This paper presents a new version of the factor-price equalization theorem. The numbers of outputs and of factor inputs are allowed to be unmatched. The domestic factor prices in a country facing the international commodity prices are uniquely determined in the neoclassical framework once the factor endowment is taken as given. This allows the factor-price equalization proposition to maintain the invariance of the derivatives of the social production possibility schedule with respect to input parameter perturbations, considering the possible repercussions in the outputs. A necessary and sufficient condition for this invariance is presented in terms of second derivatives of the social production possibility frontier with its economic interpretation. This condition is applied to the non-joint production case. The paper asserts that the equalization theorem holds true if the constant-returns to scale production functions are strictly concave except along rays and satisfy the full-rank condition on the input-coefficient matrix, and if the number of commodities is no less than that of factors. Several other varieties are also presented including the joint production case.