Before revision:  Unlike previous studies, our approach avoids making the assumption that the pricing kernel is atomless, and  we overcome one of the major difficulties encountered when one considers general economic equilibrium models in which the pricing kernel is a yet-to-be-determined unknown random variable  .
After revision:  Unlike previous studies, our approach avoids making the assumption that the pricing kernel is atomless, and we overcome one of the major difficulties encountered when one considers general economic equilibrium models in which the pricing kernel is a yet-to-be-determined unknown random variable . The method is applicable to many other models such as risk sharing model .