Fig. 2

The interplay between population and area on CO₂ emissions. a Scatter plot of the observed values of CO₂ emissions (C) and those predicted (C_P) by the Cobb–Douglas model (Eq. (3) with β_P = 0.31 ± 0.01 and β_A  = 0.45 ± 0.03). This model is a significantly better fit when compared with the urban scaling and the per capita density scaling models (Supplementary Fig. 2). b A contour plot of Eq. (3) as a function of P and A on logarithmic scale. The straight isolines/isoquants show how population and area must change in order to keep the emissions unchanged. c Scatter plot between the observed and predicted CO₂ emissions obtained from the translog model (Eq. (5) with β_P = 0.28 ± 0.02, β_A = 0.14 ± 0.05, and β_C = 0.07 ± 0.01). This model further refines the goodness of the predictions (Supplementary Fig. 2), particularly reducing the bias in urban areas with high emissions. d A contour plot of Eq. (5) as a function of P and A. We note that the isolines/isoquants of this model are not straight lines as those from the Cobb–Douglas model. We have employed base-10 logarithmic quantities in all panels