Productive public investment + sticky wages

I am currently developing a model for the analysis of public consumption and investment shocks. I assume sticky prices and wages, productive public capital which directly enters the production function and allow for the ‘time-to-build’ effects and variable private capital utilization. While solving for the steady state I would also like to leave the possibility of non-zero inflation.
I’ve tried to make a very detailed write-up of the model set-up (attached):
writeup.pdf (270.0 KB)

The equilibrium conditions are listed on page #7. Much of what follows closely resembles the lecture notes of Eric Sims on the model with sticky wages (

The problem is in trying to explicitly solve for the steady state values. I want to analyze the case when public investment and consumption constitute a certain fraction of steady-state output. The problem is that in this case steady-state public capital k^g becomes a function of steady state output (see equation 138). Then I find myself unable to solve for the real wage in equilibrium (equation 160 and 161) because I can’t figure out how to get rid of the dependence on output.

The rest of the steady state calculations (starting from equation 162) were derived assuming for simplicity that k^g = \frac{1}{\delta^g}. Obviously, this is inconsistent with what I want.

Could you please provide me with any guidance on how to proceed?

This very much sounds like a case where you need a numerical solver to solve the “final” steady state equation.