Dear Professor Pfeifer,
I read this post and other discussions on this topic but still there is something that is not clear to me. I would like to do a very simple exercise, that is to find the parameter of a policy rule that maximizes conditional expected welfare, defined as Welfare = U + \beta*Welfare(+1) where the condition is being in the deterministic steady state in the first period. Moreover, the parameter has no impact on the deterministic steady-state. I am using a 2nd order approximation and I am wondering how to find in Dynare the welfare that I have to maximize.
Maximizing the welfare ergodic mean (oo_.mean) is not correct because, as you say, this gives me the unconditional expected welfare. I think the right approach is to maximize the stochastic steady-state of welfare, where the stochastic steady state is defined as the point where the agents decide to stay in the absence of shocks, but taking into account the likelihood of future shocks: is this the method used in Schmitt-Grohe and Uribe (2004) “Optimal simple and implementable monetary and fiscal rules” and in many other papers?
If yes, is it correct to compute the stochastic steady state by i) using simult_ with a vector of zeros as the shock input starting from the deterministic steady state and ii) taking the last value of the output of simul_t ?
convergence_stoch_ss=simult_(oo_.steady_state,oo_.dr,zeros(T,n_shocks),order);
stoch_ss =convergence_stoch_ss(:,T);
Thank you very much, best wishes,
Valerio