monopolist's net revenue we must deduct from the gross receipts xp1+yp, the cost xk1+yk, where k, and k2 are the cost per unit of x and y respectively. Much the same conclusion as before may be brought out if we alter the expressions for the fares by putting in P1, for 16053', k1+1-6053', and in pa, for 3.9'18', k2+3.918'. The prices so increased may represent the fares for a greater number of miles than before. The expression which the monopolist seeks to maximise will now be the same as before.

One consequence of admitting the variation of the cost is to render the occurrence of anomalies more probable. When correlation of cost is superadded to correlation of demand, and both to monopoly, we may look out for freaks in the incidence of taxation.

I should be curious to know what "slip" has been detected in this reasoning, substantially identical with what has been already published.1 In a matter of this sort imputations of error based upon first appearances should be objected sparingly.

(2) I take next Professor Seligman's theory as to the relation between the law of cost and the pressure of taxation contained in the following passages and their context :

"Under ordinary conditions, therefore, in the case of a tax on monopolistic industry subject to the law of increasing returns or diminishing cost, the tendency is that the consumer will suffer less than in the case of an industry subject to the law of constant cost.

On the other hand, if the monopoly obeys the law of diminishing returns or increasing cost-where each additional increment of production costs more than the last the producer will be likely to add more of the tax to the price than in the case of constant or increasing returns" (p. 205).

“Given a certain elasticity of demand, we see that in the case of monopoly the tendency is that less of the tax will be shifted to the consumer when the industry obeys the law of diminishing cost or increasing returns, and that more will be shifted when it obeys the law of increasing cost or diminishing returns" (p. 208).

The debate on this question has been somewhat embarrassed by the disputants having used the principal terms in different senses. According to the definition which I have employed in the articles referred to, the law of increasing cost, synonymous with decreasing returns, holds good when the total cost of producing the quantity x of a certain commodity increases with the increase of x at an increasing rate; the law of decreasing cost, synonymous with increasing returns, holds good when the total expense of producing the quantity x increases with the increase of x at a decreasing rate. In other words, the law of increasing cost, or decreasing returns, holds good when the ratio of the last increment of cost to the last increment of produce is greater than the ratio of the penultimate increment of cost to the penultimate incre

1 The example given in my article in the Giornale degli Economisti, 1897, p. 315, differs only in details from the one here given.

2 ECONOMIC JOURNAL, vol. vii., p. 46, note.

ment of produce; with a corresponding statement for the law of decreasing cost (or increasing returns). "This definition," I intimated, "is not identical with that of some distinguished economists"; who may seem to compare the ratio of the last increment of cost to the last increment of produce, not with the ratio above stated, but with the ratio of the total expense to the total produce x, and may accordingly1 define that the law of increasing or decreasing cost holds good according as the ratio of total cost to total product increases or decreases with the increase of product.2

A geometrical illustration may put the matter in a clearer light. In the accompanying diagram the ordinate y of the curve O,Q represents the total expense required to produce any amount, x, of a certain commodity, represented by the corresponding co-ordinate.

The case represented is one in which a certain amount of expense, 001, must be incurred before any return at all is obtained. According to my definition--No. 1, we may call it-the law of increasing cost, or diminishing returns, holds good for all tracts of a curve of this sort which are convex to the axis of x, that is, in the case illustrated, throughout. According to the other definition, No. 2, the law of increasing cost holds good only for those tracts for which the slope of the curve, the inclination of a tangent at any point of the curve to the axis of x, is greater than the inclination to that axis of a line

If x is the quantity produced and f (x) the corresponding total cost, it comes to much the same whether we take as the essential attribute of increasing cost the fact that f'(x) is greater than f(x)÷x, or that f(x)÷x increases as x increases.

2 There may be other shades of meaning, especially in the case of competition as distinguished from monopoly. The difficulties presented by "increasing returns in a régime of competition are noticed in one of the articles referred to (ECONOMIC JOURNAL, 1897, p. 69).

joining that point to O. In other words, according to definition No. 2, the law of increasing cost holds good while the ratio of total cost to produce increases with the increase of produce. The relation between the two definitions is illustrated by the diagram. Beyond the point Q, at which a tangent to the curve passes through O, the law of increasing cost holds good in both senses; but on the near side of Q there is increasing cost in the first sense, but decreasing cost in the second sense. If the origin had been at O1, the axis of x being a horizontal through that point, then the law of increasing returns would prevail throughout in the second sense as well as in the first sense. any cost-curve possess either of the attributes continuously ab initio in one sense, then it will possess that attribute in the other sense also throughout.

If

The same diagram may be used to illustrate the different definitions of the law of decreasing returns; if the axis of y now denotes produce, and the axis of x corresponding cost, the curve now represents a case in which a certain amount of produce, 001, is given by nature without any outlay. For the tract O,Q, up to the limit where a tangent from the origin touches the given curve, the law of decreasing cost prevails according to the second definition, the law of increasing cost according to the first definition; after the limit Q, the law of decreasing returns in both senses.

Now as to the sense in which Professor Seligman uses the terms, the first definition is suggested by the following passage :—

“If . . . an an industry obeys the law of increasing returns or diminishing cost -where each increment in the amount produced costs less than the last" (p. 205).

But the context shows that the second, not the first, definition is in his mind :

"if he produces less, each unit will, on the supposition that he has been producing under conditions of increasing returns cost him, exclusive of the tax, more than before." (Ibid.)

Similarly, the statement that, under the condition of decreasing returns, " each additional increment of production costs more than the last," is explained away by the context. The author's diagrams. (pp. 209-210) leave no doubt as to his use of the terms. It is the cost per unit which he takes as increasing or decreasing with the law of increasing or decreasing cost. Compare his frequent use of the phrase "ratio of product to cost " (pp. 192, 211, 273, 278).

I do not complain of his employing the terms in a sense which is both useful and usual. All that I am concerned to maintain is (a) that my proposition in my sense of the terms is true, and (B), that his proposition in his sense of the terms is not so.

(a) We may follow Professor Seligman in first supposing the law of constant cost to prevail, and afterwards substituting the law of increasing and decreasing returns respectively, other things being (as

much as possible) unchanged. Only, with reference to definition I, "constant cost" must be interpreted to mean, not that the cost per unit is constant, but that the increment of the total cost per increment of product is constant; in other words, that the total cost curve is a right line, but not necessarily a right line passing through the origin, as definition II requires.

In the annexed, as in the former diagram, let the axis X represent total produce, and let the total cost curve, illustrated by the former diagram, pass through Q. Let the curve Ss represent, by its ordinate, the gross receipts (product × price) corresponding to any value of the product x. The position of maximum advantage to the monopolist is where the difference between the gross receipts and the

total cost is a maximum: that is, at a point where the tangent to the cost curve is parallel to the tangent at the corresponding point of the gross receipts curve.1 Thus, if the total cost curve is a right line BB' passing through Q (vertically above P) in order that OP should be the amount supplied, the line must be parallel to the tangent at S (vertically above Q and P) to the curve Ss.

Now let us introduce successively the incidents of increasing and of decreasing cost; neither altering the law of demand, nor the amount

1 Compare the illustration given in the ECONOMIC JOURNAL, vol. viii., p. 236, noticing that the curve BB' represents there net profits, here gross receipts. In order that there should be a maximum (the law of constant cost prevailing) the curve Ss must be concave towards the axis OX.

of the total cost nor that of the cost per unit at the point P, and accordingly neither altering the price, nor the amount supplied, OP.1

It is evident that these conditions can only be fulfilled by a curve (AA') convex to the axis of X, and a curve (CC') concave to it (corresponding to increasing and decreasing cost in my sense of the terms), each curve touching the line (BB') at the point Q.

Now let a tax of so much per unit be imposed on the monopolised article. The effect of the tax is to push up every point on the total cost curve to an extent which is determined by the following construction :-2

Through O draw the right line Ot, making an angle with the axis. of X such that at any point on the line, t, the perpendicular tp may represent the (total) tax paid on the product Op.3 To find the displacement of any point, q, on the cost curve, draw a vertical through q and measure upwards qr equal to pt intercepted between the lines OP and Ot. This construction holds for the curves of varying as well as for that of constant cost. Accordingly the new laws of cost formed by superadding the tax to the old cost will be related as shown in the diagram. The representation of constant cost will still be a right line (bb'), only inclined at a greater angle to the axis of X than the old line (BB'). The law of increasing cost will still be represented by a curve (aa') convex to the axis of X, the new curve touching the new right line at R. The law of decreasing cost (ce') will similarly retain, after the increase of cost by the last, both the character of concavity and the incident of contact with the right line representing constant cost.

Let us now compare the additions to the price consequent on the change of cost in each of the three cases. In the case of constant returns we have by construction the slope of bb' greater than that of BB', the slope of BB' equal to that of the tangent at the point S to the curve Ss, the slope of the tangent to this curve increasing as we move towards O, and diminishing as we move from it. Therefore, to find the point at which the slope of the line bb' is the same as the slope of the tangent to the curve Ss at the corresponding point, we must move towards O, diminishing OP, say to Op, at which point the monopolist's profit is a maximum for the new law of cost. In the case of increasing cost the initial slope at R is the same as that of the line bb'. Therefore, by a parity of reasoning, we must move to the left in order to reach a point at which the slope of the cost curve may be the same as that of the gross receipts curve. But as we move to the left, whereas the slope of the right line remained constant, the slope of the convex cost curve diminishes. Accordingly the point at which the slope of the cost curve becomes equal to the slope of the gross receipts

See the remarks on p. 313, par. 3, below.

2 Compare Messrs. Auspitz and Lieben's construction for the representation of a specific tax.

The tax tp the product Op x the tangent of the angle pot. 4 See note to p. 296.