Plate ix in print

The Calculus

the marginal quantity is a derivative; how the margin turned a verbal subject into a mathematical one, for good and for ill.

Behind the apparent diversity of economic theory there lies a deep and essential unity, and it is the unity of mathematics. The consumer, the firm, the whole market — each is solving the same problem, the maximizing of a magnitude under a constraint, and the laws that govern their conduct are the laws that govern any such maximum. To state the theory in words was to leave it muddled and forever disputable; to state it in mathematics was to make of it, at last, a science — though a science that paid for its rigor by narrowing what it would deign to see.

— the argument of this plate, after Paul Samuelson, 1947

The volume's flagship gave the marginal idea its mathematical body: Walras's general equilibrium, the whole economy written as a system of equations and solved at the margin. This plate steps back to ask what that mathematization meant for economics entire, for the marginal revolution did not merely change the theory of value — it changed the kind of subject economics was, turning a verbal and literary discipline into a mathematical one. The hinge was a fact about the margin itself: the marginal quantity is a derivative. To say that value answers to the last unit, to the rate at which satisfaction rises as quantity rises, is to speak, whether one knows it or not, the language of the differential calculus, in which the margin is a slope, the derivative of the total. Once economics had fixed its gaze on the margin, it had committed itself to the calculus; and the man who saw this most clearly, and built from it a whole method, was Paul Samuelson, whose Foundations recast the entire subject as a single mathematical exercise. The question is the plate's, and it is asked in two parts: how did the margin turn economics into a mathematical science, and what was gained, and lost, in the turning?

The mathematizing of economics did not begin with Samuelson, nor even with the marginalists. As early as 1838 Augustin Cournot had brought the differential calculus to bear on the workings of markets and prices, deriving with it results that no verbal argument had reached — and had been ignored for a generation, too far ahead of a discipline not ready for him. When the marginal revolution came, its mathematical wing wrote the margin frankly as a derivative and built upon it; but the revolution had a verbal wing as well, for Menger, the Austrian founder, distrusted the mathematics altogether, holding that the algebra of pleasure mistook the very character of choice, which was a matter of ranked purposes and not of measured quantities. For decades the two strands coexisted, and economics remained a largely literary subject. Alfred Marshall, the great systematizer of the age, buried his mathematics in footnotes and appendices, and is said to have counselled the young economist to translate his reasoning into symbols, work out the consequences, render them back into English — and then burn the symbols. The discipline had the calculus in its hands and held it at arm's length, half ashamed of it.1

1. The mathematizing did not begin with Samuelson. Cournot brought the calculus to bear on markets and prices in 1838 and was ignored for a generation. The marginal revolution's mathematical wing wrote the margin as a derivative; its verbal wing, led by Menger, distrusted the algebra of pleasure. Marshall, the great systematizer, buried his mathematics in appendices and is said to have counselled the young to translate their reasoning into symbols, then burn the result.

What Samuelson did, in the Foundations of 1947, was to end the coexistence and make the mathematics the substance. His governing perception was that the many separate theories of economics — of the consumer, the firm, the worker, the whole market — were not really separate at all, but one theory wearing many costumes: in every case an agent maximizing some magnitude subject to a constraint. The consumer maximizes satisfaction subject to what he can spend; the firm maximizes profit, or minimizes cost, subject to what its technology allows; and once the problem is posed in this single shape, the same mathematics solves them all. More than this, Samuelson showed that the testable content of the theory — how a quantity demanded shifts when a price moves, the direction of every such response — flows directly from the conditions of the maximum, from the bare requirement that one stands at a true peak and not on a slope.2 He gave the program a name: the search for operationally meaningful theorems, propositions that the mathematics guarantees and that observation could in principle refute. Economics, on this vision, was not a gallery of insights but a deductive structure, as orderly as mechanics.

2. In the Foundations, Samuelson made the mathematics the substance. The many theories — consumer, firm, worker, market — are one theory in many costumes: an agent maximizing a magnitude subject to a constraint. The same mathematics solves them all, and the testable content (how a quantity moves when a price changes) flows from the conditions of the maximum. He called the goal operationally meaningful theorems: propositions the mathematics guarantees and observation could refute.
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The gains were real and large, and they are why the method conquered. The mathematics imposed a discipline that words never could: an assumption hidden in a sentence stands exposed in an equation, a conclusion that does not follow can be shown not to follow, and a dispute that prose might carry on forever can sometimes be closed by a derivation. It brought generality, for a single result, proved once, holds for every case that meets its premises, where the verbal economist had to argue each anew. And it brought cumulation, the very mark of a science: results rested upon results, and a younger economist could build on an older one's theorem without re-fighting the original battle, so that the field advanced as the literary schools, forever circling back to first principles, never could. The maximizing principle gathered what had been scattered and gave economics a spine. Whatever is justly said against the method, this much stands: the mathematization is what allowed economics to become a cumulative and self-correcting body of knowledge, rather than a permanent gallery of competing intuitions, each as persuasive as its advocate was eloquent.3

3. The gains were real. The mathematics exposed assumptions hidden in sentences, closed disputes that words could carry on forever, and brought generality (one proof holds for every case that meets its premises) and cumulation (results rest upon results, as in a science). The maximizing principle gathered what had been scattered. The mathematization is what let economics become cumulative and self-correcting, rather than a gallery of intuitions as persuasive as their advocates were eloquent.

But the costs were real too, and they were the shadow cast by the very gains. A method that handles only what it can mathematize will quietly narrow its subject to what can be mathematized, and economics did just this: the historical, the institutional, the psychological, the matters of power and custom that resist the optimizing frame, drifted toward the margins of the field or clean out of it — not because they had been judged unimportant, but because they were intractable. The method prized elegance, and elegance is not truth: a model may be beautiful and false, and the assumption chosen because it lets the equations resolve — the perfectly rational agent, the frictionless market — can mislead as readily as it clarifies. It raised a barrier, shutting economics off from the educated public and from the neighbouring disciplines, and shutting out those whose insight was not mathematical. And it bred a characteristic temptation: the mistaking of rigor for truth, for a flawless deduction from false premises is still false, yet the high polish of the mathematics can lend the premises a confidence they have not earned. The aspiration to stand among the exact sciences — the envy of physics — drove the method on, and at its worst drove out judgment.4

4. The costs were the shadow of the gains. A method handling only what it can mathematize narrows its subject to what can be mathematized: the historical, institutional, and psychological drifted to the margins. Elegance is not truth — a model may be beautiful and false. It raised a barrier against the public and neighbouring fields, and bred the mistaking of rigor for truth, since a flawless deduction from false premises is false still. The envy of physics drove the method, and at its worst drove out judgment.

So the reckoning is a divided one, and the wisest of the mathematizers knew it. The mathematization was, on balance, the making of modern economics: the rigor and the cumulation were genuine, and without them the discipline could not have become a science at all, but would have stayed a province of moral and political rhetoric. Yet the losses were not illusory, and Samuelson himself, the master of the method, never confused the mathematics with the substance — knowing it for a language and a tool, knowing that a model is a deliberate simplification that helps one see by leaving things out, and that what the optimizing frame leaves out, the institutions and the history and the unmodelled depths of the human, is not made unreal by being hard to formalize. The margin, in becoming the derivative, gave economics its great strength and its great seduction at a single stroke: the power to prove, and the standing temptation to believe that what can be proved is all there is to know. The volume opened with the value of one more, a humble and almost verbal-seeming idea; it closes with that idea become the calculus, and the calculus become — for good and for ill — the very language in which economics now does its thinking.5

5. The reckoning is divided. The mathematization was the making of modern economics; without its rigor and cumulation the discipline could not have become a science. Yet Samuelson, the master of the method, never confused the mathematics with the substance, knowing it a language and a tool, and that what the optimizing frame leaves out is not unreal for being hard to formalize. The margin, become the derivative, gave economics its power to prove and its temptation to believe that what can be proved is all there is.