A theory of heterogeneous city growth

We consider an economy with three cities producing different outputs. Two cities produce intermediate goods, a type 1 producing an intermediate "agricultural" good with capital and labor only, and a type 2 producing an intermediate "industrial" good with capital, labor and human capital, and the last type 3 city produces the final good which is obtained from the two intermediate goods and labor. The asymmetric introduction of human capital allows us to prove that the three cities experience at the equilibrium heterogeneous endogenous growth rates which are proportional to the growth rate of human capital. We show that the "industrial" type 2 city is characterized by the larger growth rate while the "agricultural" type 1 city experiences the lower growth rate, and thus the type 3 city is characterized by a growth rate which is a convex combination of the two formers. This implies that the relative size in terms of output of the "agricultural" city decreases over time. This property allows to recover the empirical fact that most nonagricultural production occurs in growing metropolitan areas. But, simultaneously, as we prove that total labor employed in each city is proportional to the total population, the relative population size distribution of cities is constant over time as shown in empirical studies.