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suppose that the minimum income necessary for subsistence is Oa, and ignore any utility connected with this. Then the higher standard will be adopted when income is Ob, Ob being such that gfe=cde. So, as income advanced from a to k (on the assumptions that there is no simpler type of expenditure than the one of which the variations in utility are shown by the curve gj, and that yet a third type of expenditure is not assumed) the marginal utility of income would be traced out by the line gfcdh, those parts of the curves which are marked gd and cj remaining hypothetical. In Fig. 2 expenditure according to two standards only

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is admitted, but, of course, more than two might have to be allowed for.

A few words may be said here to prevent any misunderstanding as to the nature of the curves of marginal utilities of income in Fig. 2.1 Neither curve must be confounded with a at, some comfort would be sacrificed to appearances, though the sacrifice would yield nothing appreciable in utility till income was large enough for the type to be substantially realised. It may bo remarked that there seems to be no unanswerable reason for supposing that tho peaks of successivo curves like those in Fig. 2 could never, even for the shortest poriod, rango upwards in height from left to right.

dU 1 The equation to each of them is y when U stands for utility and for

dx income, and an assumptiop is made in each case as to the standard of living to which expenditure has reference.

successive utility curve. By the latter is meant a curve whose ordinates stand for the incremental additions made to utility as equal units of an income of a given amount are successively spent, on the assumption that each unit of income in turn is laid out so as to realise the greatest immediate utility, consistently, nevertheless, with the end finally desired; or, to put the same thing in another way, a curve whose ordinates mark from right to left the sacrifices of utility that would be entailed were the things purchased with the income instantly removed in the order in which the consumer would elect to relinquish them, all things being taken in units of equal cost. For every size of income a curve of successive utilities can be drawn theoretically. Now, for an income of any given size, say Oi, the sum of the successive utilities (apart from the utility of the minimum income Oa) would, of course, equal aiefg if gfe is the curve of the marginal utility of income as previously explained. But, for such an income, the final one of the successive utilities would be greater than ei, and the initial one would be less than ag; and similarly for all other sizes of income. The larger the income the lower is the initial one of the successive utilities (beyond the necessity minimum), because it is spent in some degree with reference to a more distant end. The extra height of the final successive utility over the marginal utility of income is caused by the fact that ex hypothesi the former, regarded as a sacrifice, cannot be reduced by modifying the expenditure of the rest of the income when the sacrifice is made. Further, it may be observed that, when the curve of marginal utility of income begins by ascending, the curve of successive utilities must ascend at first. It might be thought that the latter could not because it expresses, when read from right to left, the effects of losing successively what can be dispensed with at least cost of utility. But this reasoning is unsound, as can be demonstrated formally as follows. If x and y are two things, or two groups of things, consumed, and Ux+y stands for the utility of x and y when taken together, U, for the utility of x in the absence of y, and Uy similarly for the utility of y in the absence of x; then the reasoning implies that :

(i) Ux+y-Ur<Ux

can be directly deduced from :

(ii) Ux+y>Ux>U,

and obviously it cannot (see also page 32). When this false

inference is drawn, the erroneous assumption implying an atomistic view of experience) is made that :

Ux+y=Ux+U, However, it ordinarily happens that, when (ii) holds, (i) holds also.

It seems likely that the curves of Fig. 2 and the relations between them are such that the individual who is not very poor is generally, if not always, on an eastern slope, so to speak. When he steps to the higher type, he comes, as a rule, to such a position that money in his new circumstances is still subject to diminishing utility. So there are jumps; but there is seldom, if ever, any continuous increasing utility of income when people are not passing out of a state of extreme poverty. Nevertheless -and this is the important point—the view here espounded does involve the hard saying that the marginal utility of money may be greater to a man after his circumstances have improved.1 I am sure that this is so in an appreciable number of cases; I think it is not infrequently the case; and it may be the ruling case when certain income limits (which vary with the individual) are passed by inconsiderable amounts. It is a common experience to meet with people who have obtained a slight accession of income, and whose enjoyment of life has obviously been increased quite out of proportion to the accession of income. This cannot be explained satisfactorily if additions to income can only bring diminishing additions to utility. Such people (after taking a larger house, say) are quite likely to be more careful about casual expenditure (though possibly careless with pence for reasons to be expounded), and people do not get more careful with things as the things fall in value to them. Again, the wastefulness, or open-handedness, of many unusually wellpaid working men long after their old habits, formed when they were much poorer, have had time to crumble away, as contrasted with the penuriousness of many men in a somewhat betterpaid economic class, at least suggests that the marginal utility of money is less for the former than the latter, if the two marginal utilities may be compared (see, below, page 33). Illustrations might be indefinitely multiplied. Of course, the ultimate test is furnished by the judgment of people who have had, in a reason

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It should be pointed out that the doctrine is involved also in such a case as is dealt with in Fig. 1, which does not incorporate the fundamental notion here advanced. Such a case might be waived aside as an exception, but I should doubt whether, in view of the many very expensive items in the budgets of the rich, the number of such exceptions is so few that they may be lightly dismissed.

ably short period, wide experiences of different incomes and who realise the points at issue ; but, unfortunately, most people are not good at introspection.

II.

It behoves us now to consider the chief reasons for which the view opposed to that advanced in this paper is held. Three main reasons are ordinarily given for the view that the utility of money invariably becomes less to a man when he becomes rich. Two are empirical, and the other is a priori.

Empirically, it is pointed out that, when well-to-do, a man is apt to be more careless about pence than when he is not so well off ; which must mean, it is said, that the penny has a lower marginal utility to him in the former case. But must it? The following explanation is equally plausible for many instances, namely, that the penny ceases to figure as the unit of account in deliberations about expenditure when a man achieves wealth. The penny might have risen in utility, but, nevertheless, be neglected because of the greater aggregate utility of income. Time is too limited, in relation to the income to be spent, for the rich man to think of a penny--or, so to speak, his wealth makes his leisure too valuable for it to be worth his while to throw it away in devising plans for the saving of pennies.1 The poor man saves pounds by looking after the pence; but the rich man, by looking after the pence, would lose pounds in value. Moreover, there is this further consideration, that the rich man is encouraged to think in large units by the high average price of the things bought by him. So, despite the rich man's carelessness about pence, his money might have been endowed with an access of utility on his becoming rich. In instituting a comparison, obviously we must not take as the standard unit a sum which is disregarded in one of the cases compared.

Again, still empirically, it is maintained that the richer a man is the higher is the pay needed to induce him to do more than a given amount of work ; and that this must be because money has less utility to him when he is rich. But, assuming for the sake of argument that the first statement holds universally, the con

1 The explanation may be put crudely in this way. A man with, say, £200 a year, has 48,000 penny units of expenditure per year to think about, and he has just time for the task. But a man with £500 a year would have 120,000 units of expenditure to think about if he made the unit of account the ponny, and for so extensive a task he might not have time. He, therefore, makes his income manageable by falling back on a higher unit of account and thinking of the problem as made up of 60,000 twopenny units or 40,000 threepenny units.

clusion drawn does not necessarily follow. Greater wealth generally means that leisure has a higher value, so that a larger absolute sacrifice is made in undertaking extra work when a man has become wealthier. Moreover, in the case in which the greater wealth results from greater earnings, the person in question will obviously be unwilling to under-cut himself, so to speak, by accepting work at a low rate of remuneration; and, apart from this, he will naturally take into account that the time given to the low-paid work might reduce his capacity to earn at the highlypaid work. These considerations make it evident that any abbreviating effect which rising incomes may have on the time giren to earning can be explained without reference to the marginal utility of income.

The a priori argument puts it that the most pressing wants are satisfied first, and that as we get wealthier less pressing wants are left to be satisfied, so that in satisfying them we get less utility. This argument seems so convincing because it is self-evident in one sense, which, however, does not happen to be the sense that must be read into it when a contrast is drawn between incomes of different sizes. The mere fact that, when I have £200 a year, I buy six imitation Chippendale chairs, rather than a single genuine one which costs as much as the other six put together, proves that I prefer the six imitations to the one original chair when I have an income of £200 a year. But it does not prove that, when I have the larger income and have bought the costly chair, and compare my enjoyment of it under the new income conditions with my enjoyment of the six cheap chairs under the old income conditions—that then I must judge the former enjoyment to have been greater than the latter.1

Some of the defences of the doctrine that the marginal utility of money must always be less for the larger than for the smaller income are, it would seem, less impregnable than they appear at first sight.

III.

Another important line of defence remains. Some of my readers may be prevented from according full assent to the notions brought forward in this article by their acceptance of a principle of taxation that is steadily winning the confidence of the public, namely, the progressive principle, which recommends taxing the higher incomes at a higher rate. This principle is

1 See also pp. 29, 30.

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