In baseline New-Keynesian model with Calvo pricing, each intermediate good firm solves choosing P_{j,t|t}^\ast:
\tilde{\mathcal{L}}=E_t\sum_{k=0}^\infty(\beta\phi_p)^i\left[P_{j,t|t}^\ast Y_{j,t+k|t}-Y_{j,t+k|t}MC_{j,t+k|t}\right]
Where demand for j-firm’s intermediate good is given by: Y_{j,t+s|t}=\left(\frac{P_{t+s|t}}{P_{j,t|t}}\right)^\psi Y_{t+s|t} (note it’s assumed that update of sticky price is just the previous price with no inflation).
The FOC for \tilde{\mathcal{L}} is:
\frac{\partial \tilde{\mathcal{L}}}{\partial P_{j,t|t}^\ast} =E_t\sum_{k=0}^\infty(\beta\phi_p)^i\left[(1-\psi) Y_{j,t+k|t}+\psi \frac{Y_{j,t+k|t}}{P_{j,t|t}^\ast }MC_{j,t+k|t}\right]=0
Then if I take steady state I obtain:
\sum_{k=0}^\infty(\beta\phi_p)^i\left[(1-\psi) Y_{j}+\psi \frac{Y_{j}}{P_{j}^\ast }MC_{j}\right]=0
\frac{1}{(1-\beta\phi_p)}\left[(1-\psi)+\psi \frac{1}{P_{j}^\ast }MC_{j}\right]=0
P_j^\ast=\frac{\psi}{\psi-1}MC_j
Is this steady state right (with the deriving of the FOC)? Since in the book I’m studying from (Costa, 2016) it arrives to a different steady state for the same equation (page 75, equation 3.30):
P_j^\ast=\frac{\psi}{(\psi-1)(1-\beta\phi_p)}MC_j
I think it could be related to this topic, in which it seems there are some odd algebra steps in the book with the wage setting, since both price and wage stickiness follow similar algebraic procedures.
Many thanks!