Computing optimal policy functions in a timeless perspective: An application

As it is now well known, in the framework of DSGE models taking into accounts agents expectations, the fully unconstrained optimal policy (the Ramsey policy) has the main drawback of being time inconsistent: the authority has an incentive to recompute the optimal policy in each period in order to take advantage of the fact that they are announcing the new policy after the private agents have devised their optimal decision of the previous period under expectation that the previous policy would continue. In order to mitigate the lack of credibility attached to a time inconsistent policy, authors like Svennson and Woodford have devised the concept of optimal policy in a timeless perspective where the autority constraints herself not to use her informational advantage in first period: she will compute her policy in first period as if the optimization had been computed many periods in the past. From a formal point of view, computing the unconstrained optimal policy can be done by forming a Lagrangian with the objective function of the authority and the dynamics of the economy as constraints. One obtains a recursive solution for the state variables and the Lagrange multipliers.In the unconstrained, time inconsistent, case, one must set hte initial value of the multiplier to 1. In a timeless perspective, on the contrary, one computes them as a function of the previous values in the system as in any other period. In a real application, the previous states of the economy and the previous shocks are given by history. However, the Lagange multipliers must also be computed by recurrence. An initial value of 0 for the multiplier is set far enough in the past so that this choice has no impact anymore on today's decisions. A further difficulty comes from the fact that usually not all the variables of a DSGE model are observed and the shocks almost never. It is therefore necessary to estimate the model without optimal policy on the historical period (including the monetary policy actually followed during that period) then use the Kalman filter that was already part of the estimation procedure to recover smoothed values of unobservable variables and structural shocks. In order to finally compute the value of the Lagrange multipliers for the historical period and so obtain starting values for the optimal policy in a timeless perspective. We propose to apply this procedure to a close economy model of the US economy and a small open economy version for Canada.