Mean field games and optimal transport in urban modelling

The labour market is closely linked to the rental markets for professionals and for individuals. The purpose of this thesis is to study the interactions of these markets.In a first step, we develop and study a mean field game model that links the labour market with the rental market for professionals. In a specific framework where firms' production is assumed to have constant returns to scale, we show that the equilibria admit an explicit form that allows us to establish their existence and uniqueness. Several economic laws, including Pareto's law and Gibrat's law, hold true. Then, in a more general framework where we assume that the production has decreasing returns to scale, we establish several existence results and find the golden rule of capital accumulation. Finally, we present a numerical method to approximate the equilibria. We detail several simulations and study the influence of some parameters on the computed equilibrium by doing comparative statics.In a second step, we focus on a model linking the labour market with the rental market for individuals. It admits a spatial component and allows us to determine the distribution of residences, wages and rents. These three outputs verify three equilibrium conditions: the labour market condition, the housing market condition, and a mobility condition. The labour market condition is related to an optimal transport problem, while the other two are related to a static non-atomic game. The existence and uniqueness results we establish, exploit the fact that, at equilibrium, the distribution of residences admits an explicit form. Then, several extensions are considered such as the adaptation of the model to telecommuting. We conclude with the presentation of a numerical method developed to approach equilibria and the study, by doing comparative statics, of the influence of some parameters of the model on the calculated equilibrium.