When writing a model in its non-linear form, one needs to make sure that the steady state is entirely consistent with the model, such that in each equation the left hand side=right hand side. i.e your residuals are zero. If that is not the case, then the model can’t find a steady state since the residual errors are too large.
My question is related to how one accounts for the above when one considers a positive steady state value of a shock variable, but which has a zero standard deviation. For instance, one can have government spending in a model, which is driven by an AR1 shock process. If that shock is zero, there is still a positive steady state government spending. How does one make sure that the nonlinear equations are consistent in this case?
For instance, assume I have a resource constraint given by:
exp(c)+exp(g)=exp(y)
where
g=rho*g(-1)+epsilon
Assume epsilon is 0. The equation then becomes
exp(c)+1=exp(y), but in steady state we still have
c*+g*=y* where x* is the steady state value of each variable.
I can multiply exp(g) by g*, but then does that mean that I would have to write the non logged steady state value for g?