Smets and Wouters equations in nonlinear form

Dear colleagues,
I was wondering whether anyone has ever written a Dynare code of Smets and Wouters (2007) containing the non-linear linear equations, rather than the first order approximations? If you have heard of something like this, please let me know!
Best,
Ansgar

Dear Ansgar,
as far as I know that is not possible. The reason is the Calvo setup with price and wage markup shocks. Because of the time-varying markup in the exponent, there is no way to write the infinite sum coming from the wage and price setting FOCs in a recursive form. But you cannot enter an infinite sum to the computer. Smets/Wouters get around this by linearizing the model, which again provides a nice recursive representation that can be entered into the computer. In theory, it could be possible to get a recursive higher order representation with pencil and paper, but I am not aware of anyone who has done that. That suggests it’s (close to) impossible.

Dear Johannes,
thank you very much, yes, that sounds very familiar now that you mention it. I suppose there only two ways around this then
a.) Construct a “wedge” type shock (in case of the wage markup say a wage income tax) which up to first order has identical effects to a markup shock
b.) Move away from Calvo towards price and wage adjustment costs. Although I am sure I have heard people saying that with a Rotemberg type setup, the costs of inflatiojn are smaller since there is no price dispersion…

Best,
Ansgar

P.S.: I hope you are enjoying the wonderful weather in Germany…

a) If you do that, what’s the point in going to higher order in the first place?
b) Price adjustment costs in the Calvo setup are 0 up to first order as they are in the Rotemberg setup if there is no steady state inflation (or perfect indexing in steady state). Just have a look at Ascari’s work. But you could most probably get something similar in a Rotemberg setup with non-zero price adjustment costs in steady state.

Dear Johannes
Just found this old post of yours online; the comment that when there are price elasticity or markup shocks, we cannot write down the recursive equations needed to write down the Calvo infinite sums. Why exactly is this the case? If the markup shocks enter the infinite sums, would it not be enough to alter the definition of the auxiliary variables?

Thanks
Reuben

Yes, you would need to alter the definition of the auxiliary variables, but they will not be recursive anymore. This was pointed out by Andreasen 2012 (http://dx.doi.org/10.1016/j.euroecorev.2012.09.006) on page 1659

Smets and Wouters(2007) document the importance of real supply shocks specified as shocks to firms’ markup rates. However, with Calvo-price contracts, these markup shocks prevent an exact recursivere presentation of the equilibrium conditions which is needed for a non-linear approximation to our model.