Urban economics models cities in equilibrium. Land prices balance against commuting costs. Population density declines exponentially from the center. The city's edge is where the rent equals the agricultural value of the land. The framework is elegant, tractable, and wrong about the process — because cities do not reach equilibrium. They grow.
The growth is not a perturbation around a stable state (arXiv:2603.08338). Urban expansion is a non-equilibrium dynamical process driven by the interaction of at least three forces: population growth (which pushes demand outward), infrastructure investment (which reduces commuting costs and enables farther expansion), and market failures (which cause land to be consumed faster than optimal because individual decisions do not account for collective costs like congestion, pollution, and habitat loss).
Static models capture the shape of the equilibrium — the monocentric city with its declining density gradient — but not the trajectory. The trajectory matters because the path of growth determines the built environment, and the built environment is nearly irreversible. A subdivision built at the urban fringe in year T locks in a density, a street layout, and an infrastructure pattern for decades. The equilibrium model says what density should be; the dynamical model says what density will be, given that decisions are made sequentially, irreversibly, and with imperfect information.
The mathematical modeling shows that even without market failures, the interaction between population pressure and infrastructure investment creates oscillatory dynamics — cycles of rapid expansion followed by consolidation — rather than smooth convergence to equilibrium. Market failures amplify the oscillations and bias them toward over-expansion: the city grows faster and farther than any equilibrium model predicts, because the costs of sprawl are externalized while the benefits of proximity are privately captured.
The city is not a solved optimization problem. It is an ongoing process that never reaches the state it is approximating.