I am trying to understand the Dynare output of Ramsay plan. I ran Nk_ramsey_osr.mod with the Optimal_policy switch to 1 and the Ramsey switch to 1. The Dynare reports three values of welfare
Approximated value of unconditional welfare: -51.15717757
with initial Lagrange multipliers set to 0: -51.07941558
initial Lagrange multipliers set to steady state: -51.10785031
I wanted to clarify how these alternative welfare values relate to whether Ramsey policy is time-consistent or not. Is the initial Lagrange multipliers set to steady state value refer to optimal monetary policy timeless perspective and the policy is time-consistent? Does the value with initial Lagrange multipliers set to 0 refer to the welfare level under the optimal plan which is not time-consistent ( as discussed in Khan, A., R. King and A. Woolman 2003, Optimal Monetary Policy, Review of Economic Studies, 70, 825-860)?
The Welfare ranking is: with initial Lagrange multipliers set to 0 > initial Lagrange multipliers set to steady state> Approximated value of unconditional welfare.
Does the above welfare ranking always holds?
Thanks a lot!
Setting the Lagrange multiplier to zero in the initial period means that the planner reoptimizes in the first period, i.e., does not honor any commitments made earlier. That is the source of time-inconsistency, namely that the planner can reoptimize in the current period, making promises for the future.
With the multiplier having been set to the steady state, the interpretation is that the planner has reoptimized once in the infinite past and now honors those commitments.
The difference between conditional and unconditional welfare is about whether you condition on the values of the state variables in the current period (conditional expectations E_t) or whether you consider the unconditional expectation E.
As far as I know, there is no universal ranking of the three alternatives reported.
Thanks a lot Dr. Pfeifer! Very helpful! Related question about the OSR. The Dynare reports Approximated value of unconditional welfare and Approximated value of conditional welfare for the OSR. The unconditional welfare value under OSR should be lower than the unconditional welfare value under Ramsey. But Ramsey reports two values of conditional welfare depending on the initial value of Langrangian multiplier. Should the conditional welfare value under OSR be lower than the conditional welfare value under Ramsey only in the case in which the initial Lagrange multipliers are set to steady state values or it should also be lower than the conditional welfare value under Ramsey with the initial Lagrange multipliers set to 0? Thanks agian!