It is well-known that larger cities produce wealth through mechanisms known as agglomeration economies, which can be regarded as mechanisms by which large collections of people, who can easily travel and trade amongst each other, provide economic advantages over small collections. In this post, we will focus on the theoretical bases for agglomeration. Elsewhere, we will consider some estimates of the magnitude of the agglomeration effect.
Duranton and Puga (2004) notes that the spatial concentration of cities greatly exceeds what would be expected from land heterogeneity alone. In other words, while some locations are more physically desirable than others, if the physical quality of a location was the only factor in driving people’s choice of location to live and do business, then we would expect the population to be much more diffuse than it actually is. Cronon (1991), for instance, observes that many sites on the shore of Lake Michigan are of equal quality to the location on which Chicago is built, and yet Chicago is by far the dominant city in the region.
Furthermore, agglomeration economies are necessary to understand trade. Starrett (1978) introduced the Spatial Impossibility Theorem, which states that in the absence of agglomeration economies, there is no advantage to large-scale production over small-scale production, and there is the disadvantage of transportation costs, and therefore there should be no travel and trade. There clearly is travel and trade, and therefore there must be agglomeration economies.
Duranton and Puga (2004) consider three main agglomeration mechanisms, or micro-foundations. The first is the sharing of resources and facilities, the second is matching, and the third is the enhanced opportunities for learning. This is not a complete list of agglomeration mechanisms. We will now consider each of these three mechanisms in greater detail.
Sharing
Sharing is the first of three agglomeration mechanisms considered by Duranton and Puga (2004), and they in turn consider several ways in which sharing produces increasing returns to scale. In addition to those discussed below, they also highlight risk sharing: a larger labor market provided by a larger city helps insure a steadier demand for labor.
Indivisibility
Duranton and Puga (2004) open their discussion of sharing with the simple case of indivisibility. Some facilities, such as ice hockey rinks, must be of a minimum size and carry substantial fixed costs, and therefore it is not feasible for an ice hockey rink to serve a single person who is not engaged in trade with others.
Edwin Mills, in his contribution (Mills (1967)) to the development of the widely-used Alonso-Muth-Mills standard urban model, considers indivisibility as one of the main agglomeration economies that justifies the development of cities. An indivisible facility can be modeled as one with a large fixed cost and modest marginal cost for usage. Additionally, use of the indivisible facility is limited by two crowding mechanisms. First, a facility itself can support a limited number of customers before it becomes too crowded. Second, the larger a facility, the larger the geographic extent that it serves, and this increases per-capita transportation costs generated by the facility. Under Mills’ model, a determinant of the size of the city is the equilibrium between the economies of scale of indivisibility and the diseconomies of scale of crowding.
However, Duranton and Puga (2004) argue that, while indivisibility is a useful modeling device, it is insufficient to explain the existence of agglomeration economies, especially for very large cities. There does not appear to be any indivisibility that requires a metropolis of tens of millions of people to support. Instead, Duranton and Puga posit increasing returns to scale that result from a wider variety of differentiated intermediate inputs that a larger city can provide. They then model a city’s equilibrium size as a tradeoff between the benefits of differentiated input and the cost of crowding.
Division of Labor
Adam Smith (Smith (1776)) suggests specialization as the mechanism underlying agglomeration economies in the example of the pin factory. A larger workforce allows workers to specialize in narrower tasks, which improves productivity by three mechanisms, as explained in Duranton and Puga (2004). First, specialization allows workers to gain proficiency at a particular task in a “learning by doing” benefit. Second, specialization allows workers to avoid costs associated with task-switching. Third, specialization allows greater opportunity for mechanization of tasks.
Yang and Rice (1994) construct a general equilibrium model and find that division of labor is a necessary, though not sufficient, condition for cities to emerge. It is also necessary that transaction costs are lower than the efficiency benefit from division of labor.
Equilibrium City Size
The model of Duranton and Puga (2004) has several implications on the equilibrium sizes of cities. First, cities that specialize in sectors that show greater returns to scale tend to be larger in equilibrium size. This should not be a surprise; we observe that cities with economies based on design and services tend to be larger than manufacturing cities, which in turn tend to be larger than agricultural cities. Davis and Dingel (2014) confirm this observation on a dataset of 270 metropolitan areas in the United States.
A second consequence of Duranton and Puga’s (2004) model is that cities tend to be larger than the optimal size for their specialization. The reason is that individuals will choose their location without accounting for the crowding costs that they impose on other residents. They will only account for the crowding costs that they themselves experience. The problem stems in part from a coordination failure in city creation, which according to Henderson and Venables (2009) can be mitigated through better government institutions.
Matching
Another major agglomeration advantage offered by cities is the ability to improve matching. With a larger labor market, the probability is improved that a given worker will find a job that closely matches their skillset, thereby increasing their productivity. Duranton and Puga (2004) construct a model in which two matching benefits arise. First, due to the competition mechanism, the average mismatch–the gap between the productivity of a worker at a job for which they are perfectly suited and their actual productivity–goes down with city size. Second, the number of firms increases with city size, but at less than a linear rate, and thus the size of the average firm increases as well, and this reduces per-capita fixed costs associated with production. Although Duranton and Puga (2004) find increasing returns to scale, they increase asymptotically to an ideal per-worker productivity, and they do not increase without bound.
As is the case with sharing, Duranton and Puga (2004) model an equilibrium city size based on the agglomeration benefit of improved matching and the diseconomy of crowding that comes with larger cities, and they derive an equilibrium city size that is larger than the socially optimum city size.
Duranton and Puga (2004) consider several variations of the matching effect, including a model of Berliant, Reed, and Wang (2006). Their model finds that the probability of an employee or an employer finding a suitable match increases with city size, thus inducing these agents to be choosier about their matches. The model finds that agents are less picky than would be socially optimal, and that on account of this, cities are smaller than their ideal size. This stands in contrast to cities being too large on account of residents failing to account for crowding costs they inflict on others. Berliant, Reed, and Wang (2006) find that counties in the United States that are in the top 10% by population density, compared to those in the lower half, have productivity 0.15 to 0.3 log points higher, and this cannot be explained by differences in firm composition. However, it is not clear that improved matching is the explanation for this difference.
Learning
By offering more opportunities for face-to-face communication, larger cities facilitate more opportunities for various forms of learning. This is a well studied agglomeration mechanism, with treatments including those of Marshall (1890) and Jacobs (1969). “Learning” is a broad term, which encompasses activities ranging from formal education to informal learning of techniques of production to networking.
Duranton and Puga (2001) consider a specific learning mechanism: the tendency for young firms to perfect their production processes by tinkering. This tinkering process is aided by other firms, perhaps in somewhat different industries, that share knowledge about production. For this reason, young firms tend to thrive in cities with diverse economies. As firms mature and they have less need for further tinkering, they tend to thrive more in cities with specialized economies.
Duranton and Puga (2004) develop a model in which the agglomeration advantage of cities is in skill transfer between old, skilled workers and young, unskilled workers. A large city provides opportunity for such skill transfer that is not available in a rural area. Young workers are willing to pay a premium to live in the city to get access to this human capital, while old workers benefit from rents from transmitting their skills. As is the case with sharing and matching, Duranton and Puga (2004) model an equilibrium city size from the skill transfer scenario, based on when the marginal benefit of skill transfer is equal to the marginal crowding cost.
Conclusions
In this post, we have focused on three micro-foundations of agglomeration economies as presented by Duranton and Puga (2004): sharing, matching, and learning. This is not a complete list of proposed mechanisms. While agglomeration economies are firmly established to exist, the precise mechanisms by which they occur are unclear. Part of the challenge is that, since each of these micro-foundations are empirically similar, it is difficult to determine empirically the relative importance of these effects.
References
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