# Two-Sector Model with Adjustment Costs

Dear all,

I am currently working on a two-sector model with adjustment costs. The two sectors (G and B) own their capital stock Kg and Kb, have the same Cobb-Douglas technology and differ solely in their capital adjustment cost parameter (I use a simple quadratic cost function). When solving the model in Dynare, I happened to have a unit root, which was not desirable (I do not have any nominal variables in the model). Looking back to the equations, I am not able to identify capital and labor in each sector assuring at the same time the same wage and interest rate in the economy. In the steady state, the capital-labor shares in the two sectors are different (due to the different adjustment cost parameter). Since I would like to satisfy that both sectors produce the same good, adjusting this issue using a price of the good in one sector is not really a solution for me. I tried to to it with some productivity parameters on capital or labor, but I was still not successful. The aggregation strategy I was using is K = Kg+Kb (where Kg and Kb in each sector are not interchangable but remain in each sector) and L = 1 = Lg + Lb for the labor market. Did anyone ever had similar problems or knows some documentation on how to solve this issue?

Thank you very much in advance.

Best,
Anika

Dear Anika,
it is not easy to follow the problem. Why do your adjustment costs affect the steady state? Usually they are 0 in steady state. What determines the size of your sectors in the economy? Usually there is a CES aggregator ir similar to determine the relative shares.

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.

My hunch is that linearity is really the issue here. You might want to consult the open economy literature. There you often have a domestic and foreign intermediate goods sector that needs to be combined into a final good. You can then either assume the CES-function refers to the production function of a competitive final goods bundler or directly the preferences of the household.

Thank you very much. I tried your advice and implemented a final producer that produces the total output via a CES production function. It seems to me that I can identify the steady state capital and using the CES technology also the capital in “good” and “bad” sector. Still, the unit root does not disappear. I do not understand, what I am doing wrong. In case it makes my problem clearer, I uploaded my .mod file below.

RBC5_AC5_CES_FINPROD.mod (4.2 KB)

Again, many thanks in advance!

For some reason, the decomposition into sectors seems to be non-unique. What is strange is that `Y` does not have a unit root, while `C` and `I` do. This may give you an important clue.

Thank you very much! Unfortunately, I am still fighting with this issue, but I would like to ask, how do I find which variable the unit root belongs to? I checked some former conversations but did not find any clear hint.

The easiest is to look at the theoretical moments. If the second moment is `NaN`. you have a unit root. The `model_diagnostics` command provides more details.

Thank you very much for all your hints, I finally solved the issue!