as before the change of elasticity, 1000 units are demanded at the price 5, let us suppose the demand to be tilted up for each higher price in that neighbourhood. For example, whereas originally an output of 900 was carried off by a price of 51, let that price now become 51 x 1.1, and let the other prices be increased in the same proportion. Then we shall have, after the change of elasticity, the prices specified in the second. part of the annexed statement, corresponding to the original outputs. That is before the tax by the mere fact of diminished elasticity the price has been raised from 5 to 5.775. Now superadd the tax of 2.7 per unit, as before, and it will appear that the rise of price is from. $5.775 to $6.05, that is, $275, considerably less than the rise of price under the condition of greater elasticity, which was ·75.

But of course this manipulation of figures is "mere palpation," affording no certain warranty of general truth, and rather calculated to obscure essential points, as in the memorable instance of an arithmetical example constructed by J. S. Mill and condemned by Professor Marshall. A hundred, perhaps a thousand, empirical instances, taken impartially at random, might be required in order to elicit by a laborious elimination of chance the tendencies which may be discovered at a glance upon inspection of a few symbols.

(4) The next question is whether in general, or with what degree of generality, when a specific tax is imposed on a monopolised article, the price will rise. Professor Seligman now admits "that in ordinary cases the monopolist will shift at least a part of the burden." But he ventures "still to cling to the position" that " cases may arise in which it will be profitable for the monopolist to bear the burden himself. No part of the tax will be shifted to the consumer." He disputes my position that the tax will affect the consumer, except "in two special cases, (1) where it is not in the power of the monopolist to increase or limit his output at will; (2) where the monopolist is a sole buyer, and the supply of the article bought is perfectly inelastic; for instance, a combination of tenants dealing with landlords 2 incapable of combining."3

"That these are not the only cases, however," remarks Professor Seligman, "is clear from the argument in the text." Here is this argument-It is supposed, as in the example quoted on our page, that the monopolist will sell 1000 units at price $5, 900 units at price $51, and so on as stated in the third column of the annexed schedule. The cost per unit is supposed to be constant, viz,, $2. Professor Seligman argues:

1 Principles of Economics, Book vi., ch. 9, note on Ricardo's doctrine.

2 Pure Theory of Taxation, ECONOMIC JOURNAL, 1897, p. 227.

3 Assuming of course, that the landlords have nothing else to do with their land but to offer it to tenants. Professor Seligman seems not to accept this postulate. He says, "the landowner is not compelled to part with his land; but the tenant is compelled to occupy some apartments." (Op. cit., p. 242).

"His net profits then after a tax of of a dollar had been imposed, would be,"

"In other words the monopolist will continue to find his greatest profits in continuing to charge the original price."

He will, I rejoin, if he can only alter the price per saltum, by leaps of dollar. But surely it was not necessary in an article on pure theory to notice this obvious limitation, which may fairly be relegated to the category of friction. If the monopolist can adopt an intermediate price between $5 and $51, I submit that he will tend theoretially to do so, for the reasons which I have given in one of the papers referred to.1 Actually no doubt he may not do, so because the gain directly resulting from the change may not compensate the incidental disad vantages attending a change. This force of friction is well described by Professor Seligman, and is all the more clearly discerned by the mathematical economist, in that he perceives, as pointed out by Professor Knut Wicksell,2 that when the tax is small the gain must be very small, of the second order.3

I don't know that there remains anything worth fighting for under this head. I quite admit I never denied the efficacy of friction. Professor Seligman appears to admit the abstract theory when, in a passage which has been already quoted, he reasons: "the producer, who has advanced the tax, will increase his price only to that point where the smaller sales are compensated by the higher price, so that his net profits will still be at the maximum" (p. 204) If any difference of opinion remains, I surmise that it relates to the assumed continuity of the demand-curve (and other economic functions.) I have thought it legitimate to assume, not only with Professor Marshall, that "the demand for a thing is a continuous function," 5 but also that, like the continuous functions which we ordinarily meet with in nature, it is not continually changing its character in respect of convexity or concavity. If the gross receipts curve represented by Ss on our diagram, is concave to the axis of x at

1 See especially the diagrammatic statement at p. 236, ECONOMIC JOURNAL, 1898. 2 In the admirable study of the subject in his Finanztheoretische Untersuchungen, which I had not seen when writing before, in the ECONOMIC JOURNAL, 1897, on this topic.

3 In the symbols which we have employed above, the gain is of the order 1τ▲x; upon which, however, it may be remarked (1) that Ax, though in general of the same order, may be sometimes much larger than Tx, (2) that TAx though small in relation to the gross receipts may be considerable in relation to the net profits.

• Cf. Professor Seligman op. cit. p. 278 (note): "The error of Professor Edgeworth seems to consist in the assumption that the demand curve is continuous."

5 Preface to Principles of Economics.

the point S corresponding to maximum net profits, it may be assumed that, in general, in the great majority of cases which occur in ordinary practice, the curve will retain that character, as we move away from the point, for some finite distance. On this ground, it may be assumed as generally true, that the imposition of a tax will tend to raise price.

In the postulate of continuity lies the answer to the difficulty raised by Professor Seligman in the following passage :

"Cournot states that the tax must always be shifted (except in the cases mentioned in the preceding note.") [The cases quoted from the article in the ECONOMIC JOURNAL, 1897, p. 227, on our p. 306. Does Cournot make both these exceptions ?] "Professor Edgeworth (ECONOMIC JOURNAL, vii., p. 405) thinks that this is true in general.' Later, when hard pressed by Professor Graziani, he seeks to maintain his position by assuming 'that the change of price is small,' 'by taking Ap sufficiently small' (ECONOMIC JOURNAL, viii., p. 235). But is it fair to assume that a small change of price is more general' than a great one? And would Professor Edgeworth's elaborate formulæ all hold good, if the change of price were substantial?" (p. 276).

Certainly, the formulæ hold good for substantial changes of price as long as the conditions of a maximum continue to be fulfilled, that is, presumably for some finite distance. On the page following that which Professor Seligman has quoted, there is given an illustration, in which, as the tax is increased, the price will continue to rise up to the point where the monopolist's profits vanish. The rise of price attending increase of taxation may be interrupted when the demand curve (or some other function involved) changes its character in respect of concavity. Professor Seligman's own example affords a good instance. Taking the figures quoted above, with the omission of the tax, we have the subjoined data for the gross receipts curve. It will be seen that as the price is lowered from 6 labelled E to 5 labelled

A, the output of course being concomitantly increased, the gross receipts continually increase. But the rate of this increase is not continuous. It is less rapid from E to D than from D to C, and from C to B than from B to A. The simplest continuous curve corresponding to the data would presumably be of the form indicated by the sinuous line in the annexed diagram. The reader will be so good as to substitute in imagination a curve of this kind, instead of the grossreceipts curve Ss, in our diagram 2, on a former page. By the reasoning there given, or referred to, it appears that, with the

imposition, and gradual increase from zero, of a specific tax, the output will decrease, and the price will increase, as long as the curve is concave, say up to the point j. At that point the sort of index which we may conceive moving along the curve, stops. It may stop some time at j; or it may almost immediately fly over to the next crest, between D and C. It cannot descend to C (the law of cost being supposed constant, $2 per unit), unless the curve is very unusually complicated. It will continue to move on to the point k, where the curve again changes its curvature, after which another jump may sooner or later ensue.

Because the mathematical investigation advances by tentative steps it is not precluded from going in the direction of the rise of price as far as any other method, provided that the conditions of a maximum are secured. Without that condition, the calculus is helpless it

:

"fears to tread" where the ground is insecure; contrasted in that respect with other methods, but not to its disadvantage.

(5.) Professor Seligman draws out his arrays of figures for another pitched battle on the question whether "a tax on monopoly gross receipts must raise prices"; maintaining that, "although it is generally true that a tax on monopoly gross receipts will raise prices, this conclusion not necessarily follows." From the mathematical point of view the distinction between this case and the preceding is unimportant, with respect to the purpose in hand; as appears from Cournot's analysis. Suppose with Professor Seligman that a tax of 10 per cent. is imposed on gross receipts, then the amount which the monopolist seeks to maximise is gross receipts - total cost; or, (gross receipts -10 total cost). Accordingly, the change of price consequent on the tax will be the same as if, instead of the tax

9

1 Principes Mathematiques, Art. 41, ch. vi.

ad valorem, there had been an increase of the total cost by 11.1' per cent. The effect of such an increase of cost on price is identical with that of a specific tax in the case of constant returns, and of the same general character in the case of varying cost: as may be seen from our diagram 2, by observing that any point on the cost curve is pushed upwards as before, not now to the extent of a certain proportion of the abscissa, but to the extent of th of the ordinate. The same theoretical necessity, the same practical reservations, apply to the fifth as to the fourth issue.

Let us pause after these five rounds. I have noticed some half-adozen other instances in which my distinguished opponent's conclusions are diametrically opposed to those which are deducible by mathematical reasoning. But I think it best to confine the present discussion to issues which have been already raised; respecting which a difference of conclusion may fairly be ascribed to the difference of method rather than a mere slip on either side.

II. I go on therefore to consider some general reflexions on the mathematical method which the author has prefaced to the discussion of particular theorems.1

Here is one of these reflexions.

(1)

"The mathematical study of the pure theory often assumes a simplicity of condition which does not actually exist; it purposely neglects the all-important element of friction and constructs hypotheses irrespective of their agreement with the facts of actual life (p. 173).

I quite admit that mathematical reasoning-like all abstract reasoning-has its abuses, as well as its uses. I only enter a caveat against its being supposed that this remark is particularly relevant to the present discussion. The abstract questions which are at issue are understood in the same sense by both parties; there is no reason to suppose any difference of opinion as to the value of the right conclusions. What the impartial spectator has to consider is whether the party that dispenses with mathematical reasoning obtains the true answers. To cry out Ne sutor ultra crepidam does not prove that it is possible to make good shoes without the proper tools.

Again Professor Seligman remarks:

(2) "The chief advantage of the mathematical method is seen in the use of diagrams where an intricate point which involves the simultaneous consideration of several causes can be illustrated with greater brevity and clearness than in any other way. But when we proceed from diagrams to the higher algebra, the use of the mathematical method sometimes leads to refined calculations of more importance to the mathematician than to the economist, and of little percep tible use in solving any practical economic problems" (p. 173).

This unqualified preference of diagram to symbol appears to me to be exaggerated. When the causes to be simultaneously considered

1 Op cit. pp. 172-3.