A PRIMAL DUAL APPROACH TO COMPARATIVE DYNAMICS WITH TIME-DEPENDENT PARAMETERS IN VARIATIONAL CALCULUS

By changing the viewpoint of a variational problem so that the time-dependent parameters are the choice variables, this paper presents a dual way of establishing the effects that perturbations in such parameters have on the entire optimal trajectories of the original choice variables. The beauty of this approach is that it provides a qualitative characterization of the effects that perturbations in time-dependent parameters have on the entire optimal trajectories in a symmetric negative semidefinite matrix. Such a characterization is especially important in economics, since nearly all refutable implications in economics can be summarized by such matrices. Furthermore, a one-line proof of a generalized dynamic envelope theorem is achieved via this methodology.