# Stochastic singularity in a simple model with variable capacity utilization **URL:** \ **Category:** Stochastic simulations\ **Created:** [18 December 2023 02:50 UTC](https://forum.dynare.org/t/stochastic-singularity-in-a-simple-model-with-variable-capacity-utilization/24774 "2023-12-18T02:50:09Z")\ **Posts on this page:** 6\ **Page:** 1
**Author:** ![mariors](https://forum.dynare.org/user_avatar/forum.dynare.org/mariors/32/18071_2.png) [@mariors](https://forum.dynare.org/u/mariors)\ **Post date:** [18 December 2023 02:50 UTC](https://forum.dynare.org/t/stochastic-singularity-in-a-simple-model-with-variable-capacity-utilization/24774/1 "2023-12-18T02:50:09Z")
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](https://forum.dynare.org/uploads/short-url/l8jW0ousyTjy8rsTT0ndS1yFLST.mod) (3.7 KB) [two\_sector\_RBC\_mobile.mod](https://forum.dynare.org/uploads/short-url/iENocRgSOl8ecqvuaWfjMDpqcRO.mod) (3.8 KB) ---
**Author:** ![mariors](https://forum.dynare.org/user_avatar/forum.dynare.org/mariors/32/18071_2.png) [@mariors](https://forum.dynare.org/u/mariors)\ **Post date:** [19 December 2023 07:49 UTC](https://forum.dynare.org/t/stochastic-singularity-in-a-simple-model-with-variable-capacity-utilization/24774/2 "2023-12-19T07:49:56Z")
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](https://forum.dynare.org/uploads/short-url/4FF716wdJdbTdifeI5p3To0TH2H.mod) (3.8 KB) ---
**Author:** ![jpfeifer](https://forum.dynare.org/user_avatar/forum.dynare.org/jpfeifer/32/5044_2.png) [@jpfeifer](https://forum.dynare.org/u/jpfeifer)\ **Post date:** [19 December 2023 11:57 UTC](https://forum.dynare.org/t/stochastic-singularity-in-a-simple-model-with-variable-capacity-utilization/24774/3 "2023-12-19T11:57:36Z")
Why do you have two prices in your model and they do not show up in the budget constraint? ---
**Author:** ![mariors](https://forum.dynare.org/user_avatar/forum.dynare.org/mariors/32/18071_2.png) [@mariors](https://forum.dynare.org/u/mariors)\ **Post date:** [20 December 2023 01:13 UTC](https://forum.dynare.org/t/stochastic-singularity-in-a-simple-model-with-variable-capacity-utilization/24774/4 "2023-12-20T01:13:11Z")
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: ```auto lam + p_C = -sigma/(1-iota)*(c_A-iota*c_A(-1)); ``` the production of the consumption good ```auto C = (p_C + Z + alpha*(K_C)+(1-alpha)*L_C) ``` and the capital demand for the firm (implicitly as part of C): ```auto 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](https://forum.dynare.org/uploads/short-url/4zXToWg85CF20fegCiUI3HzixRM.mod) (3.6 KB) ---
**Author:** ![mariors](https://forum.dynare.org/user_avatar/forum.dynare.org/mariors/32/18071_2.png) [@mariors](https://forum.dynare.org/u/mariors)\ **Post date:** [20 December 2023 09:12 UTC](https://forum.dynare.org/t/stochastic-singularity-in-a-simple-model-with-variable-capacity-utilization/24774/5 "2023-12-20T09:12:05Z")
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](https://forum.dynare.org/uploads/short-url/dKcEB00DE1zRClx6h7ZqrvwyrXV.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. ---
**Author:** ![jpfeifer](https://forum.dynare.org/user_avatar/forum.dynare.org/jpfeifer/32/5044_2.png) [@jpfeifer](https://forum.dynare.org/u/jpfeifer)\ **Post date:** [21 December 2023 16:41 UTC](https://forum.dynare.org/t/stochastic-singularity-in-a-simple-model-with-variable-capacity-utilization/24774/6 "2023-12-21T16:41:40Z")
What you describe makes sense.