Is this steady state right? (Calvo pricing NK base)

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

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):


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!

It seems the Costa book is riddled with typos. Your steady state is the correct one. The steady state markup (price over marginal costs) is \frac{\psi}{\psi-1}. For \psi=6, it is 1.2.

The equation you mention has the slope of the New Keynesian Phillips curve in its place, which is clearly wrong.

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Thanks for your answer Dr. Pfeifer! Indeed, I better proceed to check carefully the algebraic outcomes when studying from this book. Again, your help is very appreciated!