This paper shows that the mathematical structure of the most widely used New Economic Geography models is identical, irrespective of the underlying agglomeration mechanism assumed (factor migration, input-output linkages, endogenous capital accumulation). This enables us to provide analytical proofs to four important and related results in the field. First, standard models display at most two interior steady states beyond the symmetric one. Second, when interior, asymmetric steady-states exist they are unstable. Third, location displays hysteresis. Finally, with forward looking agents a shock to expectations might trigger an equilibrium switch. This paper also stresses the empirical implications of the most important results derived in this study.