CH. XII.

BOOK VI. poorest, than to secure a sudden and very great diminution in the hours of labour of those who are not now weighed down by their work.

Leisure for

In this, as in all similar cases, it is the young whose faculties the young and activities are of the highest importance both to the moralist and the economist. The most imperative duty of this generation is to provide for the young the best education for the work they have to do as producers and as men or women, together with long-continued freedom from mechanical toil, and abundant leisure for school and for such kinds of play as strengthen and develop the character.

The interest of the rising

in the hours of labour of their parents.

And, even if we took account only of the injury done to the rising generation by living in homes in which the father generation and the mother lead joyless lives, it would be in the interest of society to afford them some relief. Able workers and good citizens are not likely to come from homes from which the mother is absent during a great part of the day, nor from homes to which the father seldom returns till his children are asleep. And therefore not only the individuals directly concerned, but society as a whole, has a direct interest in the curtailment of extravagantly long hours of duty away from home even for mineral-train-guards and others, whose work is not in itself very hard.

And now we must conclude this part of our study of Distribution and Exchange. We have reached very few practical conclusions; because it is generally necessary to look at the whole of the economic, to say nothing of the moral and other aspects of a practical problem before attempting to deal with it at all: and in real life nearly every economic issue depends, more or less directly, on some complex actions and reactions of Credit, of Foreign Trade, and of modern developments of Combination and Monopoly. But the ground which we have already traversed is, in some respects, the most difficult of the whole province of economics; and it commands, and, so to speak, holds the key of, that which lies yet before us.

NOTE I. (p. 151). The law of diminution of marginal utility may be expressed thus:-If u be the total utility of an amount r of a commodity to a given person at a given time, then measures its marginal utility;

du dx

d'u

and, subject to the qualifications mentioned in the text, is always dx2 negative.

NOTE II. (p. 153). If m is the amount of money or general purchasing power at a person's disposal at any time, and μ represents its total utility

Αμ dm

to him, then represents the marginal utility of money to him.

If p is the price which he is just willing to pay for an amount x of the commodity which gives him a total pleasure u, then

If p' is the price which he is just willing to pay for an amount x' of another commodity, which affords him a total pleasure u'; then

(Compare Jevons's chapter on the Theory of Exchange, p. 151.)

Every increase in his means diminishes the marginal utility of money to

du dx'

Therefore, the marginal utility to him of an amount of a com

modity remaining unchanged, an increase in his means increases

du αμ
÷
dx dm'

i.e. it increases that is, the rate at which he is willing to pay for

dp
dx'

further supplies of it. Treating u as variable, that is to say, allowing for possible variations in the person's liking for the commodity in question,

dp dx

we may regard as a function of m, u, and r; and then we have

d'p

dm dr

NOTE III. (p. 161). Let P, P' be consecutive points on the demand curve; let PRM be drawn perpendicular to Or, and let PP' cut Or and Oy

When the distance between P and P' is diminished indefinitely, PP' becomes the tangent; and thus the proposition is proved.

It is obvious à priori that the measure of elasticity cannot be altered by altering relatively to one another the scales on which distances parallel to Or and Oy are measured. But a geometrical proof of this result can be got easily by the method of projections: while analytically it is clear that dx - dy

y

which is the analytical expression for the measure of elasticity, does not change its value if the curve y=ƒ (r) be drawn to new scales, so that its equation becomes qy=ƒ (px); where p and q are constants.

If the elasticity of demand be equal to unity for all prices of the commodity, any fall in price will cause a proportionate increase in the amount bought, and therefore will make no change in the total outlay which purchasers make for the commodity. Such a demand may therefore be called a "Constant Outlay demand."

The curve which represents it, a "Constant Outlay curve," as it may be called, is a rectangular hyperbola with Ox and Oy as asymptotes; and a series of such curves are represented by the dotted curves in the following figure.

There is some advantage in accustoming the eye to the shape of these curves; so that when looking at a demand curve one can tell at once

с

whether it is inclined to the vertical at any point at a greater or less angle than the part of a Constant Outlay curve, which would pass through that point. Greater accuracy may be obtained by tracing Constant Outlay curves on thin paper, and then laying the paper over the demand-curve. By this means it may, for instance, be seen at once that the demand-curve in the figure represents at each of the points A, B, C and D an elasticity about equal to one: between A and B, and again between C and D, it represents an elasticity greater than one; while between B and C it represents an elasticity less than one. It will be found that practice of this kind makes it easy to detect the nature of the assumptions with regard to the character of the demand for a commodity, which are implicitly made in drawing a demand curve of any particular shape; and is a safeguard against the unconscious introduction of improbable assumptions.

The general equation to demand-curves representing at every point an dx elasticity equal to n is + n =0, i.e. xy"=C.

x

dy
y

portion in which the amount demanded increases in consequence of a small fall in the price varies inversely as the (n + 1)th power of the price. In the case of the Constant Outlay curves it varies inversely as the square of the price; or, which is the same thing in this case, directly as the square of the amount.

NOTE IV. (p. 167). The lapse of time being measured downwards along

Oy; and the amounts, of which record is being made, being measured by

Οι

N'
N

P

H

since PN and P'N' are equal in the limit.

If we take a year as the unit of time we find the annual rate of increase represented by the inverse of the number of years in Nt.

The rate of increase would be constant for all

dy points of the curve if Nt were constant and always=a, that is, if - xa dx for all values of x; that is, if the equation to the curve were y= - a log x.

NOTE V. (p. 180). We have seen in the text that the rate at which future pleasures are discounted varies greatly from one individual to another. Let r be the rate of interest per annum, which must be added to a present pleasure in order to make it just balance a future pleasure, that will be of equal amount to its recipient, when it comes; then may be 50 or even 200 per cent. to one person, while for his neighbour it is a negative quantity. Moreover some pleasures are more urgent than others; and it is conceivable even that a person may discount future pleasures in an irregular random way; he may be almost as willing to postpone a pleasure for two years as for one; or, on the other hand, he may object very strongly indeed to a long postponement, but scarcely at all to a short one. There is some difference of opinion as to whether such irregularities are frequent; and the question cannot easily be decided; for since the estimate of a pleasure is purely subjective, it would be difficult to detect them if they did occur. In a case, in which there are no such irregularities, the rate of discount will be the same for each element of time; or, to state the same thing in other words, it will obey the Exponential Law. And if h be the future amount of a pleasure of which the probability is p, and which will occur, if at all, at time t; and if R=1+r; then the present value of the pleasure is phR. It must, however, be borne in mind that this result belongs to Hedonics, and not properly to Economics.

Arguing still on the same hypothesis we may say that, if w be the probability that an element of happiness Ah a person will derive from the possession of, say, a piano in the element of time At, then the present value of the piano to him is

dh

dt. If we are to include all the happiness dt

that results from the event at whatever distance of time we must take T=. dh dt

If the source of pleasure is in Bentham's phrase "impure," will probably

be negative for some values of t; and of course the whole value of the integral may be negative.

NOTE VI. (p. 181). If y be the price at which an amount x of a com