Molecular simulations are carried out to determine the elastic stability limits of polyethylene and n-alkane crystals under external stresses. For polyethylene, the instability that occurs at the lowest critical stress corresponds to planes of polymer chains perpendicular to the a-axis being shifted along the c-axis (which is the chain axis); this simulation result concurs with the previous experimental result that (100)[001] crystallographic slip is the dominant plastic deformation mechanism in polyethylene crystals. For n-alkane crystals, the stability of the crystal with respect to isotropic tensile stress is greater for n-alkanes with even numbers of carbon atoms, due to more favorable packing at the chain ends. This odd-even effect in the elastic stability is related to the well-known odd-even effect in the melting temperatures of n-alkane crystals.