Calvo pricing for importing firms

General DSGE Modeling
1

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.

2

You need to expand on the notation used here.

3

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.

4

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.

5

Thanks a lot for your answer.

Regarding your comment about the price of the consumption good, imagine that I have a budget constraint as in Adolfson et al (2007) of the form:

\lambda_{t} / P_{t}(P^{c}_{t}C_t+P^{I}_{t}I_{t}+......)

with lambda as the nominal lagrange multiplier, P^c_t as the price of the consumption good, P^i_t as the price of the investment good and where P_t is the price of domestically produced goods. Adolfson et al define the real multiplier as v_t =\lambda_{t}/P_{t}, and therefore, they define v_{t,t+s} as the real stochastic discount factor,which implies that in the formulation of the calvo equation we would use the price P_{t} (i.e. the price of the domestically produced final goods) instead of just the price of the consumption good, but as you point out, in some papers the price of the consumption good is used. My guess is that in a model with three different prices, as in Adolfson et al, that is, with P_t, P^{c}_t and P^{i}_t, we should use P_t, instead of P^{c}_t, am I right?

Thanks for your help, I really appreciate it.

6

I am not sure I understand. In
\lambda_{t} / P_{t}(P^{c}_{t}C_t+P^{I}_{t}I_{t}+......)
we have the part
(P^{c}_{t}C_t+P^{I}_{t}I_{t}+......)
which is in terms of the final good or currency with price P_t. The division by P_t transforms it into units of the final good. \lambda_t is then real Lagrange multiplier for valuing the final good. In your Euler equation you would have \lambda\frac{P_t^c}{P_t} showing up, which makes the transformation explicit.

8

  1. I’m a bit confused with that part. Going back to your comment above, you said that \beta^s\frac{\lambda_{t+s}}{\lambda_t}\frac{P_t}{P_{t+s}} is the nominal discount factor, but I had understood that this was the real stochastic discount factor. I have always had this confusion about when the SDF is in real or nominal terms.

  2. But regardless of the previous point, if I have the constraint \lambda_t/P_t(P^c_tC_t + P^i_tI_t+....) and therefore, as you point out, in my euler equation I have \lambda_tP^c_t/P_t , should the stochastic discount factor for the calvo pricing process be \beta^s\frac{\lambda_{t+s}}{\lambda_t}\frac{P_t}{P_{t+s}} or \beta^s\frac{\lambda_{t+s}}{\lambda_t}\frac{P^c_t}{P^c_{t+s}} ? As defined by Adolfson et al, I think it should be the first option, but I’m not entirely sure.

Greetings and thanks for your patience

9
  1. The SDF is itself is neither real or nominal. The terminology only indicates whether it is applied to real or nominal cash flows. As such, there are different nominal or real SDFs depending on the real goods are nominal units of the cash flow.
  2. Again, it depends on how you set up the cash flow in the Calvo problem.