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CHAPTER 1
Surveying Lumpiness
Picture a bridge spanning a chasm. Removing one chunk of the span renders it worthless — indeed, it is no longer even a bridge. Because bridges are useless unless they are complete, they offer intuitive examples of lumpy, indivisible, or "step" goods. Lumpiness is found not only in large-scale infrastructure like bridges, highways, and railroad lines, but also in ordinary products and services. Some goods, like car tires or developable land, are more valuable if consumed in particular quantities or combinations. Others, like cars, jobs, houses, and pets, are often available only in difficult-to-divide chunks. Conditions like species survival or election wins depend on maintaining or reaching critical thresholds, not merely coming close. Legal rules and litigation outcomes may also exhibit lumpiness, operating in an all-or-nothing fashion, or producing results only when some threshold of compliance or deterrence is reached. And the lumpy fixed costs that attach to many endeavors — from introducing a new product to passing a new law to learning a new skill — make choices fewer and chunkier for firms, consumers, citizens, and workers than they otherwise would be.
These and many other examples will be explored in the chapters that follow. Here, I take up two foundational questions: What counts as "lumpy"? And why do we care? The answers to these questions will preview the range of aggregation and division problems taken up in this book. Many of these problems involve desired, attempted, thwarted, or contested reconfigurations — attempts to slice up things that are difficult to divide or to aggregate things that start out in pieces. Others concern the appropriate legal or practical treatment of naturally occurring or constructed lumps, whether in regulatory policy, legal analysis, informal order, bargaining settings, or the realm of self-control.
What's Lumpy?
The idea of lumpiness seems intuitive, but the term is used in more than one way and encompasses a variety of phenomena. Some distinctions and definitions will help to set the stage.
Supply, Demand, and Lumpiness
We might refer to a good as a lumpy or indivisible either because this is how the good delivers its value (in a lump, like a bridge) or because the good arrives in a lump and is accompanied by constraints (natural or constructed) that make it difficult or costly to divide (think of the full-time position that does not allow for part-time work). These are, in a sense, opposite meanings. In the first, lumpiness describes a desired end state (the completed bridge). In the second, lumpiness describes a suboptimal starting point (the full-time job). In both cases, there is a mismatch between the starting point and the desired end state, but what is necessary to span that gap differs. To build the full bridge, many smaller pieces must be assembled. The lumpy job comes preassembled, and that is exactly the problem — a slice of the job would be preferable for the employee.
One way to express this distinction is between goods that are lumpy in demand (people want full bridges) and goods that are lumpy in supply (cars and pets come in whole number units). Some goods might be described either way. For instance, we could say that an employer supplies jobs in full-time increments or demands labor in full-time increments. Regardless, lumpiness becomes interesting where what is desired (by someone) takes a different form than what is provided (by someone else). A good that is lumpy in demand, like a bridge, often must be assembled from inputs — bridge segments, labor, financial contributions, and so on — that are fragmented in supply. A good that is lumpy in supply, like a car, may need to be split into smaller use-slices to effectively meet consumer demand.
Often lumpiness is of no consequence because it can be addressed through ordinary markets or informal transactions. For example, if the smallest unit of candy that can be economically produced and sold separately is a 1.5 ounce candy bar, and if most people have no desire to purchase candy in smaller increments than this, whatever theoretical lumpiness may exist presents no difficulties. Lumpiness becomes problematic when the supplied units are much larger or smaller than desired (think of a mammoth candy bar or a single chocolate chip) and there are significant impediments to dividing up the larger unit or aggregating the smaller ones. The obstacles may stem from physical constraints or the costs of engaging in market transactions. They may even be social or psychological in nature. Philip Henry Wicksteed, writing in 1910, observes that the commercial standard of supplying ink in one-penny measures effectively precludes people from acquiring smaller quantities, given the "awkwardness and humiliation" involved in negotiating with a stationer for a smaller amount.
Lumpiness can also cause difficulties when everyone agrees that the initial (lumpy) configuration is the most valuable one, but there is more than one plausible claimant. A vivid example is the dispute over the baby that featured in King Solomon's famous decision. Babies, it turns out, are extremely lumpy. Luckily, there are alternatives to physical division, and the Solomonic outcome illustrated one of them — an award to the claimant who clearly valued the child more. As the literature on this topic has noted, indivisibilities may be addressed through a variety of techniques, including slicing the good temporally (e.g., through rotation systems); converting the good into something divisible like money, as by auctioning it off; giving claimants chances at the good that are proportionate to the strength of their claims; or giving the good to one claimant while compensating the others.
Temporal slicing of goods is an especially intriguing solution because it can bridge the gap between the physical configuration that maximizes value and the amount of the good that a particular individual wants, needs, or is entitled to receive. It works well for goods that are far more valuable when physically intact, where people do not want, and are unwilling to pay for, the whole thing. No formal slicing is necessary if people can agree to share the resource. In some cases we manage to do exactly that. People form clubs or enter communities to consume certain kinds of indivisible goods — swimming pools, tennis courts, clubhouses, and so on. Other varieties of time slicing are longstanding and familiar: library books, hotel rooms, rental cars, and so on. Entrepreneurs are now finding a multitude of ways to create small-scale market transactions that further fine-tune slicing, as evidenced by Airbnb, Uber, and many other business models. An extreme example is Recharge, an app that allows people to buy "microstays" at hotels and apartments, priced by the minute.
Consider another innovation in temporal slicing, pet sharing. Companion animals, like babies or bridges, are lumpy and can't be physically divided. But the unit in which pets arrive is not necessarily the optimal unit in which their companionship is consumed. Suppose that for one individual, Angus, dog ownership is great fun for a few days a week, but the burden continues to grow as the week wears on, and the benefits diminish apace. If the unfun days of Angus's dog-owning week could be transferred to other people who similarly experience declining returns from dog ownership (Beth and Cam, say), the dog could deliver a larger total quantum of enjoyment to its (now plural) owners.
There may be problems, of course. Time-share dog owners may shirk on bathing the dog or taking him to the vet. The dog may never get properly trained, or the constant parade of owners may produce anxiety or confusion for the dog. Some of these issues might be overcome by, for example, having a platform manager who coordinates tasks, establishes minimum time blocks, and sets care standards, but these solutions add to the costs of time slicing. BorrowMyDoggy.com, which currently operates in the UK and Ireland, enables a pet owner who retains primary responsibility for her pup to offer short-term "borrowing" in exchange for dog walking, care, or socialization, while the platform provider collects a fee that covers veterinarian access and insurance. This model offers an approximation of informal interactions over pets among friends and family, adapted to urban settings where people often lack preexisting social networks. Here, as in many other contexts, from ride sharing to home sharing, we see new models for managing lumpiness emerge as earlier (and mostly unremarked) ways of informally aligning supply and demand break down.
Some Terminology
The notion of lumpiness connects tightly to the concepts of indivisibility and complementarity. To say that a good is indivisible or that it exhibits indivisibilities does not usually mean that the good literally cannot be divided, but rather that it is considerably less valuable when divided, or that it is expensive (perhaps prohibitively so) to divide successfully. The idea of complementarity refers to the fact that certain goods and services produce more value when consumed in particular combinations. Right and left shoes are a standard example. Because most people have two feet of similar size and follow the social custom of shodding them identically, a pair of shoes typically delivers far more than twice as much value as a single shoe. Likewise, the segments that make up a full bridge span are strongly complementary; subtract just one, and the bridge becomes useless. A partially fenced yard does no better than an unfenced yard at containing animals, a car with three tires drives no better than a car with no tires, and small and scattered patches of land are useless for large-scale development.
In these familiar examples, indivisibilities are a function of complementarities. A set of tires or a pair of shoes exhibits indivisibilities not because tires or shoes are physically hard to separate from each other, but rather because splitting them up would be self-defeating — they are much more valuable when consumed together. Not all indivisibilities track complementarities in this way. Other things that we might characterize as indivisible (cars, jobs, pets, houses, and so on) might be more valuable in pieces (whether time slices or physical slices) but dividing them up is for some reason technologically or administratively difficult. I will use the term indivisibility in this book as a synonym for lumpiness. The notion of complementarity represents a general purpose explanation for why goods or services might be more valuable when aggregated in certain ways.
Two other terms associated with lumpiness are discontinuities and nonlinearities. Returns from activities like studying or voting are often discontinuous: making it over some threshold makes the difference between passing and failing, or between winning and losing an election. Nonlinearities occur when outcomes do not increase smoothly and proportionately in response to inputs. There may be increasing returns (economies of scale), diminishing returns (diseconomies of scale), sharp steps or notches at particular thresholds, or some mix of these effects. The economic tool of the production function, which maps inputs to outputs, provides traction on these ideas.
Lumpy Production Functions
Lumpiness can be understood as a certain kind of relationship between inputs (units of effort, money, or resources) and outputs (conditions, events, products, or services). Consider, for example, the connection between dollars contributed to a charity and the benefits that the charity generates in the world. If this relationship is plotted on a graph with well-being improvements on the vertical axis and dollars on the horizontal axis, what shape will the curve take?
There are many possibilities. Perhaps the relationship is linear, at least within a particular range, so that each additional dollar generates the same uptick in benefits. Think of assistance that buys increments of soup, medical care, or clean drinking water, which in turn produce a corresponding improvement in well-being among the recipient population. In other cases, a plateau may be reached after which additional dollars do less good than the dollars that went before — after every household has mosquito nets, say, the next best uses of the money may be less effective at producing marginal improvements. Conversely, there may be a snowball effect, so that as more contributions are added, each does more and more good, at least up to a point — think of class sessions added to an educational program, or inoculations against communicable diseases within a community. Or the curve may be S-shaped, with a range of increasing effectiveness followed by a range of diminishing returns.
Production functions for lumpy goods deliver outputs not in smooth, regular increments as individual units of input are added, but rather in large jumps after a series of inputs. At the extreme is a pure step good that delivers all of its utility in one large chunk or "step." Think again of a bridge. Suppose you need to span a chasm that is a thousand yards long, and the bridge material arrives in one-hundred-yard segments.
As shown in figure 1.1, value to users remains flat as the first nine segments are added, one by one. But when the tenth unit is added to create a completed bridge, suddenly value steps up all at once. There is a sharp discontinuity, illustrated by the dashed line in figure 1.1. The step not only marks out a threshold under which no benefits are provided, but also represents a plateau from which no further incremental improvements are possible. Adding more lengths to the bridge once the span is complete does no good.
In fact, such pure step goods are rare. Even a bridge can be supplied at many different quality levels, as Russell Hardin has noted. An election is also a common example of a step good — here, the inputs are the votes that either do or do not reach the critical point that enables one's preferred candidate to win. Votes short of the amount necessary to win are useless in generating the desired outcome, while extra votes beyond that level are superfluous. Of course, if one defines political objectives slightly more broadly than choosing a winner in a particular contest, the step function looks less sharp. Often we think that landslides produce at least somewhat better results for the winner than do narrow victories, while near-misses provide greater political impetus for another try than would a crushing defeat. Nonetheless, these examples provide an intuitive sense of what a lumpy or step good looks like.
Equally rare are perfectly linear goods — those with a smooth, continuous production function in which each infinitesimally fine unit of input is matched by a corresponding adjustment in output or utility. Few products can be produced, purchased, or enjoyed in literally any quantity. Often some minimum threshold must be crossed to obtain (or enjoy) the thing at all, and many goods must be transacted over in integer units (bananas, for instance). Even readily divisible goods — Wicksteed uses the example of pudding servings for children — may be relatively valueless below a certain quantity threshold.
Between the extremes of a perfectly linear good and a single-step good, we find different degrees of nonlinearity or indivisibility. Consider figure 1.2, which depicts an S-curve. This curve corresponds to a relatively lumpy good that does not take a pure step form.
Although this good does not deliver all its value in a single shot, its production function contains ranges over which the marginal effect of added pieces is sharply increasing or decreasing. The S-curve describes many collective goods that require a critical mass of participation to succeed, but that at some point plateau. It might also fit with certain kinds of land assembly projects, where value increases sharply once a certain number of parcels are aggregated, but where having all the parcels is not essential.
Lumpiness, as used in this book, refers to severe discontinuities or nonlinearities in the production function, whether or not those functions take a pure step form or intersperse segments of sharply increasing or decreasing returns with ranges exhibiting linearity. These differences in shape are important, however, because they can influence the prospects for cooperation and the risks of strategic behavior, as we will see in chapters 2 and 3.
What's in the Lump?
So far, I have spoken of "segments" or "pieces" that produce value when aggregated together. Lumpiness or indivisibility often refers to quantities of relatively fungible inputs — segments of a bridge, lengths of railroad track, tires for a car, units of work, and so on. Yet it may also refer to systems made up of heterogeneous elements, such as a machine that cannot operate without each and every one of its parts. I will use the notion of lumpiness broadly and functionally here to refer to both heterogeneous and homogeneous aggregations, given that both forms of lumpiness can generate similarly structured problems.
(Continues…)
Excerpted from "Slices and Lumps"
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Copyright © 2019 Lee Anne Fennell.
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