As the title suggests I am trying to solve for the steady state of the New Keynesian model with Calvo staggering (with the past inflation indexation). Firms consist of entrepreneurs (capital decisions) and intermediate good producers.

The model is standard otherwise, but features housing, which enters both utility functions of agents and production function, s.t. output of intermediate goods producers is determined with CRS function Y = A*F(K, N, H_e), where H_e is the housing stock purchased by the entrepreneurs (there are also patient and impatient HHs that demand housing). The housing supply is normalized to one.

I really hope the setup is clear.

The trick is with the steady state: since there are 3 factors of production, I do not know how to proceed. I have studied the paper by FernÃ¡ndez-Villaverde and Rubio-RamÃrez (2006), though their model features only two factors of production and finding wage, for instance, is pretty straightforward.

I was wondering, maybe I can proceed with the normalization? For example, apart from normalizing housing supply, I can say that aggregate output is 1 and/or labor supply (sum of individual labor supplies over agents) is 1. Is it the case? I have been trying to look for the literature on normalization, but found nothing. Hence, any help would be appreciated.

Or should I even look for the steady state given that I can calibrate the model and hence pin down st. state ratios?

I am not sure I understand the problem. If housing supply is fixed to 1, the production function only has two inputs you need to find as in the baseline case.

Housing market clearing: H_i + H_p + H_e = 1, where H_j corresponds to the housing demand of an agent j

Firms use housing in production: Y(i) = A*F(K, N, H_e)

As a result, when solving for firms in this setup (with Calvo) I have equations:

ratio of wage to return on capital,

ratio of return on housing to return on capital

real marginal cost and

production function.
and price evolution and price dispersion processes.

The problem with the second equation. I was able to find return on capital analytically, MC and optimal price and price dispersion, but still the number of variables exceeds the number of equations, even if I solve for ratios over labor.

I was thinking about any other normalization I can apply, but I was not able to find literature on rules of normalization and/or guessing values of endogenous variables.
Hence, any piece of advice from you would be highly appreciated.

That should always be the case and is straightforward.

That cannot be the case. There must be enough equations to determine the steady state. The tricky part is that with constant returns to scale, the return to capital from the Euler equation will be a function of the capital-to-labor and the housing-to-labor ratios. The housing FOC should give you a similar constraint. Pairing that with the FOC and the market clearing conditions should allow solving for labor and housing separately. However, that may require a solver as in the NK_baseline model.