The second- and third-order elastic constants of Al and Pb are calculated as the second and third derivatives of the binding energy with respect to the finite deformation parameter. The binding energy is derived from a local pseudopotential by use of second-order perturbation theory. It is shown that the binding energy satisfied not only the diagonal equilibrium condition, but also the off-diagonal equilibrium condition, i.e., the first derivative of the binding energy with respect to a volume change as well as with respect to shear deformations is zero. Accordingly, the present method of calculation is based on a stable lattice model. The results of the present calculation of the third-order elastic constants of Al are found to be in qualitative agreement with the experimental data obtained by Thomas. The complete experimental set of the third-order elastic constants of Pb is not yet available; however, the calculated pressure derivatives of the second-order elastic constants are in agreement with the experimental data of Miller and Schuele. On the other hand, the method initiated by Leigh, which is based on a rigid-band model, cannot reproduce the experimentally observed third-order elastic constants of Al. This is an indication that the pseudopotential method is to be preferred to the rigid-band method as far as the calculation of second- and third-order elastic constants is concerned.