Calvo indexation past inflation

General DSGE Modeling
1

Hello all,

I am working with a classic NKM (and labor market frictions) with price adjustment à la Calvo, where only a fraction of firms are allowed to reoptimize prices, while the others are not.

Now I would like to modify the model so that firms which instead are not allowed to optimize, imply index their unchanged prices to the previous period’s inflation rate.
Now I am stuck in that I do not know how to proceed and how to modify the 5 Calvo equations that I have in the model.

Are you aware of Papers that derive such a model, or dynare codes that I can consult?

I am very grateful for any help

2

Woodford (Interest and Prices, 2003, p. 213) does this in detail. In essence, you end up with the same NKPC but replace \pi_{t} with \pi_{t}-\gamma \pi_{t-1} (and similarly for \pi_{t+1}), where \gamma is the degree of indexation.

If you also have sticky wages, the same logic applies, where you replace \pi^{w}_{t} with its quasi-differenced counterpart, as above.

3

Dear Brasidas,

Thank you, this has already helped me a lot. I still have some problems to solve though. These are my equations in the “normal” Calvo case:

p_star=(epselon/(epselon-1))*(x1/x2);
x1=Co^(-sigma)*mc*Y+beta*teta*(pai(+1))^(epselon)*x1(+1);
x2=Co^(-sigma)*Y+beta*teta*(pai(+1))^(epselon-1)*x2(+1);
si=(1-teta)*p_star^(-epselon)+teta*pai^(epselon)*si(-1);
1^(1/(1-epselon))=teta*pai^(epselon-1)+(1-teta)*p_star^(1-epselon);

Did I understand correctly that I have to replace the variable
pai with [pai-gamma*pai(-1)]
and
pai(+1) with [pai(+1)-gamma]

Thank you for your help

4

Dear pb11,

At first glance, that looks right. The only thing I’d add is that you need to replace \pi_{t+1} with \pi_{t+1}-\gamma \pi_{t}. As I presume you have all the variables in percentage (or percentage point) deviation from steady state, there should be no constants, such as \gamma.