I want to construct a DSGE model incorporates two different producers, one producer with higher technology progress, the other with lower technology progress, that means for the first producer, its technology shock is , and for the other producer, its technology shock progress is as follows: . When in equilibrium, , and , is this construction right?

As I know, for this form of technology shock progress, its log-linearization is , that means for both producers, they have same value of technology progress in equilibrium, both of them are equal to 1. So how to realize this construction in DSGE model?

This bothers me for a long time, looking forward for your reply, thank you very much!

Indeed the linearized version looks very similar, but they are percentage deviations from different steady state values. The log-linearized version will only be identical of the \sigma_A^i are the same for i=\left\{h,l\right\}. Note also that often in linearizations, the steady state values may show up as parameters, i.e. the different levels may play a role in other equations (for example for relative outputs)