Calvo pricing for importing firms

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.

You need to expand on the notation used here.

Sorry for the lack of clarity and thanks for answering.

The first equation represents the firm’s FOC with respect to \bar P_{j,t}, which represents the optimal price for firm j, the rest is conventional notation, \theta is the Calvo parameter, \epsilon is the elasticity of substitution and \lambda_{t+s}/P_{t+s} is the real stochastic discount factor. In the other equations the meaning is the same, only that they represent the FOC of importing firms of consumption goods, which is reflected with m,c, for example \epsilon_{m,c} is the elasticity of substitution of imported consumption goods. While IM_{t+s} is the demand faced by the importing sector and S_{t+s}P^*_{t+s} refers to the nominal marginal cost.

Usually, it must be the consumption good price P_{t+s}. The reason is that the term in brackets
\left({\bar{P}}_{j,t}-\frac{\epsilon}{\epsilon-1}{MC}_{t+s}P_{t+s}\right)
is in nominal units and you use the nominal SDF
E_t\sum_{s=0}^{\infty}{(\beta)}^s\frac{\lambda_{t+s}}{\lambda_t}\frac{P_t}{P_{t+s}}
where you eliminated \frac{P_t}{\lambda_t} in the equation.