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monopoly, the consumer may bear only a very small 1 proportion of the tax, even under the law of constant (not to say increasing) cost, under which the proposition would not be true in a regime of competition. When the circumstance of rival demand is superadded to monopoly, is it to be wondered at that the abnormality, as it appears in relation to the simpler case usually contemplated, should be increased : that the consumer should not only not be damnified, but should even be somewhat benefited by the tax ?

A general idea of the modification due to the introduction of rival demand may be obtained by observing that alike in the case of independent and that of correlated demand, a tax on a monopolised article results in the diminution of the quantity of that article put on the market;? but, while in the case of independent demand the diminution of the quantity supplied is attended with a rise of price, this consequence does not necessarily follow in the case of rival demand. A tax on first class fares results in the diminution of the quantity of first class service supplied ; and accordingly there is a flow of passengers from first class to third class. The demand for third class service thus rises in the technical sense referred to in a former page, where it was stated that in monopoly this rise of demand may be attended with a fall of price. The lowering of third class fares results in a fall of the demand for first class service in this technical sense, that for every possible first class fare, the third class fare being supposed constant at its new figure, the amount of first class service demanded is less than what it would have been for each first class fare before the disturbance, the third class fare being supposed constant at its old figure. Now, it is quite consistent with ordinary presumptions that, when the demand for an article falls in this sense, its price should fall. Accordingly, the first class, as well as the third class fare, may be reduced. A fortiori, it is possible that, though the first class fare may be a little increased, yet the third class fare may be so much diminished that the consumers as a whole may gain.

Considered as a mere possibility, this statement is not open to Pro

The argument considered as ad hominem becomes a fortiori, since Professor Seligman thinks it possible that the consumer in this case may bear no part of the tax (cp. below, p. 306).

? I suppose that this proposition would be accepted by an opponent, as it is what may be expected from the analogy of competition, and is less than what those who trust that analogy accept. For a proof of the proposition I must refer to my article

“La Teoria pura del Monopolio" in the Giornale degli Economisti for 1897. 3 Above, p. 237.

* It may assist conception to imagine first-class and third-class services controlled by different managers. The steps described in the text might be made by the respective managers each endeavouring to maximise the net return in his department. But the further step, which on this supposition would be likely to occur, namely, the continued reduction of the first-class fares in order to steal custom from the third-class department, would be stopped by the directors of the common concern, who would not allow a gain to the first-class department to be purchased by a preponderant loss to the third class department.

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fessor Seligman's raillery in the passage above quoted. The plausibility of his objection is obtained by substituting a downright indicative" the tax will reduce ... the price”... “the result of the tax is to cheapen”...-for the potential mood in which I had couched my proposition, a tax on one commodity may benefit the consumers of both ... "the consequences of the new tax may be," . .. and so on. I added the following caution :

“Of course I do not suppose so delicate an adjustment-such a frictionless movement towards the position of maximum profit-to be realised in the concrete management of an English railway. But I think that it may be of scientific interest to establish the theoretic possibility of the paradox.”' ?

The proposition in question is to be taken in the same spirit as the paradox of Mill, that an improvement in the production of an export may be detrimental to the exporting nation. What should we think of a free trade writer who remarked on Mill's theorem that it would surely be a grateful boon to weary and perplexed ministers of commerce, since now all they had to do in order to foster commercial prosperity would be to injure the manufacture of exported commodities! Mill's theorem is useful as presenting an extreme and striking instance of a general truth which, if not indeed paradoxical, is yet not so familiar, but that it is desirable to call attention to it, the important truth that the interests of the parties to international trade are not so completely identical as some free traders have supposed. So our paradox calls attention to the truth that taxation in a regime of monopoly is more diversified and irregular in its consequences, and I think it may be added, likely to be less detrimental to the consumer, than an equal impost in a regime of competition. The extreme exemplifications of these truths are not designed to ease “perplexed and weary ministers,” but to startle indolent and prejudiced economists from their dogmatic slumber, and incite them to reflect that maxims learnt too well from the study of familiar cases cannot always be applied without modification beyond the sphere of experience on which they were founded.4

These preliminary considerations will, I hope, dispose the student to attend to the mathematical ratiocination by which I have elsewhere deduced the theorem under consideration. Addressing the general reader rather than the mathematician at present, I will not repeat this second part of the proof. I confine myself to the third stage, the verification, which consists in instancing laws of cost and of demand which actually fulfil the theory. 1 ECONOMIC JOURNAL, vol. vii., pp. 230 and 232.

2 Ibid., p. 231. 3 Political Economy, Book iii., ch. xviii., $ 5.

* On the meaning and use of paradoxes, compare De Quincey, Works, Ed. 1889, i, p. 199, and vii., p. 206. “Several great philosophers have published, under the idea and title of paradoxes, some first-rate truths, on which they desire to fix public attention, meaning, in a short-hand form, to say: 'Here, reader, are some extraordinary truths, looking so very like falsehoods, that you would never take them for anything else if you were not invited to give them a special examination.'” 5 Giornale degli Economisti, 1897. No. 34.–VOL. IX.

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(B) Let us put P, as the price of a first class ticket for a certain journey, or number of miles, and p, as the price of a third class ticket for the same journey. At these demand-prices let the number of the first class passengers be x, that of the third class passengers be y. Then, agreeably to the general laws of demand, P, must be so related to x that, other things being the same, P, decreases as x increases, (and conversely); and p, must be similarly related to y. Also as first class and third class service are rival commodities, an increased supply of third class service, while the amount supplied of first class service remains constant, will be attended with a decrease in the first class fare at which that amount of first class accommodation is demanded. And the numbers of the first class passengers will be similarly related to the third class fares. These conditions, and any others that may reasonably be required, are fulfilled over a considerable range of prices 2 by the following laws of demand :-3

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We may begin by supposing the cost constant-a very possible case, as Cournot has remarked.4 Then the profit, which it is the object of the monopolist to maximise, the net monopoly revenue, in the phrase of Professor Marshall, is of the form x * P, +y * P2 --C, where C is a constant. It may be shown first that this net profit is a maximum, when x = 20,000, y=100,000 corresponding to the fares P, = £5 x :9= £4 10s. ; P, = £1 ~ 2.3'18' = £2 6s. 4:36'd. ; secondly, that if there is imposed on first-class travelling any tax of so much per ticket, not exceeding •16885 £5 or about 16s. 10?d. per ticket, it will become the interest

| Professor Irving Fisher in his Mathematical Investigations has suggested the question whether the prices of two articles, x and y, for which the demand is correlated, must be regarded as the partial differentials, with regard to x and y respec. tively, of a certain function which represents the total utility afforded by any quan. tities of x and y. I answer this question in the affirmative (See Giornale degli Economisti, 1897, p. 314 note), with the same reservations as the conception of total utility requires in the case of a single variable, in particular that it should not be measured from the extreme point of privation : and accordingly take p,dx +p,dy in my example as a complete differential of a function which I need not write out, but may call U. It may facilitate conception to consider the case in which x and y are not articles of consumption but factors of production, for instance the carriage of different kinds of goods, for which P, and P2 are the respective fares. Then U may stand for the sum of the producers' surpluses enjoyed by the customers of the railway on the assumption that each producer will push his expenditure on each factor of production up to the margin of profitableness.

? As to the range over which the formulæ are applicable, see note to p. 291.

3 From these simultaneous equations we can obtain x and y in terms of P, and pa. As stated ECONOMIC JOURNAL, 1897, p. 54.

* Principes Mathematiques, Art. 30.

of the monopolistic company to lower the fares both for first class and third class passengers.

The statement may be conveniently altered by putting for * - 20,000, and y for y = 100,000. Then the expression which is to be maximised becomes 20,000€ x 5(1.6053 – 28 – 3(8 - 96)** – 1n) + 100,000n (3-9°18' - 2(n - 6975)} – }) (+ a constant). And it has first to be shown that this expression is a maximum when &=1 and n=1. The reader to whom this sort of investigation is not familiar may be advised to substitute in (the variable part of the above-written expression values for & and n at first very close to unity, c.g., for $, 1.001, or 1 - .001, and for n values about equally distant from unity; then gradually enlarging the divergence from unity to realise that for a considerable distance on both sides of unity the result of substituting different values of $ and is to make the expression smaller than what it is for &=1 and n=1: that is, 3.218' x 100,000.1

Consider next the consequence of imposing a moderate tax of so much per ticket on first-class fares, say, £5 x 7, where is a fraction not exceeding :16885. The amount to be maximised is no longer now xpı + YP2, but the same - 578; that is, if we employ & and n as before, the same expression as before, minus 100,000 x T. The value of this modified expression when & =1 and n=1 is 100,000 (3.2'18' – ). It will be found that for any assigned value of → (up to the limit mentioned), there can be found a value of less than unity, and a value of n greater than unity, such that the monopolist's revenue, as modified by the tax, should be a maximum for those values, while both the prices-both the first class and second class fares—are less than what they were before the imposition of the tax.

Here, as before, the reader may be advised to begin with small quantities. To any small value of 7 there may be expected to correspond two values of & and n in the neighbourhood of unity, rendering the (modified) monopolist revenue a maximum. For example, to T= .0017155.. x £5, a little more than twopence per ticket, there corre

1 If at be the difference between the assumed value of g and unity, and on the difference between the assumed value of n and unity, then, a; and an being small, the increase of the monopolist’s net profit consequent upon the change from g and n is approximately – }(3:344 +24549+ 6311. An"); as may be shown by expanding the surd or irrational terms in the expression for the profit according to the algebraic rule for extracting the square root (Cf. Todhunter's Algebra for Beginners, ch. 28), 'and neglecting powers of ag and an above the second. The above expression for the in crease of the profit is negative, whatever the signs of ag and an: showing that the profit corresponding to $=1, n=1 is greater than for any other values in the immediate neighbourhood. In whatever direction we step from the position defined by the equality of g and of n to unity, we descend and continue to descend to a considerable distance in every direction -for instance up to 8 = 96, 9 remaining ; 1 up to n=6975, & remaining 1, and much further in the positive directions of & and n. The stoppage at those points has of course no economic significance : it was adopted merely for the sake of arithmetical convenience; otherwise it would have been better to use cube roots where now square roots are used.

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spond the values 3 = .999, n=1.0015845238.. ; and it may be found by actual substitution, or better by general reasoning, that the loss to the monopolist through the decrease of his receipts is .000,00086 x £100,000 nearly, while his gain in having to pay tax on a smaller number of first class tickets is double that amount, viz., .000,0017 x £100,000 nearly. The monopolist is, therefore, better off with the new values of $ and n than he would be (after the imposition of the tax) with the old values, and, as it will be found, with any other values of

and n. But these values of Ś and n correspond to lower values of Pi and

P2 (first and third class fares) than existed before the tax; as may be seen if these values are substituted for & and

in the expressions for P, and p, respectively. As we increase 7 and therewith decrease § and increase n,

the general relations which have been indicated persist : the monopolist gains more by escaping part of the tax than he loses by the diminution of the receipts, as the values of & and n move further away from unity (the proportion of the gain to the loss becoming greater as the absolute quantities become greater), while both the fares continue to diminish. Thus for the limiting value of t, viz., 16885.. we have A$—:04 and An= .05248. And by actually substituting .96 for $, and 1.05248 for it is found that the new receipts are less than the old receipts by about .002 x £100,000. Against this loss is to be set off the gain of saving the tax of £5 X.16885 on 04 x 20,000 first class tickets : that is a gain of .00675 · · x £100,000—a gain more than three times greater than the loss. At the same time the first class fare is diminished by (1 05248—2 x .04—3.008) ~ £5, that is diminished by .0129 x £5 nearly, = ls. 31d.; and the third class fare is diminished by £2 (3025+.05248—:55)—} .04, = £.0516, = 1s. Old, nearly.

The net gain of some £475, which we have found to attend the lowering of both fares, might well be a substantial percentage of the net profits, supposing these to be, say, about 7 per cent. of the gross profits, which were originally £321,818.18. Say the net profits (per month or year) are about £20,000, the monopolist gains about 2 per cent. on his net profits by making the adjustment described.

We have so far been supposing the total cost to be a fixed amount, say about £300,000. But the reasoning is not materially altered when we suppose the cost variable. To take a simple instance, let the cost consist partly of a constant sum, and partly of two additional amounts respectively proportional to x and y, the number of first class and that of third class tickets. To obtain now the expression for the

1 The clue to the investigation is given by the following equations. Let the new & (corresponding to the maximum after the tax)be 1+ 4% (where ag is a small quantity positive or negative), and similarly let the new

be 1+ Δη.

Then the values of a and an in term of 7 are given by the following (simultaneous) equations

Anti-ssista=0.8&=0. These equations are approximately satisfied by the example in the text. They become less and less exact as ag and an are increased.

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