Smets and Wouters equations in nonlinear form

[Ansgar]

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

[jpfeifer]

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.

[Ansgar]

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...

[jpfeifer]

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.