In a microscopic model of fractional quantum Hall liquids with electron-electron interactions and confinement, we calculate the edge Green's function via exact diagonalization. Our results for υ=1/3 and 2/3 suggest that, in the presence of Coulomb interaction, "extelalal" parameters such as the sharpness of the edge and the strength of the edge confining potential, which can lead to edge reconstruction, may cause deviations from universality in the edge-tunneling I-V exponent. In particular, we do not find any direct dependence of this exponent on the range of the interaction potential as suggested by recent calculations in contradiction to the topological nature of the edge.