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

**URL:** <https://forum.dynare.org/t/is-this-steady-state-right-calvo-pricing-nk-base/17749>\
**Category:** General DSGE Modeling\
**Created:** [29 March 2021 04:31 UTC](https://forum.dynare.org/t/is-this-steady-state-right-calvo-pricing-nk-base/17749 "2021-03-29T04:31:22Z")\
**Posts on this page:** 3\
**Page:** 1

<div class="post-metadata">

**Author:** ![dsge\_modeling](https://forum.dynare.org/letter_avatar_proxy/v4/letter/d/9de053/32.png) [@dsge\_modeling](https://forum.dynare.org/u/dsge_modeling)\
**Post date:** [29 March 2021 04:31 UTC](https://forum.dynare.org/t/is-this-steady-state-right-calvo-pricing-nk-base/17749/1 "2021-03-29T04:31:22Z")

</div>

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](https://books.google.com.co/books?id=rPIgvgAACAAJ&printsec=frontcover&redir_esc=y)) 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](https://forum.dynare.org/t/calvo-wage-stickiness-nk-model-optimal-wage-derivation/17675) 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!

---

<div class="post-metadata">

**Author:** ![jpfeifer](https://forum.dynare.org/user_avatar/forum.dynare.org/jpfeifer/32/5044_2.png) [@jpfeifer](https://forum.dynare.org/u/jpfeifer)\
**Post date:** [29 March 2021 08:13 UTC](https://forum.dynare.org/t/is-this-steady-state-right-calvo-pricing-nk-base/17749/2 "2021-03-29T08:13:25Z")

</div>

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.

---

<div class="post-metadata">

**Author:** ![dsge\_modeling](https://forum.dynare.org/letter_avatar_proxy/v4/letter/d/9de053/32.png) [@dsge\_modeling](https://forum.dynare.org/u/dsge_modeling)\
**Post date:** [30 March 2021 01:47 UTC](https://forum.dynare.org/t/is-this-steady-state-right-calvo-pricing-nk-base/17749/3 "2021-03-30T01:47:30Z")

</div>

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!