A Problem of Log-linearization

[lssp]

Dear friends,

I'm a new dynare user, and I have encountered a problem of log-linearization in a paper of DSGE model.

This problem is displayed in the attachment. Would you please have a look? I'd really appreciate that.

Thank you so much!

[jpfeifer]

This is a typical case of an equation that can not be expressed in percentage deviations from steady state because the steady state is 0. In that case, people just use a linearization instead of a log-linearization. Because the equation in the document is already linear, you can enter it in the way stated in the document.

[lssp]

Dear Professor,

Thanks for the reply. That's very helpful for me.

I have do the linearization according to you advice in the attachment. Is that right?

and I have finished a code in log-linearization form. It can give the result, but the result is not reasonable.

Could you please tell me what the probable problem may be ?

Sorry to bother you.

Thank you so much!

[jpfeifer]

Almost. To get from the first to the second line, you expanded the fraction by n^{H,h}/n^{H,h} and defined
(n^{H,h}_{t+1}-n^{H,h})/n^{H,h} as the hatted variable. But that means that a n^{H,h} as a prefactor in the second part is missing.

[lssp]

Almost. To get from the first to the second line, you expanded the fraction by n^{H,h}/n^{H,h} and defined
(n^{H,h}_{t+1}-n^{H,h})/n^{H,h} as the hatted variable. But that means that a n^{H,h} as a prefactor in the second part is missing.

Dear professor,

I'm sorry that I do not quite understand the last sentence in your reply.What do you mean by prefactor is missing?

and would you please tell me which one in the attachment is the right form?

Thank you so much!

[jpfeifer]

You can do both, but they have different interpretations. In the first case, \hat n is defined as n_t - \bar n, i.e. is the linear deviation from steady state. In the second case, \hat n is defined as (n_t - \bar n)/(\bar n) and is the percentage deviation from steady state. Assuming you log-linearized the rest of the model and \hat n was defined to be percentage deviations in the other equations, only the second one is correct.