In this talk we will present results regarding the regularity and long time behavior of the one-dimensional, local, first-order mean field

games system and the planning problem. In particular, we will show the existence of classical solutions without the need to assume blow-up of the cost function near small densities. We also characterize the long time behavior of the solutions, proving that they satisfy the turnpike property with an exponential rate of convergence, and that they converge to the solution of the infinite horizon system. The approach relies on the elliptic structure of the system and displacement convexity estimates.