  Global sustainability relies on our capacity of understanding and guiding
urban systems, and their metabolism, in an adequate way. It has been proposed
that bigger and denser cities are more resource-efficient than smaller ones
because they tend to demand less infrastructure, consume less fuel for
transportation and less energy for cooling or heating in per capita terms. This
hypothesis is also called Brand's Law. But as cities get bigger, denser and
more resource-efficient they also get richer. And richer inhabitants buy more,
increasing its resource demand and associated environmental impacts. To fully
understand the nexus between population size or density and the environmental
impacts generated by a city, one needs to take into account both direct and
indirect impacts. Facing the lack of empirical evidence on consumption-based
emissions for cities, in this paper we propose a mean-field model to derive
emissions estimations out of well-established urban metrics (city size,
density, infrastructure, wealth). We aim at understanding if Brand's law holds
true after adopting a consumption-based approach to urban emissions. The
proposed model shows that when considering consumption-based emissions, in most
cases Brand's law falls apart - bigger cities have greater purchase power,
resulting in greater consumption of goods and higher associated GHG emissions.
The model also shows that decoupling between population and emissions is
possible and dependent on the decoupling level between income and impacts. In
order to achieve it, a shift in consumption patterns of most cities is of
utmost importance, so that each new monetary unit added to the GDP, or any
other income variable for that effect does not result in a proportional
increase in GHG emissions.
