are numerous, diagrams are apt to become helpless. Thus in order to treat our first question diagrammatically, it would have been necessary to resort to solid geometry. And, as there are only three dimensions of space, even solid geometry would be inadequate to illustrate the problem of three classes of fares. Even with respect to the other issues, where we are concerned with only one commodity, the use of symbols appears to me to have some advantage. I propose to illustrate this statement by restating in symbolic language, specially addressed to the mathematical reader, solutions of the four problems(2) to (5) inclusive-which have already been treated otherwise.

Let us begin with Cournot, by considering an indefinitely small tax or addition to taxation, i in his notation, or, as we might say, Ar. It follows from Cournot's reasoning that the increment of the price consequent on an increment of taxation may be expressed in terms of the following quantities: (1) the price, say p; (2) the rate at which the total cost increases with the product, say c; (3) the rate at which the increase of the total cost attending an increment of product increases with the increase of product, say c'; (4) the elasticity, or rate at which the amount demanded diminishes as the price is increased, say e; and (5) the rate at which the elasticity increases with the increase of price, say e'. Substituting these symbols in Cournot's expression for the change of price consequent on a tax (in his Art. 38, or rather in the expression which he gives for the change of price consequent on an increase in the cost of production equivalent to a specific tax in his equation (4), Art. 31) we have, mutatis mutandis 2 {-2e-c'e2-e(p-c)}ApeAT.

To apply now this formula to the problems in hand. We see at once that to a (positive) increment of taxation corresponds an increase of price. This proposition holds good alike for specific and ad valorem taxes-our (4) and (5) (compare Cournot, Art. 41). And what is true of an indefinitely small increase of taxation, is true of a finite

ΔΡ increase, so long as the denominator in the expression for continues Δτ

positive; that is, I think we may say "in general "-in the long run of cases-to some finite distance from the point at which we started, as shown by Cournot, Art. 32. As he says, "this method of demonstration should be borne in mind, as it will be frequently recurred to." Bearing it in mind with respect to the remaining problems Ap above designated, (2) and (3), we need only examine how is affected by the incidents in question, namely, variations in the law of cost, and variations in the elasticity.

1 Principes Mathematiques, ch. vi., Art 38.

AT

2 It will be observed that is F(p) is identical with our e, his (p) with our -ce, his p'(D) with our c, his "(D) with our c', his F" (p) with our e'.

Considering the formula above given, we see that both the numerator and the denominator of the expression for being essentially ΔΡ AT

ΔΡ positive, must decrease with the increase (and increase with the

AT

decrease) of c', other things being the same. The only other relevant things are e, e', p and c. And the only significant question is whether we have any reason to think that any of these quantities is likely to be greater or smaller when c' is greater, in general in the long run of cases. I submit we have no ground for thinking that there is any correlation between c' and any of those variables. Accordingly in the long run the rise of price consequent on any assigned increase of taxation is likely to be greater the smaller c' is. A particular case of this proposition is that the rise of price is likely to be greater when c' is negative than when c' is positive: in other words, higher when the law of increasing, than when the law of decreasing, returns prevails. That is, understanding those laws as I have defined them. There is no direct connexion between increasing and decreasing returns in the other But the proposition which has been enunciated, is true also in the second sense so far as the attribute, which forms the first definition, is apt to be attended with the attribute which forms the second definition 2-that is, possibly, very far.

sense.

We come lastly to problem (3). How is affected by the increase

AT

or decrease of e? It is quite a relief, after the monotony of contradiction, to have to admit that I have committed a slip at this point. In my former version of the theory in the expression for "the increase of price due to a small tax" corresponding to the expression for Ap just now written, I put, for the sake of simplicity, a single symbol, B, for what I now call - c'e2. And that was all right for the immediate purpose in hand. But in applying the formula to enunciate the effect of a change in e, I treated B as constant, forgetting that it involved e.

The following is, I now think, a more correct statement. In any given case it is impossible to say whether the increase of elasticity conduces to the increase or the decrease of the efficacy of a tax to raise price; unless we are given not only c' (which may be supposed), but also e', involving the curvature of the demand curve, which is not, I think, usually given, even as to sign, much less with the quantitative precision which would often be necessary for the present purpose.

5

But the expression for AP, though perfectly indeterminate for any

AT'

1 The negative of this denominator being, as pointed out by Cournot, Art 31, "necessarily negative, according to the well-known theory of maxima and minima." 2 Cf. above, p. 295 and p. 302.

3 ECONOMIC JOURNAL, vii., p. 227, note 2.

4 Ibid., p. 228, note 4 continued from p. 227.

5 As to this unknown element, see ECONOMIC JOURNAL, 1897, p. 232, note 2.

particular case may afford, I think, a certain presumption for the long run. Suppose the sign of c' to be given, e.g. +, the law of decreasing returns prevailing. Then in the long run of cases-while e' is now positive, now negative in sign, now large, now small in absolute quantity for a majority of those cases an increase of e would be attended with an increase in the denominator of the above-written Ар

expression for p or rather what it becomes when both numerator

Δτ

and denominator are divided by e, and, therefore, a decrease of the ratio under consideration. Conversely, when the law of increasing returns prevails, c' being negative, an increase of elasticity is likely to be attended with an increase in the efficacy of taxation to raise price.

To the extent of the former clause I have to retract my original statement. But I am still able to affirm universally, without reference to the law of cost, the contradictory of Professor Seligman's theory that "the greater the elasticity of demand the more favourable-other things being equal-will be the position of the consumer"; if the situation. of the consumer is tested, as it ought to be, not so much by the rise of price as by the loss of consumer's surplus. Employing the proper criterion of the consumer's welfare, we may affirm, "the greater the elasticity of demand the more unfavourable-other things being equal-will be the situation of the consumer.' "" 1

For the purpose of obtaining propositions in Probabilities, as the preceding may be described, I submit that symbols seem to have an advantage even over diagrams-not to speak of numerical illustrations. Thus it may be objected to our diagrammatic proof 2 of propositions (2) that some a priori knowledge derived from analysis is required to guarantee the legitimacy of our supposition that the law of cost only is varied, other things being preserved constant. A diagrammatic proof of proposition (3) would be even more precarious.

(3) One more of Professor Seligman's general reflexions :

1 The loss of consumers' surplus consequent on raising the price from p to p+Ap is (Cf. Giornale degli Economisti, 1897, p. 314) Ap; where x is the amount produced, which, by Cournot's equation (3) of ch. V,=e(p-c). Therefore the loss of (p-c)e AT

consumers' surplus

=

ep-c)Ap

2

(p-c)AT

1

This expression for the loss of consumers' surplus is always positive, both the numerator (=AT÷e) and the denominator (for the reason given in note 1 to p. 312) being positive. The loss is in the long run greater the greater e is, since one term of the denominator becomes less as e becomes greater; while other term of the denominator which involves e may be treated as inoperative, on an average, in the long run of all possible values (positive and negative) of e', in our ignorance of e. Thus the loss of consumers' surplus is likely to be greater the greater the elasticity.

* Above, p. 297.

e

"It may even be doubted whether the mathematical method has independently discovered any important principle susceptible of practical application that could not have been also expressed in every-day language."

Those who have followed the preceding discussion may be disposed to admit that, if the mathematical method does not itself discover important practical principles, it may at least be usefully employed to test the principles which a distinguished practical economist regards as important. If it is worth his while to employ some pages of economic analysis and numerical examples in endeavouring to prove those principles, it is worth our while to employ some lines of symbol in endeavouring to disprove them. Thus it is argued by Professor

Seligman :

"If a high tax, for instance, be imposed on the passenger tickets of a railway, subject to the law of increasing returns where the most profitable business happens to be the passenger traffic, and where an increase of fares would mean a considerable falling off in travel, the resulting abandonment of several passenger trains would mean a considerable increase of the percentage of fixed to operating expenses, and therefore a great fall of profits. The railway will therefore add as little as possible of the tax to the fare."

If this thesis deserves attention, so does the anti-thesis, that if the law of increasing returns holds good1 and the demand is elastic,2 a tax on railway tickets is likely to be particularly hurtful to passengers. In these days when the system of monopoly is, so to speak, on its trial, a certain practical interest attaches to Professor Seligman's theory that, whereas

"the condition most favourable to a monopoly is that of decreasing cost or increasing returns"... "the tendency is . . . that less of the tax will be shifted to the consumer than under any other proportions in the ratio of product to cost" (p. 210).

This is, indeed, a "comforting doctrine to the consumer" (Ibid). The doctrine is certainly important, if it is true. If it is not true, though supported by such high authority, is not the contradiction of it important?

These negations are also affirmations, but not very confident ones. It is as if an opponent should prophesy that the last week of April or May would be the coldest part of the month. The reply is that what we know about the matter points in a contrary direction: there is a constant cause making for greater heat-namely, the position of the earth relatively to the sun-in the latter part of each month; though doubtless that tendency may be counteracted by unpredictable vicissitudes of weather. What if the more abstract part of political economy, like the more sublime part of astronomy-that which contemplates the mechanism of the heavenly bodies external to our system-were not at present susceptible of direct practical application, the mathematical 1 Even in the author's sense. Above, p. 302. 2 Above, p. 313.

theory of economics might still confer a benefit analogous to that which the mathematical theory of astronomy conferred when it discredited the pernicious pretensions of the astrologers. There are those who think that even of the received economic analysis the most important function is negative. Thus Mr. Leslie Stephen :

"Political economy, as I venture to think, has been especially valuable in what I have called its negative aspect. It has been more efficient in dispersing sophistries than in constructing permanent theories. Economic writers have exploded many absurd systems. They have so far cleared the way for an application of sounder methods. But the complexity of the problem is so great. . . ."1

The sort of sophistry which has been eradicated from the general field of economics by the received organon finds a still virgin soil in the nooks and corners of which the cultivation requires the implements of mathematics.

The trenchancy of this criticism is not inconsistent with the diffidence which is proper to an inexact science, and the respect which is due to a high authority. For on the one hand, the region of hypothetically abstract theory, to which this polemic is confined, forms the one territory of economics in which issues may be fought out without compromise, there being a right diametrically opposed to the wrong. And on the other hand, it is no discredit to the ablest combatant, when he is unprovided with the proper weapons, to succumb. F. Y. EDGEWORTH.

MORTALITY IN EXTREME OLD AGE

In times past longevity was a favourite subject among authors on vital statistics. Curious lists with particulars as to very old persons were drawn up and discussed, and the aspects of longevity under various circumstances traced. Thus the well-known German statistician, Süssmilch, author of the voluminous work Die Göttliche Ordnung, deal at length with this question (1.c. §§ 481 seq.). The famous naturalist A. v. Haller, likewise, in his Elementa Physiologia Corporis Humani, deals with longevity generally as well as with "longævitas hominis before and after the flood, pointing out several causes, as heredity, frugality, or contemplative habits of life. Later on C. W. Hufeland wrote his renowned Makrobiotik (1796), William Barton his Observations on the Probabilities of the Duration of Human Life (1793), and James Easton his Health and Longevity (1799).

A hundred years ago there was thus a series of observations on longevity at hand, which might have led to interesting conclusions, if they had only been correct. Unfortunately this was not the case; many of the most famous records of longevity have been reduced into absurdity, or they have at least lost much of their romantic flavour. 1 1 Life of Fawcett, p. 149.