Productive activities in cities and human cognitive capacities

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Bettencourt, Lobos and West. model the potential links between variations in β for productive activities in cities and human cognitive capacities. Here is the relevant quotation from their appendix:

"The exponents β are commensurate for many social quantities, but there is no strong indication that they must be identical for different urban systems. Can we determine β from the formulation of a maximization or minimization principle, as was done in biology for the properties of networks of resource distribution? There is little doubt that human interactions in a city may be represented in terms of networks, and it is difficult to foresee their general structural properties. What we do know is that if a city provides an enlarged space of opportunities for effective interactions between people but also that the number and intensity of such interactions is constrained by time and effort and by limits on individual cognition. This is what Milgram, writing on the experience of living in cities (3), referred to as information saturation. This observation can be used to produce an estimate of the values of β.

First, consider the total number of effective contacts C between individuals in a population of size N. The maximal value that C can take is C = N(N − 1) / 2, implying a bound on . This upper bound corresponds to every individual in a city knowing everyone else, which is clearly not realistic as cities grow large. Instead, consider that the quantities Y of Table 1 are proportional to the number of effective contacts so that C(N) = C₀N^β. Let's now define P as the ratio of productive contacts per capita between the largest city with population Nmax and the smallest city with population N_min, so that

. [1]

P expresses by how much an individual's time, effort, and cognitive ability can be expanded in response to the greater demands of the largest city, relative to those of the smallest town. If we assume P = 10-100, and N_max / N_min = 10⁷, we obtain β = 1.14-1.28, which is in qualitative agreement with the observations."

Table 1. Scaling exponents for urban indicators vs. city size

Y	β	95% Confidence	Adj-R2	Observations	Country–year
New patents	1.27	1.25,1.29	0.72	331	U.S. 2001
Inventors	1.25	1.22,1.27	0.76	331	U.S. 2001
Private R&D employment	1.34	1.29,1.39	0.92	266	U.S. 2002
Supercreative employment	1.15	1.11,1.18	0.89	287	U.S. 2003
R&D establishments	1.19	1.14,1.22	0.77	287	U.S. 1997
R&D employment	1.26	1.18,1.43	0.93	295	China 2002
Total wages	1.12	1.09,1.13	0.96	361	U.S. 2002
Total bank deposits	1.08	1.03,1.11	0.91	267	U.S. 1996
GDP	1.15	1.06,1.23	0.96	295	China 2002
GDP	1.26	1.09,1.46	0.64	196	EU 1999–2003
GDP	1.13	1.03,1.23	0.94	37	Germany 2003
Total electrical consumption	1.07	1.03,1.11	0.88	392	Germany 2002
New AIDS cases	1.23	1.18,1.29	0.76	93	U.S. 2002–2003
Serious crimes	1.16	[1.11, 1.18]	0.89	287	U.S. 2003
Total housing	1.00	0.99,1.01	0.99	316	U.S. 1990
Total employment	1.01	0.99,1.02	0.98	331	U.S. 2001
Household electrical consumption	1.00	0.94,1.06	0.88	377	Germany 2002
Household electrical consumption	1.05	0.89,1.22	0.91	295	China 2002
Household water consumption	1.01	0.89,1.11	0.96	295	China 2002
Gasoline stations	0.77	0.74,0.81	0.93	318	U.S. 2001
Gasoline sales	0.79	0.73,0.80	0.94	318	U.S. 2001
Length of electrical cables	0.87	0.82,0.92	0.75	380	Germany 2002
Road surface	0.83	0.74,0.92	0.87	29	Germany 2002

Data sources are shown in SI Text. CI, confidence interval; Adj-R2, adjusted R2; GDP, gross domestic product.

From: 2007 Growth, innovation, scaling, and the pace of life in cities. Luís M. A. Bettencourt, José Lobo, Dirk Helbing, Christian Kühnert, and Geoffrey B. West PNAS 104(17):7301-7306.