Failure in DSGE modeling including homework sector (Benhabib et al.,1991)r

During advanced macroeconomics course, I have been trying to represent the outcomes discussed in Benhabib et al.(1991), and matlab told me the steady state cannot be find. It will be appreciated if someone well-behaved in matlab operation can achieve original illustration.

The paper`s information are as follows.

Homework in Macroeconomics: Household Production and Aggregate Fluctuations
Jess Benhabib, Richard Rogerson, Randall Wright
Journal of Political Economy, Vol. 99, No. 6 (Dec., 1991), pp. 1166-1187

I doubt someone has a full replication lying around. Maybe you could share your try in the hope that people can improve upon it.

Thanks for focusing on my question! Here is my replicating code. code.pdf (20.5 KB)