Hello everyone

I have a question about the Calvo pricing process. When we take the FOC, we get the following expression:

E_t\sum_{s=0}^{\infty}{(\beta\theta)}^s\left[{\lambda_{t+s}\left(\frac{{\bar{P}}_{j,t}}{P_{t+s}}\right)}^{-\epsilon}\frac{Y_{t+s}}{P_{t+s}}\ \left({\bar{P}}_{j,t}-\frac{\epsilon}{\epsilon-1}{MC}_{t+s}P_{t+s}\right)\right]=0

the term P_{t+s} that divides Y_{t+s}, is there because we use the stochastic discount factor in real terms, but in an open economy context with incomplete exchange rate pass-through, should the pricing process be like this:

E_t\sum_{s=0}^{\infty}{(\beta\theta)}^s\left[{\lambda_{t+s}\left(\frac{{\bar{P}}_{j,t}^{m,c}}{P_{t+s}^{m,c}}\right)}^{-\epsilon_{m,c}}\frac{IM_{t+s}}{P_{t+s}}\left({\bar{P}}_{j,t}^{m,c}-\frac{\epsilon_{m,c}}{\epsilon_{m,c}-1}S_{t+s}P_{t+s}^\ast\right)\right]=0

or like this?

E_t\sum_{s=0}^{\infty}{(\beta\theta)}^s\left[{\lambda_{t+s}\left(\frac{{\bar{P}}_{j,t}^{m,c}}{P_{t+s}^{m,c}}\right)}^{-\epsilon_{m,c}}\frac{IM_{t+s}}{P_{t+s}^{mc}}\left({\bar{P}}_{j,t}^{m,c}-\frac{\epsilon_{m,c}}{\epsilon_{m,c}-1}S_{t+s}P_{t+s}^\ast\right)\right]=0

I think it should be the first way, assuming all firms are owned by households, but I’ve seen some papers using {P_{t+s}^{mc}} instead of {P_{t+s}}.

Thanks in advance for your comments

Ana.