A simple neoclassical model

[jpfeifer]

Sorry, but I am not well-versed in that branch of the literature. But usually the utilization rate is not a choice in the input market as capital and labor chosen.

[macroresearch123]

Dear Professor Pfeifer,
First of all, I apologise that this question is not directly related to Dynare use, rather theoretical. I hope you won’t mind with that as you help us in so many directions.
As I can estimate the model now for which I received your help earlier (in previous posts), I would like to add Gertler and Karadi (2009)’s financial friction model into this. Just to remind you my earlier model : It has household sector (HH) ( with nominal wage rigidity and it produces capital and decides capital utilization also), Final production sector, Monopolistically competitive intermediate goods sector (nominal price rigidity) and government & the central bank.

Now I would like to separate the capital production from HH first of all. And I want the intermediate goods sector to decide capital utilization rate. While dealing with the intermediate sector, my understanding/the way I am proceeding is,
I explicitly use per unit capital value (Q) and the total cost for capital hiring is , gross interest rate*Q K. This interest rate comes from the bank’s lending rate.
So while minimizing total cost in input markets, the firm’s control variables are Labour (N), capital (K) as well as utilization rate (U)?

The firm's production function is, Yt = At (u e k)^alpha L^(1-alpha), where u is capital utilization rate, e is capital quality shock and K is capital. Then , intermediate firm’s profit function contains two control variables: firm j’s output price (P(j)) and U. Here in profit function, the capital utilization comes through the production function . This implies, intermediate firm (j) is deciding capital utilization twice : once in cost minimization problem in factor market and second in profit maximization in output market?
Is it correct/consistent?

Regards

Do you have a pdf of the equations? It sounds like this wouldn't make sense to have them optimizing over capacity utilization twice. Intermediate firms choose them in your model, right? That should give you the price of utilization and the household side should give a supply condition. You might want to lay out the context of the model -- why you're adding these mechanisms -- in the future so it's easier to follow the question

[sadia]

Thank you macroresearch123 and Professor Pfeifer. Yes, capital utilization rate is chosen by intermediate goods sector. Can I kindly refer you to the equations of Gertler and Karadi (2009)'s, henceforth GK, financial friction model? It is slightly different than my context, in the sense that, in GK, the intermediate goods sector is perfectly competitive in output market. So, in the profit maximization problem, intermediate firm chooses L, u and K. In this case, how do I explain the FOC with respect to u? Price of capital utilization ? Depreciation rate is a function of u. Final goods price is determined in monopolistically competitive retail sector. Capital price Q (Tobin's Q) is determined in a separate capital producing sector, not in the household sector.

If I say instead that, intermediate sector is monopolistically competitive, then it minimizes costs in input markets and maximizes profit in output market by choosing its output price from the profit function. So there are two optimization problems in the intermediate sector. Now where does the u fit in, if I assume similar to GK (2009) that intermediate sector chooses capital utilization? And capital is produced by capital producing sector.

May be its a very simple issue that I got confused with due to my naive understanding. Sorry about this. Thanks in advance for your response.