# Stochastic singularity in a simple model with variable capacity utilization

Greetings:

I am examining a simple 2-sector RBC model with variable capital utilization. Here households choose the total capital stock subject to a second-order investment adjustment cost, but can then freely divide it between sectors. Thus, sector-specific capital K_C and K_I are choice variables, even though the total capital stock is predetermined.

I am obtaining indeterminacy for this specification, even though a specification without variable capital utilization does not produce this issue (first attachment below).

As best as I am aware, variable capital utilization by itself should not induce indeterminacy per se, so I believe there is some error. Yet, I have not been able to pinpoint it. Any leads would be extremely appreciated!

two_sector_RBC_simp.mod (3.7 KB)
two_sector_RBC_mobile.mod (3.8 KB)

Hi:
Greetings:

So, it does not seem that variable capital utilization is relevant. Here is a stripped down 2-sector RBC model–no variable capital utilization or investment adjustment costs–just allowing the rental rate of capital to be differentiated between the sectors (together with factor mobility) seems to generate indeterminacy. And, yet, I thought this was a well-established baseline that should be determinate?

Best regards.

two_sector_RBC_diff_R.mod (3.8 KB)

Why do you have two prices in your model and they do not show up in the budget constraint?

Hello and thanks for the response!

The price P_C refers to the price of a consumption basket, whereas p_C is that of an individual consumption good. However, the monopolistic competition is a distraction here since in this simplification households consume all varieties; the absence of variety effects mean that P_C and p_C move one-for-one.

So, really we obtain a two-sector RBC baseline with prices p_C in consumption and p_I in investment. We normalize the investment good price to one, so that in log deviations p_I = 0. Moreover, we let C denote consumption expenditures, so that C = p_C + c_A, where c_A is the consumption good.

The price p_C also shows up in the first order condition for consumption:

``````lam + p_C = -sigma/(1-iota)*(c_A-iota*c_A(-1));
``````

the production of the consumption good

``````C = (p_C + Z + alpha*(K_C)+(1-alpha)*L_C)
``````

and the capital demand for the firm (implicitly as part of C):

``````r_C + K_C = C
``````

I have also rewritten the mod file without using consumption expenditures C and dropping the redundant P_C, in case it helps. (And this does not affect the indeterminacy result).
two_sector_RBC_diff_R.mod (3.6 KB)

Hi. I think I figured out the issue. With capital mobile between the sectors, it does not make sense for the rental rate to vary. If it did, then households would allocate all capital to that sector. So, for capital to be allocated to both sectors, then the rental rate would have to be the same.

Accordingly, for the rental rate to vary and to maintain a single investment decision, it must be the case that households first investment, then split the capital between consumption and investment sectors at the end of the period. Thus, the capital in each sector is predetermined at the time the aggregate shock hits and firms make their production decisions. The rental rate then adjusts in each sector to equate capital demand with the (fixed) level of capital.

I adjusted the timing in the adjusted mod file, and the Blanchard-Kahn conditions hold.
two_sector_RBC_diff_R.mod (3.6 KB)

So, this seems to be the timing assumption in “Demand Shocks as Technology Shocks” by Bai, Rios-Rull, and Storesletten.

This is my tentative conclusion, but any additional input/context is welcome.

What you describe makes sense.