Figure 6.

The scaling of urban quantities with population size for MAs in Europe. These data include all urban systems in the EU and Switzerland with more than two cities above 500 000 people (102 functional cities in 12 nations: Austria, Belgium, the Czech Republic, France, Germany, Italy, The Netherlands, Poland, Spain, Sweden, Switzerland and the UK). Because the data have been centred in each nation (see text) the expected scaling relation has intercept zero in a logarithmic plot. We show the simplest prediction of urban scaling theory as a yellow line in each panel, setting superlinear exponents at β = 7/6 and sublinear at β = 5/6. With these choices, there are no free parameters and a direct test of urban scaling can be performed without statistical uncertainties arising from best fits in small data samples. (a) Shows the results for GDP and a nearly exact agreement with theory (best fit, R² = 0.90; table 1). (b) Shows the results for urbanized area, the best fit gives a slightly larger β than predicted by theory (red line, R² = 0.88), but fails to describe the largest cities. Urban scaling fits most of the data well and correctly predicts the urbanized area of London and Paris. Employment (c) is also linear as expected (R² = 0.97). Finally, patents in (d) are noisier, but the best fit (red line, R² = 0.30) and prediction from urban scaling theory (yellow line) are statistically consistent and, in particular, predict well innovation rates for Paris. Data for patents are not provided for the UK and are inconsistent for Poland, so that these two nations are not included in the analysis of figure 1d. (Online version in colour.)