Hello! I have two questions:

Would a costpush shock make sense when I’m not distinguishing between output gap (ytilde) and welfare relevant output gap (denoted by x) as in Gali’s notation? I understand costpush shock would be zero in this case according to the textbook. But I am still able to run this with a positive markup shock in Dynare, I cannot understand why would that be happening.

I have a twosector NK model, and I have derived a second order approximation to household’s utility function analytically. When I try to use it for optimal policy, I get the following error:
“You cannot include exogenous variables in the planner objective. Please define an auxiliary endogenous variable like eps_aux=epsilon and use it instead of the varexo.”
I am unable to understand why this would be happening (Please find my Dynare file attached, in case need be).optpolicy.mod (11.7 KB)
I would really appreciate some guidance on this. Thank you very much!