The spatial evolution of economic activities and the emergence of cities

This paper examines the spatial agglomeration of workers and income in a continuous space-time framework. Local markets feature spatial spillovers and both exogenous and endogenous amenities. Workers relocate to maximise their instantaneous utility, constrained by mobility costs. In the limit of infinite workers, short-run equilibria are described by a partial differential equation (PDE). The PDE reveals spatial dynamics influenced by initial conditions, path dependence, and metastability (persistence), where prolonged stability is disrupted by sharp transitions to new distributions. We characterise conditions for spatial agglomeration in stationary equilibria and demonstrate that social utility consistently increases over time, suggesting efficient spatial allocations. Numerical results replicate key patterns, such as city formation, dependence on historical spatial patterns, and nonlinear out-of-equilibrium dynamics.