In the Smet and Wouters(2008) and other DSGE model for a business cycle, my understanding is that factor share parameter does not affect the response of output and inflation to any shocks because they rely on a Cobb-Douglas production function. Am I right?
before 2000, the labor share is stable, so usual DSGE model calibrate factor share parameters using a stable long-run average of the labor and capital share. However, as the labor share has been decreasing, if we change this parameter to a lower value, I think steady state would change, but it could not affect the effects of technology and demand shocks. Since business cycles are interpreted as deviation from steady state, change in steady state does not matter for business cycle. is my understanding right?
No, generally the approximation point will affect the slope of the policy functions as well. It would not expect it to matter too much, but there should be an effect.
Thank you for your response. Actually, I want to incorporate a IST shock into a DSGE model with a CES production function. However, since my understanding is that only labor argumeting progress can deliver a balance growth path under a CES function, I have to use a C-D function in order to consider non-stationary IST shock. am I right?
Not really. You will run into trouble if you try to do stochastic simulations over long periods of time with shocks constantly hitting. In that case, you would be continuously moving away from the initial approximation point and the approximation accuracy would deteriorate.
But if you only want to simulate a one-time permanent shock, you don’t need to worry about this.
Thank you for your response. I have another question. In some business cycle models that deal with technology shocks, the technology shock is assumed to be non-stationary, and through this assumption, the model captures the permanent effects of technology shocks. In these cases, detrending is applied before log-linearization. In contrast, other business cycle models assume that technology shocks are stationary and do not involve a separate detrending process. Is the difference between these approaches simply a matter of distinguishing between permanent and temporary technology shocks? However, even when detrending is applied in the first case, it does not explicitly differentiate between permanent and temporary technology shocks. In that case, what is the purpose of treating the shocks as non-stationary and applying detrending? Especially, when trying to include an IST shock in a CES production function, a balanced growth path cannot be achieved, so the IST shock cannot be treated as non-stationary, and therefore it cannot be used in the data to estimate the relative price of capital. If, instead, we assume the IST shock to be stationary in a CES production function and estimate it using the cycle of the relative capital price, is there any difference in the IST shock compared to the first case?
It’s more a matter of whether you want to explicitly model trend shocks and how they affect business cycles. After all, a permanent technology shock will e.g. have an impact on consumption growth.
The detrending with unit root shocks is not necessary for simulations. You can simulate permanent shocks. Things are different if there is a drift. In that case, you need to detrend the model for an accurate approximation to take place. Similarly, if you want to estimate the model, you generally need to detrend.
You are right that permanent IST shocks with a CES production function imply that a BGP generally does not exist. The problem is how to deal with that. You can of course assume only temporary shocks. The big question is whether that is actually compatible with the data.