Poblem with auxiliary variable transformation

Hello everyone, I’m trying to replicate the paper Rannenberg(2016)“Bank Leverage Cycles and the External Finance Premium”.In the online appendix,he use an auxiliary variable

I don’t know why to do this. But indeed when I do not use this auxiliary, BK condition does not hold.After using this auxiliary variable,the model can be solved.I wanna know the reason to do this transformation. Is it correct? Thx!

@romero could maybe help.