Multi-sector NK model with different growth rate

General DSGE Modeling
1

Dear users,

I am trying to working on a two-sector NK model with different growth rate, The model has a investment good which is a CES function comined with two sector’s goods:
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in (Carvalho et al. 2021) they assume the steady-state levels of productivities are the same across sectors,so detrending the Investment good can be done.My question is :if two sector’s productivities exhibt different growth rate, will the model have a steady-state?

I thank you in advance for your help.
Jason

2

My guess is that the answer is no. Depending on the elasticity of substitution, one sector would become increasingly bigger or smaller than the other one.

3

Thank you for your reply!