Consumption Based Asset Pricing

I am working on some consumption based asset pricing models. I am modelling consumption growth in several different ways. An obvious one is to model consumption growth as an AR(1) process:

$g_{t+1} = \phi_0 + \phi_1 g_t +\epsilon_{t+1 }$

Where $g_{t+1}$ is consumption growth. What are the implications of having $\phi_1=0$? What about $\phi_1=1$?

In which case is consumption growth iid?

If $\phi_1=0$ then consumption growth is iid. If If $\phi_1=1$ then consumption growth is has a unit root and is not stationary, and so will be the model.