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.

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.

[quote=“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.[/quote]

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?

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.