Mean field games (MFGs) are dynamic games with infinitely many infinitesimal agents. In this context, we study the efficiency of Nash MFG equilibria; namely, we compare the social cost of an MFG equilibrium with the minimal cost a global planner can achieve. We find a structure condition on the game under which there exist efficient MFG equilibria and, in case this condition is not fulfilled, quantify how inefficient MFG equilibria are.