Figure 3

Elasticity of scale and the transition from decreasing to increasing returns to scale of GDP. (a) Dependence of the elasticity of scale (\(\varepsilon\)) on population (N) and incoming commuters (S) calculated from the translog model for US counties (upper panel) and Brazilian municipalities (lower panel). The color-coded background represents the elasticity of scale for each location in the plane population versus incoming commuters (on base-10 logarithmic scale) and the gray circles indicate the values of \(\log N\) versus \(\log S\) for each US and Brazilian urban area. The solid lines in both panels separate the region of decreasing returns to scale (\(\varepsilon <1\): bluish region) from the increasing returns to scale one (\(\varepsilon >1\): reddish region). We notice that urban areas are spread around the trend defined by \(S \sim N^\gamma\) (or \(\log S \sim \gamma \log N\), with \(\gamma = 0.87 \pm 0.01\) for the US and \(\gamma = 1.06 \pm 0.01\) for Brazil – white dashed lines) and that about 63% of urban areas in the US and Brazil display increasing returns to scale. The spread of the relationship between \(\log S\) and \(\log N\) allows urban areas with similar population sizes (as illustrated by the star markers) or similar incoming commuters (as illustrated by the square markers) to have significantly different values of elasticity of scale. (b) Probability distribution of the elasticity of scale estimated for all urban areas in the US (purple) and Brazil (green). The average elasticity of scale \(\langle \varepsilon \rangle\) is slightly larger for Brazil than the US (\(\langle \varepsilon \rangle =1.04\pm 0.001\) and \(\langle \varepsilon \rangle =1.02\pm 0.001\), respectively), while the Brazilian distribution is significantly more skewed to the right when compared with the US case (skewness \(=1.1\) and skewness \(=0.3\), respectively). (c) Relationship between the elasticity of scale and the GDP of urban areas in the US (purple) and Brazil (green). The GDP of Brazilian municipalities has been converted from Brazilian Real (BRL) to their equivalent values in US Dollars (USD) in this plot. We notice that the elasticity of scale is approximately proportional to the logarithm of urban GDP (as indicated by the dashed lines) with the constant of proportionality (\(\kappa\)) larger for Brazil than the US (\(\kappa =0.124\pm 0.001\) and \(\kappa =0.099\pm 0.001\), respectively). We also observe that the elasticity of scale of urban areas with similar GDP is always larger for Brazilian municipalities than US counties.