Shocks and steady state

I am a novice and I have reached a contradiction. please correct my mistake. Thanks in advance
Productivity comes in two forms in the models(As far as I have seen):

Of course, I know these two are one, but the models use one of these two forms

Now when we want to calibrate the steady state:

According to the first method:

Log(A)=rho_A*log(A) Therefore A=1

According to the second method: A=1

Now my question is about the other shocks. For example, shock to government spending in the RBC model(DSGE_mod/RBC_baseline.mod at master · JohannesPfeifer/DSGE_mod · GitHub)
If we use the following form:


g_ss=g and g=gshare*y;
That is, g_ss is not necessarily equal to 1. It depends on gshare
That is reasonable

But if we use the second form:
In the steady state:
Log(g)=rho_g*log(g) Therefore g=1

That is, in other equations, if we have g, we put it equal to 1(in steady state).
What is my mistake??

It should be
\log g_t=(1-\rho_g)\log g^*+\rho_g \log g_{t-1}+\varepsilon_t^G

Hi @michaelm, I’m also a novice :slightly_smiling_face:
As a cross-reference, You may also have a look at: