I am wondering if I implement a Ramsey problem correctly: I follow Schmidt-Grohe and Uribe 2004 "Optimal fiscal and monetary policy under sticky prices" which can be downloaded from Schmidt-Grohes website http://www.columbia.edu/~ss3501/research/sticky-jet.pdf. As Appendix B shows the FOCs for t >= 1 and t=0 differ for example for \lambda_t: (B.9 and B.22):
B.9:
\lambda_t^b-\frac{\rho(\nu_{t-1})\lambda_{t-1}^b}{\pi_t}-\beta\frac{\lambda_t^p}{\lambda_t^2} E_t \lambda_{t+1}\pi_{t+1}(\pi_{t+1}-1)+\frac{\lambda_{t-1}^p}{\lambda_{t-1}}\pi_t(\pi_t-1)-\lambda_t^c\gamma(\nu_t) = 0
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lam_b-rho(-1)*lam_b(-1)/pi-BETA*lam_p/lam^2*lam(+1)*pi(+1)*(pi(+1)-1)+lam_p(-1)/lam(-1)*pi*(pi-1)-lam_c*gamma = 0;
B.22:
\lambda_t^b-\beta\frac{\lambda_t^p}{\lambda_t^2} E_t \lambda_{t+1}\pi_{t+1}(\pi_{t+1}-1)-\lambda_t^c\gamma(\nu_t) = 0
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lam_b-BETA*lam_p/lam^2*lam(+1)*pi(+1)*(pi(+1)-1)-lam_c*gamma = 0;
When implementing that in a deterministic setting I did the following:
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time_0*(lam_b-BETA*lam_p/lam^2*lam(+1)*pi(+1)*(pi(+1)-1)-lam_c*gamma)+(1-time_0)*(lam_b-rho(-1)*lam_b(-1)/pi-BETA*lam_p/lam^2*lam(+1)*pi(+1)*(pi(+1)-1)+lam_p(-1)/lam(-1)*pi*(pi-1)-lam_c*gamma)=0;
var time_0;
periods 1;
values 1;
I pass on the steady state and calculate the residuals of the start and end point. However, in that line above (with time_0) there is a mistake since plugging the starting values in the code leads to evaluating them at the time-0 restriction. If I then run a simulation without any further shocks the system converges to the steady state with transitional dynamics (since with the starting values there are positive residuals). How do I implement it correctly that the FOC are satisfied but if I start at the steady state without any shocks I stay there?
Best,
Peter