the abscissa and the ordinate.1 But such a change does not alter the character of a curve in respect of convexity or concavity. If it was convex or concave at any point before the change, the transformed curve will have the same character at the corresponding point. The character of the return, whether increasing or decreasing, in the primary sense, depends on the material conditions of production, not on the accidents of price. With respect to the distinction in the secondary sense, we may employ a theorem given in my former treatment of the subject,2 that in such a figure as our 1 A above, if a tangent (not shown in the figure) is drawn from the origin to the curve, the point of contact is the limit at which the returns cease to be increasing in the secondary sense and become decreasing. This relation, too, may be considered as an invariant, not varying with a change of scale.

But money can no longer be ignored when we consider price as varying with the amount put on the market by the individual entrepreneur; as it is proper to conceive in a regime of monopoly. We must now distinguish z the amount of product in kind due to the employment of the factor x,3 and the money-value of that product depending on the law of demand.

7. Factors and other coefficients.-In general, we may presume that, as shown above in Fig. 1, to any assigned amount of outlay there corresponds a definite amount of product, and conversely. In this presumption it is taken for granted that the entrepreneur applies his outlay to the best of his ability in order to obtain the greatest possible profit. To that end he may have to adjust any number of variables, such as the time of trains, the place of porters, and so forth. We ought to distinguish this sort of coefficient, which does not enter into the expression for outlay from factors-of-production usually regarded as, in the phrase of a judicious writer, "factors of expense." 5

*

1 If in Fig. 1 A the ordinate represent not x the amount of a factor, but § the money value thereof, the curve will then be a cost-curve of the kind employed by Auspitz and Lieben.

Similarly in Fig. 1 в the abscissa may be taken to represent outlay in money. 2 Loc. cit., p. 294.

66

Supposed to be purchasable by the monopolist at a constant price. 4 Cf. Marshall, Principles, ed. v. p. 152: taking farmers as they are with the skill and energy which they actually have." Cf. also the passage cited in the sequel (p. 97), with reference to Prof. Carver's views.

* These gratuitous coefficients may be identified with the "parameters u, v, w... which Dr. Zotoff in his elaborate note on the Mathematical Theory of Production (ECONOMIC JOURNAL, Vol. XXXIII. p. 115) introduced and eliminated.

5 Johnson (and Huebner), Railway Traffic and Rates.

Here might appropriately follow the discussion of plural factors of production. But it is better first, still with special reference to a single simple factor, to advert to the grounds on which different definitions are preferred.

8. The two species compared.-Things which are insignificant for the purposes of action and pleasure do not obtain names. What then is the purpose with reference to which the names now in question have been imposed? The essential fact, I submit, is that the attribute designated Diminishing Return is the criterion of a maximum; not only of a quantity such as z, the product considered as a function of x, the factor used, but also of a quantity such as bzcx (where b and c are constants), denoting the net product.1

Moreover, it is to be conceived that an analogous condition is fulfilled by the gratuitous coefficients above noticed,2 though the vocabulary of the economist may fix attention on the paid factors of production. For instance, in the case of a given amount of labour and capital to be applied to an optional amount of land,3 the condition which must be fulfilled by the law of production in order that the product should be a maximum is the same whether the land is free, or subject to a rent per acre.4

How comes it, then, that the secondary definition is so largely employed by economists? For one reason, there is often no difference in the denotations corresponding to the different connotations. This occurs when the cost-curve represented in Fig. 1 A is convex (to the abscissa), ab initio. This coincidence of fact may explain the frequent use of different definitions by the same writer.6

1 Cf. below, p. 74. In the abstract b and c may be not prices, but quantities of some commodity other than money, in particular the commodity produced. 2 Above, subsection 7.

As in Prof. Carver's instructive example above cited.

4 Let the product be z, = = f(x, l), where x is the amount of working capital, I of land employed; and let the net product be f(x, l) — c1x — cal, where c1, c2 are constants (cf. note 1 above). The criterion of a maximum, namely, that the second term of variation should be thoroughly negative, is the same whether c, is zero or not.

Cf. below, p. 157.

The coincidence is thus affirmed by one who was among the first to discern the principle of Diminishing Returns-West :-" each additional quantity of work bestowed on agriculture yields an actually diminished return, and, of course, if each additional quantity of work yields an actually diminished return, the whole of the work bestowed on agriculture in the progress of improvement yields an actually diminished proportional return."-Essay on the Application of Capital to Land, pp. 6-8, quoted by Prof. Cannan, ECONOMIC JOURNAL, Vol. II. p. 63.

But this identity is not always to be supposed. Rather, the curve in Fig. 1 A is typical of many modern industries in which an initial outlay is required. What is the rôle of the secondary definition in such cases? Let us consider this nice question with reference to three kinds of economic regime: (a) perfect competition, (b) monopoly practised by a perfectly self-interested monopolist, (c) monopoly practised, or at least regulated, by the State.

1

(a) In the first case it is proper to suppose a constant price at which each entrepreneur strives to sell that amount of product which brings him in a maximum profit. If in Fig. 1 A the constant price is represented by the inclination (to the abscissa) of a straight line through the origin 1 (the axis OZ now representing the money-value of any quantity of product z), then the amount produced by an individual whose cost-curve 2 is OLPQR will be the abscissa to the point on the curve, which is such that a tangent to the curve at that point is parallel to the line OS; provided that the curve is convex (to the abscissa) at that point.3 Now at first sight it might appear that this condition could be satisfied by the convex portion of the curve in Fig. 1 a, between P and Q, if the price were suitable. But the condition will be found to imply that the total gain obtained from the production is less than the total loss incurred; which is, normally and in the long run, absurd. Accordingly, we are concerned (in a regime of competition) only with that part of the curve which fulfils the secondary as well as the primary definition, the tract beyond Q.4 When we speak of Increasing Return in the present connection we are mostly not thinking of the concave portion of a curve Mr. Flux, whose book has been cited above on behalf of the primary definition, seems to adopt the secondary one in his article on "Law in Palgrave's Dictionary.

1 In accordance with the Auspitz-and-Lieben construction, noticed in the ECONOMIC JOURNAL, Vol. XVII. p. 226.

2 In the sense above explained. Good examples of (the materials for) such a curve are given by Cunynghame (Geometrical Economics); referred to by the present writer (ECONOMIC JOUrnal, Vol. XV. p. 67), and distinguished from a supply-curve, individual or other.

3 R in Fig. 1 A, is meant to represent this point, corresponding to the seventh dose of labour (thirty-five days of labour), in accordance with the data of Table I. A.

In Fig. 1 B (the abscissa of any point on) OS may stand for the cost, the amount of outlay x multiplied by a constant; while the ordinate to the curve is the total yield (in money or other scale commensurate with the cost).

For a more complete analysis the reader is referred to Prof. Pigou's stupendous article on " Producers' and Consumers' Surplus " in the ECONOMIC JOURNAL, Vol. XX. [Restated in Wealth and Welfare; with reference to which see II., 323, 433, and contexts.]

such as that in Fig. 1 A; but of something quite different, which might be illustrated as follows:-Let the curve in Fig. 1 A represent the cost-curve for an individual typical entrepreneur. With the increase of production in virtue of "external economies," the curve, or that tract of it with which we are concerned, may be lowered as a whole in such wise that each amount of product, x, corresponds to a lower cost. Similarly, Diminishing Return now has a signification other than the convexity of a curve such as that in Fig. 1 A.

It may be worth remarking that when we contemplate the working of a competitive regime as bearing on the interest of the community, from the point of view of the philosophic statesman, then we welcome the phenomenon of Increasing Return (or deprecate its contrary) as tending to (or from) some quantity which it is proposed to maximise. But the criterion of such a maximum is analogous to our primary conception.

(b) When we leave the case of perfect competition, the sort of return which is diminishing in the primary but not the secondary sense-the (convex) tract of the curve in Fig. 1 A between the points P and Q-becomes more significant. Suppose that the general expenses of a Company, like that of the canals in France, were defrayed by the Government, then, even though the ruling price, determined, say, by competition, were inadequate to the total expenses, it might be the interest of the Company to produce an amount between (the amounts corresponding to) P and Q. Something similar occurs in the case of two competing railways obliged in the struggle for survival to leave out of 2 account past expenses of construction. As we continue to remove the conditions proper to the regime of competition, the importance of the point Q, at which Diminishing Return in the secondary sense sets in, becomes less conspicuous. Suppose that in the case put by Professor Carver 3 the farmer has a limited amount of capital-and-labour, say thirty-four days' labour, to apply to plots of land which he can have for nothing. The arrangement which he will find most profitable is to cultivate two such plots of land, applying to each seventeen days' labour; since thus on the assumptions of our Table II. he would produce (twice 317 =) 634 bushels; whereas, by applying the whole thirty-four days' labour

1 J. S. Mill sometimes expressed himself as if the greatest average well-being was the summum bonum. But the better opinion, I think, is that of the philo. sophic Sidgwick that the end of political action is to maximise the quantum of happiness.

2 The case of duopoly; below, p. 117. 3 Distribution of Wealth, ch. ii.

to one plot, he would have produced less than 560 bushels (the produce of thirty-five days' labour according to Table I.).

66

In ordinary monopoly the outlay would not be limited thus absolutely, but by the necessity of limiting the production in order to keep up the price. The limit may be exhibited by one of Auspitz and Lieben's Constructions. In our Fig. 1 A let the curve represent cost to a monopolist of any amount z produced. And substitute (in imagination, not shown in the figure) for the straight line OS a curve passing through O concave to the abscissa; the ordinate representing the total value of the abscissa, z. Then the point of maximum profit to the monopolist may well prove to be a point in the tract between P and Q. Thus it by no means seems to be a universal truth that with a given road-bed and with a given equipment in the way of depôts, offices, machine shops, etc., and with a given labour force, an increase in the rolling-stock will, between rather wide limits, enable the road to carry more freight and passengers; but this increase in its capacity will not be proportional to the increase in the rolling-stock." 1 If we represent the outlay on the "given road-bed" by OL in Fig. 1A (not drawn to scale), and the outlay in rolling-stock by increments along OX above L, it is not certain that this outlay will be pushed up to a point corresponding to in the figure, as the above statement implies. If the demand for passenger-service is very inelastic, it might be the interest of the Railway to restrict the supply within such limits that the increase of carriage room would present Increasing Return (in the secondary sense contemplated). Nay, it is quite possible that Increasing Return in the primary sense may rule ; the monopolist may arrest production at a point below even P in our figure, a point beyond which he would lose by the fallingoff of demand more than he would gain in cheapness of production. This is possible but not probable. For there is a correlation-though not an identity-between the criteria of maximum

1 Carver, Distribution of Wealth, ch. ii. p. 86. Cf. p. 88: "An increase of the rolling-stock would (except in very exceptional circumstances) increase, but not proportionately, the carrying capacity of the road."

2 As pointed out by the present writer, ECONOMIC JOURNAL, Vol. XVII. p. 236. For instance, it is possible that railways in America are deterred from lowering passenger fares, not so much by the cost of increased accommodation as by the belief that the demand would not keep up (cf. Johnson and Huebner, Railway Traffic and Rates, Vol. II. p. 207). They may be wrong in that belief as Weyl and others urge (cf. Johnson, American Railway Transportation, p. 150); but it is possible that they may be right. But owing to the circumstance of Joint Cost (for freight and passenger) and Discrimination (between passenger fares) a clear-cut concrete example is not to be expected.