What does this equation mean?

In 2007, Bettencourt, Lobo, Helbing, Kühnert and West published a cross-sectional analysis of American metropolitan statistical areas that has since become the standard citation for the claim that cities obey scaling laws. Measured against city population, quantities tied to innovation and economic output scale superlinearly — new patents scaled with an exponent of 1.27, inventors 1.25, private research and development employment 1.34, GDP in a range of 1.15 to 1.26, total wages 1.12, even serious crime 1.16 — while quantities tied to physical infrastructure scale sublinearly: length of electrical cable 0.87, road surface 0.83, gasoline stations 0.77, gasoline sales 0.79 [ 1 ] . In the compact notation the paper made famous, a quantity Y relates to population N as $Y(N) = Y_0 \, N^{\beta}$. where $\beta$ $\approx$ 1.15 for the socioeconomic quantities and $\beta$ $\approx$ 0.85 for the infrastructural ones. A city twice the size of another does not simply have twice the patents and twice the roads; it tends to have somewhat more than twice the patents and somewhat less than twice the road surface. The paper’s own framing is worth stating precisely, because it is often flattened into a slogan: doubling a city’s population is associated with roughly a fifteen percent bonus in per-capita innovation and wealth, and a corresponding roughly fifteen percent saving in per-capita infrastructure, at the same time. This is the opposite of how biological organisms scale — an elephant’s metabolism runs slower, gram for gram, than a mouse’s, and larger organisms generally do less per unit of mass, not more. Bettencourt and colleagues note explicitly that cities invert this pattern: the pace of social and economic life increases with size rather than slowing down, which is the empirical basis, and the important qualifier, behind any comparison between cities and biological organisms — the resemblance is in persistence and inheritance, argued through the rest of this article, not in metabolic scaling.