Oil-in-water microemulsions of the zwitterionic surfactant tetradecyldimethylamine oxide (TDMAO) that are formed in the presence of various alcohols as cosurfactant have been investigated by means of small-angle neutron scattering and interfacial tension measurements. From these experiments the sum of the bending moduli 2 kappa + <(kappa)over bar> has been determined and found to be in the range of 2-5 kT. It is found that the admixture of the alcohol leads to relatively constant values for longer chain alcohols whereas short chain alcohols induce a significant reduction of 2 kappa + <(kappa)over bar> that is proportional to the concentration of added alcohol. This behavior of the bending elasticity corresponds to the phase behavior in the ternary systems (TDMAO/alcohol/water) where for the systems with high values for the bending elasticity formation of lamellar phases is observed, whereas for systems of low rigidity bicontinuous isotropic phases are formed with structures that are characterized by a low persistence length.