Marshall’s argument therefore proceeds from the notion of maximizing total utility, by way of the law of diminishing marginal utility, to the conclusion that the marginal utilities of commodities bought must be proportional to their prices.

But now what is this ‘utility’ which the consumer maximizes? And what is the exact basis for the law of diminishing marginal utility? Marshall leaves one uncomfortable on these subjects. However, further light on them was thrown by Pareto.

Pareto started off, originally, from the same utility theory as Marshall; the argument we have just summarized would have been quite acceptable to him also in the first stage of the development of his ideas. But instead of proceeding afterwards, as Marshall did, to concentrate attention upon the demand for a single commodity (and thus to investigate the relation between the curve of diminishing marginal utility and the demand curve), Pareto turned his attention to the problem of related --complementary and competitive-- goods. Here he made an extension of the earlier analysis; or rather, something which started as an extension but ended as a revolution.

For the purpose of studying related goods, Pareto took over from Edgeworth,* a geometrical device --the Indifference Curve. When we are concerned, like Marshall, with one commodity only, we can draw a total utility curve, measuring amounts of that commodity along one axis, and total amounts of utility derived from those various amounts of commodity along the other axis. Just in the same way, when we are interested in two commodities, we can draw a utility surface. Measuring quantities of the two commodities X and Y along two horizontal axes, we get a diagram in which any point P represents a collection of given quantities (PM and PN) of the two commodities. From every such point, we can erect an ordinate in a third dimension whose length represents the amount of utility derived from that particular collection of quantities. Joining the tops of these ordinates, we get a ‘utility surface’ (Fig. I overleaf).

* Mathematical Psychics, pp. 21-2.

In principle, this is simple enough; but three-dimensional diagrams are awkward things to handle. Fortunately, having once visited the third dimension, we need not stay there. The third dimension can be eliminated, and we can return to two.

Instead of using a three-dimensional model, we can use a map (Fig. 2). Keeping quantities of the two commodities X and Y along the two axes, we can mark off on the horizontal diagram the contour lines of the utility surface (the broken line in Fig. (I). These are the indifference curves. They join all those points which correspond to the same height in the third dimension, that is, to the same total utility. If P and P are on the same indifference curve, that means that the total utility derived from having PM and PN is the same as that derived from having P’M’ and P’N’. If P" is on a higher indifference curve than P (the curves will have to be numbered so as to distinguish higher from lower), then P"M" and P"N" will give a higher total utility than PM and PN.

What will be the shape of these indifference curves? So long as each commodity has a positive marginal utility, the indifference curves must slope downwards to the right. For if X has a positive marginal utility, an increase in the quantity of X, unaccompanied by any change in the quantity of Y (that is to say, a simple movement to the right on the diagram), must increase total utility, and so bring us on to a higher indifference curve. Similarly, a simple movement upwards must lead on to a higher indifference curve. It is only possible to stay on the same indifference curve if these movements are compensated --X increased and Y diminished, or X diminished and Y increased. The curves must therefore slope downwards to the right.

The slope of the curve passing through any point P has indeed a very definite and important meaning. It is the amount of Y which is needed by the individual in order to compensate him for the loss of a small unit of X. Now the gain in utility got by gaining such an amount of Y equals amount of Y gained times marginal utility of Y; the loss in utility got from losing the corresponding amount of X equals amount of X lost times marginal utility of X (so long as the quantities are small). Therefore, since the gain equals the loss, the slope of the curve

= (amount of Y gained)/(amount of X lost) = (marginal utility of X)/(marginal utility of Y)

The slope of the curve passing through P measures the ratio of the marginal utility of X to the marginal utility of Y, when the individual has quantities PM and PN of X and Y respectively.

Have we any further information about the shapes of the curves? There ought, it would seem, to be some way of translating into terms of this diagram –the principle of diminishing marginal utility. At first sight, it looks as if such a translation were possible. As one moves along an indifference curve one gets more X and less Y. The increase in X diminishes the marginal utility of X, the diminution in Y increases the marginal utility of Y. On both grounds, therefore, the slope of the curve must diminish. Falling curves, whose slope diminishes as we move to the right, will be convex to the origin, as they have been drawn in the diagram.

But does this quite necessarily follow? As far as the direct effects just taken into account are concerned, it must; but there are other indirect effects to take into account too. The increase in X may affect not only the marginal utility of X, it may also affect the marginal utility of Y. With such related goods the above argument does not necessarily follow. Suppose that the increase in X lowers the marginal utility of Y, and the diminution in Y raises the marginal utility of X; and that these cross-effects are considerable. Then the cross-effects may actually offset the direct effects, and a movement along the indifference curve to the right may actually increase the slope of the curve. This is no doubt a very queer case, but it is consistent with diminishing marginal utility. Diminishing marginal utility and convexity of the indifference curves are not the same thing.

3. We come now to the really remarkable thing about indifference curves the discovery which shunted Pareto’s theory on to a different line from Marshall’s, and opened a way to new results of wide economic significance.

Suppose that we have a consumer with a given money income, who is spending the whole of that income upon the two commodities X and Y, no others entering into the picture. Suppose that the prices of those commodities are given on the market. Then we can read off the amounts that he will buy directly from his indifference map, without any information about the amounts of utility he derives from the goods.

Mark off a length OL along the X-axis (Fig. 3), representing the amount of X which he could buy if he spent all of his income upon X; and an amount OM on the Y-axis, representing the amount of Y he could buy if he spent all his income upon Y; and join LM. Then any point on the line LM represents a pair of quantities of the two commodities which he could buy out of his income. Starting from L, in order to acquire some Y, he will have to give up X in the proportion indicated by the ratio of their prices; and the price-ratio is indicated by the slope of the line LM.

Through any point on the line LM there will pass an indifference curve; but usually the line LM will intersect the indifference curve. If this happens the point cannot be one of equilibrium. For, by moving along the line LM in one direction or the other, the consumer will always be able to get on to a higher indifference curve, which gives him greater utility. He is therefore not maximizing his utility at that particular point.

It is only when the line LM touches an indifference curve that utility will be maximized. For at a point of tangency, the consumer will get on to a lower indifference curve if he moves in either direction.

Tangency between the price-line and an indifference curve is the expression, in terms of indifference curves, of the proportionality between marginal utilities and prices.

4. Thus we can translate the marginal utility theory into terms of indifference curves; but, having done that, we have accomplished something more remarkable than a mere translation. For, in the process of translation, we have left behind some of the original data; and yet we have arrived at the desired result all the same.

In order to determine the quantities of goods which an individual will buy at given prices, Marshall’s theory implies that we must know his utility surface; Pareto’s theory only assumes that we must know his indifference map. And that conveys less information than the utility surface. It only tells us that the individual prefers one particular collection of goods to another particular collection; it does not tell us, as the utility surface purports to do, how much the first collection is preferred to the second.

The numbers which we give to the indifference curves are indeed wholly arbitrary; it will be convenient for them to rise as we go on to higher curves, but the numbers can be 1, 2, 3, 4..., 1, 2, 4, 7..., 1, 2, 7, 10..., or any ascending series we like to take.

Pareto’s little piece of geometry thus resulted in a conclusion of wide methodological importance. It is necessary, in any theory of value, to be able to define just what we mean by a consumer of ‘given wants’ or ‘given tastes’. In Marshall’s theory like that of Jevons and Walras, and the Austrians) ‘given wants’ is interpreted as meaning a given utility function, a given intensity of desire for any particular collection of goods. This assumption has made many people uncomfortable, and it appears from Pareto’s work that it is not a necessary assumption at all. ‘Given wants’ can be quite adequately defined as a given scale of preference need only suppose that the consumer has a preference for one collection of goods rather than another, not that there is ever any sense in saying that he desires the one collection 5 per cent. more than the other, or anything like that.

Now of course this does not mean that if any one has any other ground for supposing that there exists some suitable quantitative measure of utility, or satisfaction, or desiredness, there is anything in the above argument to set against it. If one is a utilitarian in philosophy, one has a perfect right to be a utilitarian in one’s economics. But if one is not (and few people are utilitarians nowadays), one also has the right to an economics free of utilitarian assumptions.

From this point of view, Pareto’s discovery only opens a door which we can enter or not as we feel inclined. But from the technical economic point of view there are strong reasons for supposing that we ought to enter it. The quantitative concept of utility is not necessary in order to explain market phenomena. Therefore, on the principle of Occam’s razor, it is better to do without it. For it is not, in practice, a matter of indifference if a theory contains unnecessary entities. Such entities are irrelevant to the problem in hand, and their presence is likely to obscure the vision. How important this is can only be shown by experience; I shall hope to convince the reader that it is of some considerable importance in this case.

5. Acting on this principle, we have now to inquire whether a full theory of consumer’s demand at least as thoroughgoing as Marshall’s cannot be built up from the assumption of a scale of preference In constructing such a theory it will be necessary every time to reject any concept which is at all dependent upon quantitative utility, so that it cannot be derived from the indifference map alone. We start off from the indifference map alone; nothing more can be allowed.

In undertaking this reconsideration we lose the help of Pareto; for even after Pareto had established his great proposition, he continued to use concepts derived from the earlier set of ideas. The reason was, perhaps, that he did not take the trouble to rework his earlier conclusions in the light of a proposition which he only reached at a rather late stage of his work in economics.* However that may be, he missed an opportunity.

6. We have now to undertake a purge, rejecting all concepts which are tainted by quantitative utility, and replacing them, so far as they need to be replaced, by concepts which have no such implication.

The first victim must evidently be marginal utility itself. If total utility is arbitrary, so is marginal utility. But we can still give a precise meaning to the ratio of two marginal utilities, when the quantities possessed of both commodities are given.* For this quantity is represented by the slope of an indifference curve; and that is independent of the arbitrariness in question.

If an individual is to be in equilibrium with respect to a system of market prices, it is directly evident that his marginal rate of substitution between any two goods must equal the ratio of their prices. Otherwise he would clearly find an advantage in substituting some quantity of one for an equal value (at the market rate) of the other. This is therefore the form in which we must now write the condition of equilibrium on the market.

It may be observed that in this formulation we have, as yet, scarcely departed from Marshall. The marginal rate of substitution of X for Y is what he would have called the marginal utility of X in terms of Y. We may transcribe Marshall if we like, and say that the price of a commodity equals the marginal rate of substitution of that commodity for money.

7. The second victim (a more serious one this time) must be the principle of Diminishing Marginal Utility. If marginal utility has no exact sense, diminishing marginal utility can have no exact sense either. But by what shall we replace it?

By the rule that the indifference curves must be convex to the axes. This may be called, in our present terminology, the principle of Diminishing Marginal Rate of Substitution.* It may be expressed in the following terms: Suppose we start with a given quantity of goods, and then go on increasing the amount of X and diminishing that of Y in such a way that the consumer is left neither better off nor worse off on balance; then the amount of Y which has to be subtracted in order to set off a second unit of X will be less than that which has to be subtracted in order to set off the first unit. In other words, the more X is substituted for Y, the less will be the marginal rate of substitution of X for Y.

But what is the exact reason why we must replace diminishing marginal utility by precisely this principle-the principle of diminishing marginal rate of substitution? As we have seen already,* they are not exactly the same thing. The replacement is therefore not a mere translation; it is a positive change in the foundation of the theory, and requires a very definite justification.

* See above, p. 16.

The justification is this. We need the principle of diminishing marginal rate of substitution for the same reason as Marshall’s theory needed the principle of diminishing marginal utility. Unless, at the point of equilibrium, the marginal rate of substitution is diminishing, equilibrium will not be stable. Even if the marginal rate of substitution equals the price ratio, so that the acquisition of one unit of X would not yield any appreciable advantage; nevertheless, if the marginal rate of substitution is increasing, the acquisition of a larger quantity would be advantageous. It is instructive to set this out on the indifference diagram (Fig. 4).

At the point Q on the diagram, the marginal rate of substitution equals the price-ratio, so that the price-line touches the indifference curve through Q. But the marginal rate of substitution is increasing (the indifference curve is concave to the axes), so that a movement away from Q in either direction along LM would lead the individual on to a higher indifference curve. Q is therefore a point of minimum, not maximum, utility, and cannot be a point of equilibrium.

It is clear, therefore, that for any point to be a possible rate of equilibrium at appropriate prices the marginal rate of substitution at that point must be diminishing. Since we know from experience that some points of possible equilibrium do exist on the indifference maps of nearly every one (that is to say, they do decide to buy such-and-such quantities of commodities, and do not stay hesitating indefinitely like Buridan’s ass), it follows that the principle of diminishing marginal rate of substitution must sometimes be true.

However, for us to make progress in economics, it is not enough for us that the principle should be true sometimes; we require a more general validity than that. The law of diminishing marginal utility used to be assumed generally valid (with perhaps some special exceptions), and on that general validity important economic conclusions were based. We shall have to investigate those conclusions afresh; but, if they are to have any chance at all, they need as their basis a property of the indifference map which is more than sometimes true.

What were in fact the grounds upon which economists used to base their general principle of diminishing marginal utility? Usually an appeal to experience; though to experience of that uncomfortably vague sort which does not offer any opportunity for actual testing. Critics have not been lacking to point out that this procedure was not very scientific, and the doubts which have been thrown by our present discussion upon the intelligibility of the `law of diminishing marginal utility’ itself can only strengthen the case against the traditional procedure. If, however, we throw over diminishing marginal utility as being in any case dubious, and now certainly irrelevant, can we base upon similar `experience’ a general principle of diminishing marginal rate of substitution? Again, I suppose, we might get away without being challenged; but one would like a surer foundation.

8. We can, I think, get that surer foundation if we reflect on the purpose for which we require our principle. We want to deduce from it laws of market conduct --laws, that is, which deal with the reaction of the consumer to changes in market conditions. When market conditions change, the consumer moves from one point of equilibrium to another point of equilibrium; at each of these positions the condition of diminishing marginal rate of substitution must hold, or he could not take up such a position at all. So much is clear directly; but to proceed from this to the law of diminishing marginal rate of substitution, as we need it in economic theory, an assumption is necessary. We have to assume that the condition holds at all intermediate points, so that there are no kinks in the curves between the two positions of equilibrium. (If there are kinks in the curves, curious consequences follow, such that there will be some systems of prices at which the consumer will be unable to choose between two different ways of spending his income.) The general principle of diminishing marginal rate of substitution merely rules out these oddities; by that principle we select the simplest of the various possibilities before us.

The road which lies before us now begins to be distinguishable. If this is the true foundation of the principle of diminishing marginal rate of substitution among consumption goods, other principles can be discovered whose foundation is exactly similar. These principles can be enumerated, and their consequences worked out. Some of them deal with production, and will be considered in Chapter VI below; the rest are extensions, into one field or another, of the principle elicited in this chapter. That there are a great many such extensions appears at once when we consider how wide is the variety of human choices which can be fitted into the framework of the Paretian scale of preference. What begins as an analysis of the consumer’s choice among consumption goods ends as a theory of economic choice in general. We are in sight of a unifying principle for the whole of economics.

9. But this is running ahead. Before we can explore these long avenues much preparation is needed. One necessary piece of preparation may conclude this chapter.

During most of the above discussion we have made the extreme simplification that the consumer had his choice restricted to expenditure on two sorts of goods. It is high time that we abandoned this simplification, for if our theory were confined to this simple case there would not be much to be said for it. It is in fact one of the main defects of the indifference-curve technique that it encourages concentration upon this simple case, concentration that can easily prove very dangerous.

In order that equilibrium should be stable, when expenditure is distributed among many commodities, it is necessary that no possible substitution of equal market values should lead the consumer to a preferred position. This means not only that we must have a diminishing marginal rate of substitution between each pair of commodities, but also that more complicated substitutions (of some X for some Y and some Z) must be ruled out in the same way. We may express this by saying that the marginal rate of substitution must diminish for substitutions in every direction. This is a rather complicated condition, but it will appear, as we proceed, that it leads directly to conclusions of great importance.

On the same grounds as before, we shall assume that the marginal rate of substitution diminishes in every direction at every position with which we shall be concerned in our analysis. I do not think this could be established introspectively, or from ‘experience’, but it can be justified in the same way as we have justified the simpler condition. It becomes clear now, however, that it is a fairly drastic hypothesis, which gives us a good deal to go on, and from which we can expect to deduce some positive results.