Endogenous city size in urban search models: the case of high reallocation costs

A large literature underlines the fact that city sizes are heterogeneous and urban sprawl is not optimal (i.e. cities are too large). Surprisingly, we do not have a clear understanding of these two facts in urban search economics (see Zenou (2009)). Indeed, this literature systematically considers the city size as exogenously fixed to unity. Thus, this article aims at answering the following questions: How to endogenize city size in urban search models' What are the relationships between city size and labor market? Is city size optimal? Motived by empirical studies (among others Brueckner and Largey (2008)), we show the existence of a density externality in a urban search model with high reallocation costs a la Zenou (2009). In presence of this density externality, we demonstrate that the land market equilibrium leads to an endogenous city size. Moreover, solving the labor market equilibrium, three other results appear. First, the global unemployment rate is affected negatively by the endogenous city size. Second, we prove that the endogenous city size is directly determined by the transport cost of workers, the housing price elasticity and the search efficiency of workers. It is also indirectly driven by unemployment benefits, the productivity of agents, the separation rate, the cost of a vacancy and labor market frictions. Third, the endogenous city size in the decentralized city is optimal if and only if the social planer does not take into account the firms' welfare and if he weighs equally the welfare of workers and landlords. This counterintuitive result is mathematically explained using Sandholm (2001, Theorem 5.4). Otherwise, the size of the city is not optimal which results in the need of a new policy on the housing price elasticity. More precisely, if the social planer is egalitarian (i.e. it weights equally the welfare of workers, absent landlords and firms), it determines a lower city size that the decentralized city. However, if it is not egalitarian, the efficiency of urban sprawl is conditioned by the importance of the workers' welfare and firms' welfare in the social criterion. Namely, if the worker's welfare (respectively firms' welfare) is important in the social criterion, the optimal city size is larger (respectively lower) than the city size in the decentralized economy.