The previous plate gave the slope but not yet the use of it. Diminishing marginal utility — the gradient, Gossen's first law — describes how the worth of one good falls as one acquires more of it; but a person in a market does not face one good. He faces many, and a purse that will not stretch to all he wants, and the question that presses on him is not how the worth of any single thing declines but how, across the whole array of things he might buy, he ought to divide his limited means. To this question Gossen gave a second law, and it is in some ways the more consequential of the two, for it is nothing less than the rule that defines the optimum — the formal statement of what it means to allocate a scarce resource as well as it can be allocated. The first law describes a slope; the second tells you where, along all the slopes at once, to come to rest.
The rule is this: divide your spending so that the last unit of money laid out on each good yields the same marginal utility — or, stated with a little more care, so that the marginal utility you get per unit of money is equal across every good you buy. Suppose it were not. Suppose the last pound spent on bread brought you more additional satisfaction than the last pound spent on wine. Then you are misallocated, and you can see at once how to improve: spend a little less on wine and a little more on bread, and you gain more satisfaction from the bread than you lose from the wine, and your total rises. But as you buy more bread its marginal utility falls — you slide down the gradient — and as you buy less wine the marginal utility of wine rises, and the gap between them narrows. You keep shifting until the gap closes, until the last pound on bread and the last pound on wine bring exactly the same return; and at that point, and only at that point, no further shifting can help you. That balance, struck across every good at once, is the optimum.
The argument is so simple that it can feel like a sleight of hand, and it is worth pausing on what does the work. The whole of it rests on a single test: at the best allocation, there is no profitable move left to make. If, anywhere in your basket, a unit of money would buy more satisfaction in one use than in another, then you have not yet arrived, because the move that captures the difference is still open to you; you are at your best only when every such move has been used up, which is to say only when the last unit of money is equally productive of satisfaction wherever it is spent. And the reason you can always get there — the reason the margins can always be brought level — is the gradient itself. Because the marginal utility of each good falls as you buy more of it, every shift toward a good lowers its marginal return and every shift away from a good raises it, so that the act of reallocating is self-correcting: it closes the very gaps it is chasing. The first law is thus not merely the companion of the second but its mechanism. Diminishing marginal utility is what makes the equalising of margins possible at all.
Spend so that the last penny on bread brings exactly the satisfaction of the last on wine, and on coal, and on all the rest — for if it did not, you could move a penny to where it bought more, and would not yet be at your best.
Stated for a shopper dividing a wage among groceries, the principle can look like a small piece of consumer arithmetic; but its reach is very nearly the whole of economics, for the same rule governs the allocation of every scarce resource among every set of competing uses. A firm dividing a fixed outlay among its inputs should arrange it so that the last pound spent on labour adds as much to output as the last pound spent on materials or on machines — the condition of least-cost production. A person with a fixed stock of hours should spread them across his tasks so that the last hour given to each yields an equal return — the condition of a well-spent day. A government with a fixed budget should fund its programmes so that the last pound of each buys the same increment of public good; an investor should hold his assets so that the last pound in each contributes equally to the return he seeks. In every case the form is identical: take a scarce thing, spread it among its uses, and spread it so that the marginal return is the same at every margin. Lionel Robbins defined economics itself as the study of the allocation of scarce means among competing ends, and the equimarginal principle is precisely the rule by which that allocation, done well, is everywhere carried out.
It is worth being exact about the claim, because the equimarginal condition is not merely a serviceable rule of thumb but the very definition of an economic optimum. To say that an allocation of scarce means is optimal is to say that no rearrangement of it can yield a further gain — and that, as we have seen, is the same as to say that the marginal returns have been equalised across all its uses, for an unequal margin is precisely a rearrangement waiting to be made. When the discipline later wrote its theory in the calculus, this is the condition that the mathematics produced: maximise satisfaction subject to a budget, and the first-order conditions that fall out are exactly that the marginal utility of each good, divided by its price, be equal across all goods — and equal, moreover, to a common value, the marginal utility of money itself, which measures what one more unit of the budget would be worth. The Lagrange multiplier of the formal treatment is nothing but Gossen's intuition given a symbol. The retired Prussian had seen, in words and in pictures, the precise content of constrained optimisation a full century before it became the working language of the field.
An optimum, in the economist's exact sense, is only this: an arrangement of scarce means from which no reshuffling can wring a further gain — which is to say an arrangement in which all the margins have been brought level.
An obvious objection arises, and it is right to meet it. No one, buying groceries, computes the marginal utility per pound of bread and wine and coal and sets them equal; the thing is absurd as a description of how anyone actually shops. This is true, and it does not touch the principle, because the equimarginal condition is not a claim about the calculation a person performs but a characterisation of the allocation a person reaches when he allocates well. A shopper who simply buys a little more of whatever pleases him most for the money, and a little less of whatever pleases him least, and adjusts as he goes by feel and habit and the memory of past regret, is groping toward the very balance the principle describes, and will tend to arrive somewhere near it without once naming a marginal utility. Whether people in fact reach the optimum, and by what means — careful reckoning, blind habit, or trial and painful error — is a real and contested question, and a later plate must take it up. But the principle defines the target whether or not anyone consciously aims at it: it says what the best allocation is, and leaves for separate inquiry the matter of how nearly, and by what route, the best is reached.
The rule in its clean form supposes a world more tractable than the one we inhabit, and three of its idealisations are worth marking. It supposes, first, that goods come in units fine enough to be adjusted by the last small increment — but many do not: one cannot buy nine-tenths of a refrigerator, and where goods arrive in indivisible lumps the margins can be brought only approximately level, a complication a later plate will pursue. It supposes, second, that one buys at least a little of everything, so that there is a last unit of each to set equal — but often the best course is to buy none at all of some good, whose very first unit is not worth its price, and at such a corner the tidy equality gives way to an inequality the optimum simply tolerates. And it supposes, third, that marginal utilities can be compared across goods at all, that the satisfaction from bread and the satisfaction from wine are magnitudes of one kind that can be set equal — an assumption of measurable, cardinal utility that a later generation would find it could mostly do without, restating the optimum instead as a tangency of curves that asks only which of two bundles a person prefers. Through all three qualifications the core holds intact: to allocate a scarce resource well is to leave no margin standing higher than another, because a higher margin is an opportunity not yet taken.
With the second law the marginal account of the buyer is complete. The gradient gives the shape of a single want; the equimarginal rule disposes a limited purse across many wants to their best collective satisfaction; and out of the two together comes the whole of the demand side of a market — the consumer's optimum, and, as price varies and the optimum shifts, the downward-sloping curve of demand itself. But a price is made by two sides, not one, and the marginal principle that governs the buyer governs the seller no less: a firm, too, weighs one more against its cost, and produces up to the margin at which the two meet, and traces thereby a curve of supply. The demand blade was now ground; the supply blade waited; and it remained for someone to set the two together, to show that price is determined by neither alone but by the meeting of both, and to write the whole in the clean language of the calculus. That someone was Alfred Marshall, and his two blades are the subject of the next plate. As the table says of every quantity, first to last: the argument remains open.