as are equal to those of the earlier time in nominal value, a Crusoe does not find any part of the later services marked off thus from the rest as principal. For in his world there is no such equation as ab a'b' (see the second paragraph of § 43); and for him the equation world of the market be

=

As b and b' do not exist in

ab" — a'b'' = a"b"" of the comes ab" ab"" = a"b"". his world, the services of the later time in his world that correspond to the services a' in the world of the market, that is, the services that constitute the principal of the later time, are services equal to the services a, when contemporary with a, in value to Crusoe himself; and the formula for the rate of interest with Crusoe becomes b'b" bb""

b"

- I instead of

b'

I.

CHAPTER V

RELATIONS OF THE INTEREST PROBLEM REPRESENTED GEOMETRICALLY

$52. As I have said, the relations of the many factors involved in our problem are hard to explain clearly by means of language only. They can be dealt with most conveniently in the symbols of algebra, to which therefore I have had recourse. Many minds, however, including my own, are not quite satisfied with their grasp of such relations until they have visualized them in terms of relations in space. For many of us, therefore, it is fortunate that the relations involved in the interest problem can be represented geometrically.

$53. It is to be especially noted that the pairs of curves which in Diagram III below coincide at G, O, X, E, L, and U are not the same curves we had in Diagram II (§ 41), which bound the estimated cost and the estimated value of the various units of advances (to nature), but those we had in Diagram I (§ 25), which bound the same factor-lines in the case of services.

In Diagram I we represented the cost or the value of any unit of services by a mere line instead of by an area because we assumed the unit of services whose cost or value was in question to be infinitesimal in size. In our present diagram, however, Diagram III, the marginal unit of services considered is in every case one of appre

90

ciable size, so that its cost and its value

which are

equal,1 of course, since we are at the margin-will have to be represented by an area instead of by a line.

Since the curves bounding the cost and the value of various units of the supply of services coincide theoreti

The capital letters indicate points. The small italics indicate lines corresponding to the factors designated by the same symbols in the algebraic sections of Chapter IV. Thus b' is the line EC; b is the line GC; á is the line IJ, etc.

DIAGRAM |||

cally at only a single point, as in Diagram I (§ 25), we ought theoretically to represent the cost and the value of a unit of services of appreciable size by two different areas such as the shaded areas of Figure I and Figure II below respectively. For the study of our particular problem, however, it will do no harm to represent the

1 Except, of course, for the slight difference suggested in the following paragraph.

curves as coinciding in a horizontal line long enough to bound the top of the shaded areas representing respectively the cost and the value of the marginal unit of the services in question, so that the area becomes a rectangle. The assumption involved in doing this is, of course, merely that the cost and the value of the infinitesimal unit at the very margin hold true also of all parts of a unit of appreciable size. For example, the assumption is that in the case of the advance to nature of the services required to produce eight suits of clothes having a market price of $12.50 each, the cost and the value

FIGURE I

FIGURE I

of the infinitesimal unit of these services at the very margin holds true of all the other units of them; and no one will contend that the error of such an assumption can lead us astray in respect to the main factors of the interest problem.

The base-lines of Diagram III are those of three successive years, 1912, 1913, and 1914, as marked. For the illustration of the main factors of the problem the part of the diagram covering 1912 and 1913 is sufficient; but the part covering a third year is added to illustrate the factors involved in the compounding of interest, which we shall consider in § 59 below.

54. The diagram illustrates the case of the advance

to nature involved in locking up in the storehouse of nature's causal nexus the services required to make eight suits of clothes, each having a market value corresponding to the price $12.50. It is drawn to scale, each objective unit being represented by a sixteenth of an inch laid off across the page from left to right, as from C towards D or from I towards J, and each subjective unit being represented by a sixteenth of an inch laid off up the page, as from C towards G or from I towards O. Of the two curves coinciding from E to F, the lower bounds the cost to the advancer1 of successive increments of the services of 1912, the advance of which is illustrated by the diagram covering the two years; the upper bounds the value to him ditto. Of the two curves at G, on the other hand, the lower bounds the cost to the market of 1912 ditto, and the upper the value to the market of 1912 ditto. The pairs at L and O are the corresponding curves for the year 1913; but note that though the person for the curves at L is the same advancer we had for the curves at E, that is, any advancer of 1912 who at his margin is considering advancing the services CD to nature for one year, the market for the curves at O is not the same market we had at G but the market of 1913.

The services advanced- or whose advance is under consideration are, as I have said, those required to make eight suits each having in 1912 a market value corresponding to the price $12.50. They are measured

1I mean either the actual or the potential advancer according to whether he actually decides to advance the services or not. As we are considering the situation when he has reached the margin of his advancing, we must admit that he is just as likely barely to fail to make the advance as barely to make it.