Help with log-linearisation


I am trying to solve a model but have difficulty log-linearising this particular functional form (Armington aggregator). If anyone know,could you please help me. This is the equation:

p_t^m = chi * (a_t^m)^(-1/mu) * [chi*(a_t^m)^(1-1/mu) + (1-chi) * (a_t^x)^(1-1/mu) ]^(1/(1-1/mu) -1).

chi and mu are paramters. p_t^, a_t^m and a_t^x are endogenous variables.

Thank you so much,

Just do a first order Taylor approximation


and then expand the fractions

where x_hat is now percentage deviations from x_0. Taking the partial derivatives f’ is straightforward and follows standard algebra rules.