Inflation in Equilibrium

Suppose we have the following market-clearing condition for the goods market within a New-Keynesian setting:

$$c_t = (1-\alpha)(1-\frac{\phi}{2}\pi_t^2)y_t + \alpha (1-\frac{\phi}{2}\pi_{t-1}^2)y_{t-1},$$

where $$\pi$$ denotes inflation. Can I consider $$\pi_t \equiv\pi_{t-1}$$ here in this equation? I know we do this in the case of steady-state. Is it the same for inflation here in the equilibrium too?

Moreover, as a general question, can we have GDP ($$y_t$$) from the previous period in the market-clearing condition as I have here? I came up with this equation since part of the labor income from the previous period is invested in period $$t$$.

• Is it a homework problem? What do you mean by "can I consider..." - do you want to assume that always..., or there exists a case such that...? Whether the equilibrium is a steady state or nonstationary depends on the model. There may be multiple equilibria. Depends on the model whether past GDP is in the market clearing condition or not. Mar 31 '21 at 6:58
• Thank you for the answer. No it's not homework. I mean "is there a goods market-clearing condition that includes the previous GDP?" In my model, there are two agents. One agent invests part of her wage at time $t$, and at time $t+1$, the profit is shared between two agents.That is why when I write down the equilibrium for total consumption, there is profit that actually comes from output of the previous period. Mar 31 '21 at 10:04