I am currently working on a growth model where I have three growth trends, namely: demographic growth (N(t)), labor-augmenting technical change (lL(t)) and energy-saving technical change (Ae(t)).

I am trying to stationnarise my model. For most of the aggregates this is straighforward: I detrend by the efficient labor N(t).Al(t), meaning that the growth rate of the variables is (gn.gal). However, what to do with a variable that has 3 trend growth? Indeed, energy is impacted by N, Al and Ae. Is it (gn.gal)/gae?

That one is tricky, therefore the long pause. If I remember correctly, then Uzawa’s theorem tells us that only (asymptotically) labor augmenting technology (or something that can be represented as such) results in a balanced growth path. That would indicate that even if you try to detrend the model appropriately, it will not be stationary in the usual sense.

Thank you for your answer. Indeed, it is tricky and I am aware of Uzawa’s theorem. Does that mean that Henriet et al. (2014) balanced growth path is not really one, since balanced growth is not compatible with every configuration for technological progress in a neoclassical framework ? How do they make simulations on Dynare if the BGP is not the usual one ? Do you have any hint?

Jones (2008) says that “if a neoclassical growth model exhibits SS growth, then technical change must be labor-augmenting, at least in SS”.

I was thinking by only assuming labor-augmenting technology at SS, my energy-augmenting technology be zero at SS (but not during the transitory period). Can I do that?

Yes, that would be feasible. But then you would only detrend your model by labor augmenting technology and keep the rest. The remaining question is whether you are considering perfect foresight or stochastic simulations. The former should be able to handle this. For the latter, it might not be possible.

Yes that is what I have in mind, approaching Ae(t) as a trend deterministic growth (such as demographic growth) rather than a technical progress. However, I was thinking in doing stochastic simulations… If not possible, I will only do deterministic ones.

Why is it not possible with stochastic simulations? I thought shocks are 0 at SS with stochastic simulations.

A deterministic trend is by construction not subject to stochastic shocks. The problem is that stochastic simulations (with the exception of extended path) rely on a smooth approximation around the steady state. But in steady state your trend is no more existing. That may introduce problems.

What if I approach Ae(t) as not being a trend but rather a stochastic shock; being zero at SS (hence “no trend of Ae(t) at SS”) and other during certain periods (using the shock and stoch_simul command) ?

Ok ok. What I retain is that whatever type of simulation I use (deterministic vs stochastic) I only detrend the model in order to have variables per efficiency labor units.

I will try the stochastic one, and see what happens. I have some stochastic shocks in the model as well.

Oh one last thing: what you mean is that one cannot add multiple stochastic growth trends, only one (that would be the labor-augmenting technical progress)?

I was not talking about what you can have in the original model. I was only saying that the detrending for entering the model with a well-defined steady state into Dynare can only be done for labor-augmenting technology. For the other growth trends, no well-defined steady state would exist.

In the NK_baseline.mod, we have a Cobb-Douglas-production function which allows representing both growth trends as a compound labor augmenting technology growth process. The second paper features neutrality of money in steady state. So the money growth trend does not affect real variables and Uzawa’s theorem does not apply.

So in your mod.file we truly have only one growth trend (combining labor and investment growth) then?
Can I do the same with my model if I suppose the following cobb-douglas function exists?

Yes right. I still don’t get how those authors managed to do the thing I am trying to do with Dynare, or their model is just not “stationary” in the usual sense.

You can verify in section 2.5 of that paper that you need particular restrictions on the growth trends to have a BGP, i.e. you cannot have arbitrary different trends. If you accept those restrictions, then everything is fine.

Ok thank you. That is what I wanted to do from the beginning. What do you mean that “you cannot have arbitrary different trends”? I thought the paper was incorporating 2 trends (labor-augmenting and energy-augmenting). Or are you referring to the fact that growth trends have to be constant? Or to the fact that prices and energy taxes have to grow to the energy-augmenting growth rate?