The Method — The Margin, Plate vii

A firm considering another unit of output, a worker considering another hour, and a household considering another purchase face a common question: what changes if a little more is done? The gains already secured and the costs already incurred do not answer it. The decision turns on the additional benefit and the additional sacrifice that the proposed change would bring. This is the marginal method, the habit of directing attention to the difference between available courses of action. In its familiar smooth form, the rule is to expand an activity while marginal benefit exceeds marginal cost and stop where the two meet. Its force comes from an elementary implication: if an available increment adds more than it takes away, the present position can be improved. Its careful use begins with the word available. A limit may prevent the next increment, an indivisible project may require a larger leap, and a locally attractive choice may belong to a venture that should never be begun. The method is powerful when its objective, alternatives, and limits are kept inside the reckoning.

The equality of marginal benefit and marginal cost describes the summit of a particular kind of problem. Suppose an activity can be varied continuously, its total benefit and total cost can be expressed on a common scale, and its net benefit is a smooth, concave function of its quantity. At an interior optimum, a small increase or decrease is feasible. If marginal benefit exceeds marginal cost, increasing the quantity improves the result; if marginal cost exceeds marginal benefit, reducing it improves the result. Equality marks the point at which neither small movement helps, and concavity makes that point a global maximum. For example, let total benefit be 12q − q² and total cost be 2q, for a nonnegative quantity q. Net benefit is 10q − q²; marginal benefit is 12 − 2q and marginal cost is two. They meet at five units, yielding a net benefit of twenty-five. Four units yield twenty-four, as do six. The equality finds the summit because the net gain rises before it and falls after it. A derivative equal to zero without those further conditions would identify only a stationary point.¹

1. The quadratic example is authored. For the modern mathematical distinction between stationary conditions, constraints, and global optima, see Stephen Boyd and Lieven Vandenberghe, Convex Optimization (2004), §§ 4.2.2–4.2.3, pp. 138–142. Maximizing a concave function is equivalent to minimizing its negative, a convex function. The example's second derivative of net benefit is −2.

When a resource has several uses, the same comparison asks whether moving a little from one use to another would improve the result. A household dividing its expenditure between bread and wine can compare the additional satisfaction from a dollar in either place. If an interior allocation yields more from the last dollar on bread, shifting a little expenditure from wine to bread raises the household's utility. Under the smooth conditions of this model, an optimum equalizes marginal utility per dollar across the goods actually bought. Jevons makes the corresponding argument for labor: the next increment should go to the activity yielding the greater additional utility, until no advantageous transfer remains. The quantities being compared must match. A return per hour cannot be set against a return per dollar without accounting for the conversion; a financial return under one risk cannot simply be equated with a return under another. And the utility comparison belongs to a particular person's preferences. It does not require treating one person's unit of satisfaction as equal to another's, or assuming that the shopper consciously performs the calculation. The condition describes an allocation within a model; how people actually reach their choices remains a separate inquiry.²

2. W. Stanley Jevons, The Theory of Political Economy, third edition (1888), chapter V, “Symbolic Statement of the Theory” and “Distribution of Labour,” including note 109. The expenditure version is the same constrained-allocation argument expressed per unit of money; its interior conditions do not imply that every good must be purchased.

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A binding limit can stop the activity while an additional unit would still be worth more than its direct cost. If the illustrative activity is capped at three units, three is the best feasible quantity even though marginal benefit there is six and marginal cost is two. The gap of four measures the immediate gain from a small relaxation of that cap: permission or capacity to do a little more has value. The constrained optimum has balanced the relevant scarcity as well as the direct expense. A household may similarly buy none of a good whose first increment gives too little satisfaction for its price; there is no purchased last unit whose marginal utility must be equalized with the others. Indivisibility produces another departure from a literal equality. If our quantity can take only whole numbers, the fifth unit adds three to benefit and costs two, while the sixth adds one and costs two. Five remains best, although neither of those finite increments has benefit exactly equal to cost. In both cases the instruction survives as a comparison of feasible changes. The equation is the special smooth, interior expression of that comparison.³

3. Boyd and Vandenberghe, § 4.2.3, especially p. 142, and § 5.5.3, pp. 243–246, for inequalities, active constraints, and complementary slackness. The cap and integer examples are authored. At q = 3 the derivative of net benefit is 4; the discrete increment in benefit from q − 1 to q is 13 − 2q.

A good marginal quantity also needs comparison with the alternative of undertaking the activity at all. Suppose the same project requires a payment of thirty whenever production is positive, and that payment can be avoided by choosing zero. Conditional on operating, five units still give the highest return above variable cost: the fixed payment has not changed the slope. Yet the best operating result is now a loss of five, while staying out yields zero. If the avoidable payment is twenty, operating at five instead yields a gain of five. The marginal condition is identical in the two cases; the decision to enter differs. Once a payment has irreversibly been made and cannot be recovered, the comparison changes again, because the payment then lies on both sides of the present choice. This is why the opportunity cost in The Alternative matters. The relevant expense depends on what can still be avoided or recovered. More generally, with increasing returns or several separated peaks, a favorable small adjustment may lead only to a local maximum. The best feasible alternative can lie across a region that no sequence of improving tiny steps will traverse.⁴

4. In the authored entry example, maximum net benefit before the fixed payment is 25. Subtracting an avoidable 30 gives −5; subtracting 20 gives 5; abstaining gives 0. A payment already sunk is common to the present alternatives. Compare The Alternative. Boyd and Vandenberghe, § 4.2.2, supplies the precise convexity condition under which a local optimum is also global; it cannot simply be assumed for a project with a fixed entry cost.

The method can improve a reckoning only after the reckoning includes the consequences that matter to the question being asked. A firm's extra revenue and extra private expense identify its profit incentive. A community assessing the same output may also count an injury imposed on neighbors, or a benefit for which the firm receives no payment. The two marginal balances can therefore recommend different quantities without any error of differentiation. Likewise, a public budget cannot allocate every dollar by a single return until it has specified what outcomes count, how different claims are to be weighed, and which constraints must be honored. Uncertainty requires a further choice of how possible outcomes enter the objective, together with evidence about their likelihood. These are substantive judgments that the equality of two margins cannot supply. The virtue of the method is to make the consequences of such judgments traceable: name the feasible action, identify what it changes, include the alternatives it displaces, and compare that change against the chosen purpose. A precise calculation inherits the quality of those decisions.⁵

5. The distinction here is between applying a decision rule and specifying its objective and feasible set. This volume's The Efficiency separates Pareto efficiency from distributive judgment; The Externality examines the difference between private and social cost. No common monetary measure or particular social weighting is presumed in this paragraph.

The marginal method thus carries one inquiry across many problems: is there an available change whose gain exceeds what it gives up? In a smooth interior problem with suitable curvature, the inquiry ends at an equality. At a boundary it ends against a constraint; among indivisible alternatives it compares finite increments; across a decision to enter it must also examine the whole commitment. These are ways of applying the method to the choice that exists, rather than asking the world to supply the choice that makes an equation convenient. The distinction between a rule for an optimum and a description of conduct matters just as much. People may be uncertain, mistaken, habitual, or unable to make the adjustment the model permits. Even when each person does choose well, their separate choices still have to fit together. One buyer's intended purchase is another person's proposed sale, and the resources each wishes to use are also wanted elsewhere. The next plate asks how those many marginal reckonings can be reconciled in a common system of prices. It is the equilibrium.