The article by Arrow and Kehoe lays out some of Scarf's most prominent work. For example, there is a theory of the optimal holding of inventories called the (S, s) theory: basically, the idea is that firms and stores don't re-order more supplies every day. They re-order it in batches. They wait until the quantity on hand falls to some lower level s, and then place an order for a fixed amount which raises the quantity on hand up to the higher level S. The theory of how far apart s and S should be will depend on various measures of volatility and risk. The question of when or if (S, s) theory was the right way to think about inventory problems was a hot topic in the 1950s. Scarf provided an answer that was as nearly definitive as these things get in economic theory, arguing that it was.
It turns out that the (S, s) theory isn't just about business inventories. More broadly, it's a theory about how economic agents make decisions when there are costs of adjustment, which can often lead to a situation where there are long periods where nothing much seems to happen, followed by sharp changes. For example, this pattern often arises in business investment of many kinds, in hiring and firing decision by firms, in consumer purchases of big durables like cars and houses, and even in small-scale decisions like taking a larger fixed amount of cash out of the ATM machine, rather than going by the machine every time you need $20. It further turns out that these sharp and lumpy changes can be related to overall macroeconomic business cycles.
For those who want an overview of (S, s) theory, useful starting points in JEP would be the article by Andrew Caplin and John Leahy in the Winter 2010 issue, "Economic Theory and the World of Practice: A Celebration of the (S, s) Model," and farther back, the article by Alan S. Blinder and Louis J. Maccini in the Winter 1991 issue, "Taking Stock: A Critical Assessment of Recent Research on Inventories."
Scarf was a major player in many of the central topics of research into economic theory for several decades after the 1950s. Arrow and Kehoe discuss his work on describing the "core" of an economy, on how to calculate a fixed point as the equilibrium of an overall economy (not just an individual market), how the presence of increasing returns affected the existence of equilibrium, and other issues. I would only embarrass myself by trying to summarize this work here, but I'll note that while Scarf work was often deeply technical and mathematical, he also had a gift for suggesting straightforward phrases and analogies that clarified what was at issue.
Scarf's article in the Fall 1994 issue of JEP took up the topic of "indivisibilities," which overlaps with aspects of these issues of determining an optimal outcome in the presence of increasing returns and lumpy choices. Scarf wrote:
I am, I believe, not alone in thinking that the essence of economies of scale in production is the presence of large and significant indivisibilities in production. What I have in mind are assembly lines, bridges, transportation and communication networks, giant presses and complex manufacturing plants, which are available in specific discrete sizes, and whose economic usefulness manifests itself only when the scale of operation is large. If the technology giving rise to a large firm is based on indivisibilities, then this technology can be described by, say, an activity analysis model in which the activity levels referring to indivisible goods are required to assume integral values, like 0, 1, 2, . . . , only. When factor levels are specified and a particular objective function is chosen, we are led directly to that class of difficult optimization problems known as integer programs.Of course, this problem of considering big lumpy changes is conceptually similar to the inventory problem, which also involved thinking about lumpy changes. Scarf uses a series of numerical examples in JEP to argue that when there are indivisibilities, the optimal answer will be a "neighborhood system"--which is to say that there often is not a single correct answer, but rather a group of closely related possibilities. Here's his conclusion in the JEP article. For those not initiated into economics, it may not carry a lot of meaning. For those of us who have drunk the economics Kool-Aid, it's an example of Scarf's facility for using the language of technical economics with a mixture of concreteness and fluidity that keeps the economic themes front and center:
But let us leave this example with only two discrete choices concerning types of plants, and remember that in a large manufacturing enterprise there will be many discrete choices involving a large menu of tasks and machinery, each of which has its own capacity, set-up cost and marginal cost. The equipment may be placed in a number of different locations on the shop floor; the work may be passed from one piece of machinery to another with complex requirements of scheduling and precedence, and the tasks may alter from one job lot to another as the product specification varies. Demands may be revised capriciously and unexpectedly over time; output may be shipped to many different regions. The enterprise may have a host of competitors or none at all. In the absence of internal market prices, combinatorial arguments and quantity tests are necessary to regulate the flow of activity inside the enterprise in an optimal fashion.
My message boils down to a simple straightforward piece of advice; if economists are to study economies of scale, and the division of labor in the large firm, the first step is to take our trusty derivatives, pack them up carefully in mothballs and put them away respectfully; they have served us well for many a year. But derivatives are prices, and in the presence of indivisibilities in production, prices simply don't do the jobs that they were meant to do. They do not detect optimality; they aren't useful in comparative statics; and they tell us very little about the organized complexity of the large firm. Neighborhood systems are the discrete approximations to the marginal rates of substitution revealed by prices. They are relatively easy to compute, seem to behave pretty well under continuous changes in the technology, and will ultimately lead to even better algorithms than we have now.
We know much more about the structure of neighborhood systems than I have been able to describe here—not enough, perhaps, to derive a really satisfactory theory of the internal organization of the large firm at the present time. But my own intuition is that this is an important way to proceed. I am confident that serious, ultimately useful insights about the large firm will eventually be obtained by thinking very hard and long about indivisibilities in production.
