals m divided by n”, where we assume that m and n are the smallest integers for
which this ratio is true.
This means that m and n have no factors in common
apart from one. Squaring both sides and rearranging gives you “2 times n squared equals m squared”.
This means that m squared is an even integer, since it’s equal to two times n, which
must be an even number, whether n itself is even or odd. Therefore m itself must be even;
it can’t be odd because an odd number times an odd number yields another odd number.
Therefore m is divisible by two.
Let’s call this number which is half of m
“k”, so that “k equals m divided by 2”: . Then m squared, which we
already know equals two times n, is the same thing as four times k squared. This means in
turn that “m squared equals 2 times n squared equals 4 times k squared”, since
four times k squared is equal to m squared, Which by our assumption that the square root
of two is rational–equals two times n squared, so that “n squared equals 2 times
k squared”.
But this means that n has to be even, since it is
equal to two times another integer, and two times any number, even or odd, is an even
number. Thus m and n share the common factor two; but we began by assuming that they had
no factors in common apart from one.
We have a contradiction, and therefore have proven
that the square root of two can’t be expressed as the ratio of two integers.
Therefore it is an irrational number.