The rule of the previous plate carried one assumption lightly, almost without remark, and it is time to examine it, for it is the first of the rule's premises that the world routinely breaks. The assumption was that one can always move smoothly — take a little more or a little less, add an increment or shed one, and find the optimum where the last small step ceases to pay. But a great deal of what must be decided does not come in small steps at all. A bridge is built or it is not; there is no such thing as three-fifths of a bridge, no marginal span one can add and test against its marginal worth. A factory is opened or it stays shut; a railway is laid or the valley goes without; a worker is hired or not hired, there being no way to engage a third of a person. Such things are indivisible — they come in whole, unbreakable units — and for a decision about an indivisible thing the smooth marginal calculus offers no purchase, because the smallest available step is not an increment but a lump. This plate is about the lumps, and about the surprising amount of trouble they make for the tidy apparatus the preceding plates have built.
The first trouble is the most direct. When the choice is whether to build the bridge, one cannot proceed by the marginal comparison — by asking whether one more unit of bridge is worth its cost — because there is no one more unit; there is only the whole bridge or no bridge. The decision is all-or-nothing, and it must be made by a different test: not the marginal one of weighing the next increment, but the total one of weighing the entire benefit the bridge would confer against the entire cost of providing it, and building if, and only if, the first exceeds the second. This is a genuine departure from the marginal rule, and it is worth marking, because the marginal rule is so often taught as though it were universal. For divisible things — how much bread to buy, how many hours to work, how much of a continuously adjustable input to employ — the margin is sovereign. But for indivisible things, the lump that must be taken whole or not at all, the relevant comparison is once again of totals, and the marginal optimum gives way to a plain yes-or-no verdict on the lump as a whole. The two rules are complementary, the one for smooth choices and the other for lumpy ones, and a great deal of practical error comes from applying the smooth rule where the lumpy one belongs.
The deeper trouble is that the lump does not merely sit outside the marginal calculus; it deranges the curve the calculus runs along. The indivisible thing is typically a large fixed cost — the bridge, the factory, the rail network, the costly first copy of a piece of software — an expense that must be borne once, in full, before a single unit of output is produced, and that does not rise as output rises thereafter. Spread that fixed cost over the units produced, and the more units there are, the smaller each unit's share of it becomes: the average cost of production falls as output expands. This is the phenomenon of increasing returns to scale, and it stands the ordinary picture on its head. The whole marginal apparatus, from the falling gradient to the rising supply curve, was built upon diminishing returns — upon the premise that doing more of a thing grows harder and dearer at the margin. Where there are large fixed costs, doing more grows cheaper, and two consequences follow that the competitive model cannot absorb. The first is that pricing at marginal cost, as the rule prescribes, no longer covers the fixed cost and leaves the firm at a loss. The second is graver still: the largest firm, spreading its fixed cost over the most units, has the lowest cost of all, and can undersell every smaller rival until it stands alone.
Where the great cost is paid once and for all, before the first unit is made, every unit after lowers the average, and the firm that makes the most makes each most cheaply — so that the very economies the lump creates are the slow death of the competition the margin assumed.
It was the young Italian economist Piero Sraffa who, in a famous essay of 1926, pressed this difficulty into a direct accusation against the reigning theory. Marshall's apparatus had supposed that a competitive firm faces a smoothly rising cost curve — that as it expands, its marginal cost climbs, until that cost meets the price and the firm expands no further. Sraffa asked where the rising cost was supposed to come from, and found that the answer would not hold. If a firm's costs rise as it grows, he argued, it is generally because the firm relies on some factor it cannot freely multiply — a particular site, a special skill, a scarce input peculiar to it — but a firm with privileged command of a scarce factor is not the helpless price-taker that perfect competition requires; it has a foot already in monopoly. If, on the other hand, costs are constant or falling as the firm grows — the far more common case, once fixed costs and the economies of scale are admitted — then nothing stops the firm from expanding indefinitely, and competition again dissolves into the dominance of the few. Either way, Sraffa concluded, the smooth competitive supply curve of the textbooks rested on a contradiction: the conditions that would make it rise are the very conditions that destroy the competition it presupposes.
Sraffa's quiet, technical demolition had consequences out of all proportion to its modest length, for it pointed the way out of the impasse it had exposed. If the competitive model could not survive realistic costs, then economists should stop assuming competition and study, instead, the markets that actually exist — markets in which firms are not price-takers but face downward-sloping demand curves of their own, each with some power over its price, each selling a product a little different from its rivals'. Within a few years the suggestion had borne fruit on both sides of the Atlantic: Joan Robinson in Cambridge and Edward Chamberlin at Harvard published, in the same year of 1933, the two founding treatises of the theory of imperfect, or monopolistic, competition. The tidy Marshallian world of countless identical price-takers gave way to a messier and more realistic one, populated by firms with brand names and market power, setting their own prices, advertising their differences, and earning, behind the shelter of their fixed costs and their economies of scale, returns that no purely competitive firm could keep. The indivisible lump, in short, did not merely complicate a calculation; it redrew the map of how markets are understood to work.
Sraffa's question was quiet and technical, but its answer dismantled the competitive ideal: if costs fall as a firm grows, no firm is the helpless price-taker the model required, and the market becomes a contest of the few, not the harmony of the many.
It would be comforting to treat all this as a catalogue of special cases, exceptions to a rule that holds well enough in general. The discomfort is that the exceptions are nearly everywhere. Modern production is fixed-cost production: the factory, the power station, the rail and road and pipe networks, the research that yields a drug or a chip, the writing of a program that thereafter copies for nothing — each loads a vast expense onto the front of the enterprise and spreads it thinner with every unit sold. Whole industries are natural monopolies, in which one network is cheaper than two and competition would be sheer waste, so that we tolerate a single provider and regulate it rather than pretend rivals could sensibly coexist. The all-or-nothing decision is the daily bread of anyone who must choose whether to undertake a project, rather than how much of it to do. And increasing returns, far from being a curiosity, may be the central fact of economic growth, the reason that large, dense, specialised economies pull ever further ahead of small and scattered ones. The smooth, divisible, diminishing-returns world in which the margin reigns unchallenged is not the world we mostly inhabit; it is a convenient approximation to one corner of it.
None of which overturns the margin so much as it fences it. The marginal rule is not refuted by the existence of lumps; it is bounded by them, and within its bounds it remains exactly as powerful as before. Even amid indivisibilities the margin keeps its place between the lumps: having decided to build the factory, one still decides at the margin how much to produce in it; having laid the network, one still prices the marginal call or carriage by the marginal rule. Many of the indivisibilities, moreover, are softer than they first appear — capacity can often be added in smaller increments than the whole-factory picture suggests, a network can be extended a street at a time, and what looks like an unbreakable lump dissolves, on closer inspection, into a coarse but genuine divisibility. And where the lumpiness is real, the apparatus can be adapted to it, at the cost of its elegant smoothness, by the methods that handle choices among whole numbers rather than continuous quantities. The honest summary is that the margin is the right instrument for the smooth and divisible, the total comparison the right instrument for the lumpy and indivisible, and increasing returns the one genuinely revolutionary case, where the diminishing returns the whole apparatus assumes are not merely bounded but reversed.
So the indivisible marks the first true frontier of the margin's empire — the place where the smooth calculus loses its footing, the all-or-nothing comparison takes over, and increasing returns can dissolve the very competition the marginal picture took for granted. It is a real limit, and an important one, but it is a limit on the margin's scope, not on its truth: where the margin applies, it applies as firmly as ever. There remains, however, a more unsettling question than whether the margin's smoothness holds — a question not about the world the rule describes but about the people the rule is supposed to describe. The whole apparatus assumes that decisions are made at the margin: that a firm setting its output actually weighs marginal cost against marginal revenue, that the businessman has the marginal quantities before him and equates them. But does he? Does anyone? Or do firms merely blunder along by rules of thumb and habit, arriving at outcomes that look, after the fact, as though margins had been equated, without anyone having equated anything at all? That question detonated, in the middle of the twentieth century, into one of the angriest controversies the discipline has known. The next plate enters the quarrel. As the table says of every quantity, first to last: the argument remains open.