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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.

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