How to handle large additive equations when steady state is difficult to compute


I am trying to replicate a paper that estimates a DSGE model about renewable energy policies. The model contains four types of firms, one of those produces energy from a non-renewable fossil fuels stock. See the Lagrangian of the firm in the image at the end.

Where the equation multiplied with lambda is a constraint, that says that future resources are the depreciated past resources less the energy production. The FOCs are also depicted in the image at the end. The constraint is omitted there.

Eliminating lamda allows for the first two FOCs to be combined neatly, but when eliminating lambda in the third equation, this yields a long additive equation with expectations.

I tried two things:

a) Put the model in levels into dynare and trying to solve for the steady state. However, quickly it dawned on me, that if I try to get the steady state of S, the constraint yields an imaginary number (see last part of image below).

b) I was advised to log-linearize the model, such that all steady state variables are zero, but then for such additive equations, still many orignial steady state values are required. The next advice was then to calibrate these steady state values instead of computing the steady state.

My first question is, given the steady state equation for the stock of non-renewable resources, can the model still work while containing such an equation? Maybe with a restriction that the stock may only drop to zero?

If it can, how do I implement this in dynare? Do I take additional steps such that the steady state becomes computable or do I indeed calibrate the steady state values?

Any help would be greatly appreciated.

Please step back for a moment and recheck your computations. An imaginary object does not make sense. My hunch is that there is a sign error that needs to be resolved first.

Dear Professor Pfeifer,

Thank you for your reply. The equations above are, with exception of the last one, equations provided from the Appendix of the paper I am replicating and not my own computations. I may be able to state my question more precisely.

Again, the stock of non-renewable resources evolves according to the following equation:

S_t = (1-\delta^s) S_{t-1} - A_t^{ef}(N_t^{ef})^{\theta}(K_{t-1}^{ef})^\nu(S_{t-1})^{1-\theta-\nu}

If this equation needs to be included as a model equation (which I am quite sure it needs to be), would the implied steady state of S make it impossible to compute the model?

If the model is not computable, as it is written down here, do you know of any Dynare codes that involve firms using a non-renewable resource as input factors?

That equation only has on trivial steady state at 0 from what I can see. Starting from a given value, S can only decrease. So this does not strike me as a model that can be locally approximated.