Thank you very much for the response.
My original goal was to build an RBC model with two sectors that differ in their exogenous default probability, which will enter the steady state and gives me a different capital-labor share. Since I wanted to figure out the technical issue behind it and which remedies to this problem are possible, I focused on a version of the RBC model without default or more complicated features, but only with convex adjustment costs and a sector specific adjustment cost parameter (which also enters the steady state). Once this is the case, the capital-labor ratios are different and I cannot find a unique interest rate that clears the market in the steady state. I tried to adjust them using a productivity parameter on capital (the “raw” capital stock can be used more efficiently in the production function, instead of K_t^G, a^G*K_t^G, then I have an additional parameter that I could adjust to match at least the “effective” capital- labor share). However, then by construction, the firm with higher adjustment costs or higher probability of default is always more efficient in using their “raw” capital stock in production, which I am not very happy with from the persepective of interpretation.
But even then, I have to make an initial condition on capital in one sector and I am also running into the unit root problem in Dynare.
I was looking at some papers on formal and informal sector, which also have two production sectors producing the same output. Still, all papers that I found assume either different production functions (f.ex. CRS and linear), or use different structural parameters (f.ex. different alphas in the Cobb-Douglas function).
Just a quick question about what you mentioned at the end with the CES aggregator. I was only using a Cobb-Douglas production function and aggregating only by summing up output, capital and leisure etc in the two sectors, f.ex. Y = Y^G + Y^B. Is there a better way to do it? I only know CES production functions, but never saw a case where one used a CES function to aggregate over the sectors? Or am I confusing something?
Again, many thanks for your help.