# Numerical method for solving deterministic model

Hi, I am currently working on my Master thesis on climate policy analysis and I am trying to solve a dynamic deterministic general equilibrium model. This model is a classical Cass Koopman Ramsey model with a geometric growth of TFP and in which I added a durable goods producers sector. The representative household can choose either to consume this durable (in a CES) or to rent it to firms. (If needed I can give more details about the formal model equations.) I would like to ask, how dynare would numerically solve this ?

I think that it needs a starting condition for stocks variables and a terminal condition, which would be the steady state of my optimality equations system, and then it finds the root (Newton algorithm) of a systeme of N equations with N unknowns.

If this is the case and if my understanding is correct, I first need to rewrite my system as a function of deflated variables only which have a steady state. But which variables should I deflate ? I read in a some papers that in such model, labor is constant at steady state, but are there other variables behaving like this ?

Many thanks in advance for your help which would greatly help me in my research.

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1. Yes, your understanding is correct.
2. Yes, the most straightforward approach is to detrend the model. You could proceed without that, but you would still need to know the growth rates of the variables.

Thanks a lot for your answer, that makes me much more confident. I could detrend my model assuming a steady would exist but I assumed that at this steady state the prices of my 2 goods and labor supply/demand are constant. Capital, consumption and output are constant once they are deflated by labor productivity. I think it is a classic hypothesis am I still right ?

Yes, that is correct. It is well-known that there are particular preferences that are consistent with a BGP. They ascertain that labor supply is constant. Typically, the only tricky part in detrending is the Lagrange multiplier on the budget constraint. See

Thanks a lot !

I still have two last questions : 1) one about the Lagrange multiplier and 2) another about steady states in bigger models with both exogenous productivity growth and learning by doing elements.

1. I could rewrite the household’s program as a function of stationnary variables. So, my budget constraint is slightly modified, a (1 + g) appears in front of the capital (from last period) returns. Is it still the case that the lagrange multiplier has a trend ? Since my budget constraint is divided by the productivity now, it does not have the same interpretation as before right ? Here is my budget constraint in level :

2. My ultimate goal is to build a simplification of the determinist E-QUEST model from the European Commission : E-QUEST - A Multi-Region Sectoral Dynamic General Equilibrium Model with Energy: Model Description and Applications to Reach the EU Climate Targets

Do you think that perfect foresight solver is adapted to bigger model like this one with, say, 10 sectors and 3 geographic areas ?

-I am thinking in particular about the learning-by-doing elements in the model. What if in each sectors there is both an exogenous growth of labor efficiency and a learning by doing factor ?

-The factor added to account for this, is the aggregate capital. This is an externality, it is taken as given by firms.
-So I understand that, if I add such elements, I also need to deflate this aggregate capital by the exogenous productivity to stationnarize the model, right ?

Again thank you for your help