Quadratic welfare loss

Dear Dynare users,
I have derived a quadratic loss function of a sligthly complicated model. I want to be sure that my algebra is correct: I have written a Dynare code to simulate the model and verify that, up to a second order, the following objects are identical:

W=logC-1/(1+phi)*N^(1+phi)+beta*W(1);               % Welfare     
W_appr=-(y^2 + eps/lambda*pi^2 + G)+beta*W_appr(1);   % Quadratic welfare that I have derived

where G is a linear function of other quadratic terms. I have simulated the model, but up to a second order, the IRF of W and W_appr are slightly different. Up to a 1st order, the IRF are identical.

In order to be sure that this is the correct approach, I have also written a code (which I attach) in order to simulate the Gali’s textbook NK model. Again, the IRF of welfare and approximated welfare are slightly different at the second order. Notice that, compared to Gali’s quadratic welfare, I have also included terms independent from policy: these terms are of course necessary to compare W and W_appr.

My question is: is this the right approach to verify in Dynare whether a quadratic welfare loss derived by hand is correct? Becuase, either this is not the correct approach, but then I really would like to understand why. Or this is the right approach but my code is wrong (maybe it is wrong the non-linear Calvo Block. Or maybe I have made some mistakes when I have included terms independent from policy in the welfare function).

Thank you so much,

Valerionk.mod (3.8 KB)

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At second order, the two should be identical. Is there a reason you approximate the objective manually?

Yes, I approximate it manually becuase I want to know the exact formula of the several coefficients of the welfare loss. In this way I can better interpret the role of every single friction in the model. But I also would like to know if my algerba is correct. So, if I have understood correctly your answer, my approach is correct: if the two objects are different it means that I made some mistakes, even in the basic textbook framework.

You might find https://github.com/JohannesPfeifer/DSGE_mod/tree/master/Born_Pfeifer_2018/Welfare helpful as it implements the nonlinear objective underlying the results of Gali’s textbook results derived from a quadratic function.

Thank you Johannes,
I have used your code to check if my non-linear Calvo block is correct. However, still I am puzzled because up to 2nd order the following two variables are different (and they are supposed to be equal):

Welf=log(C)-N^(1+phi)/(1+phi)+betta * Welf(+1);
Welf_gali=Uss-(1+phi)/2 * y^2 +a - epsilon_p/(2*lambda)*pi^2+betta * Welf_gali(+1);

where Uss is utility in steady state.

I have figured out that the problem is the behavior of log price dispersion, variable s in my code. According to Gali’s and Woodford’s books, up to 2nd order log price dispersion should be:

s2=epsilon_p/(lambda * 2)*pi^2;                                        

However, in my model “s” (log price dispersion) and “s2” (defined above) are two different objects.

Indeed, if I write my quadratic welfare loss as:

Welf_appr=Uss-(1+phi)/2*y^2 +a -s                       +betta * Welf_appr(+1);

then “Welf_appr” and “Welf” are exactly the same object, up to a 2nd order.

Still, I don’t understand why the standard relation between price dispertion and inflation does not hold here.

I attach my code, let me know if you have any idea about this problem.

Thank you so much,


nk2.mod (4.6 KB)

That’s hard to tell without going through the full derivation. Have you checked e.g. the Lombardo/Vestin (2008) paper? I found the ECB working paper version quite helpful.

That paper helps, thank you. The problem was that price dispersion is not equal to:

epsilon_p/(lambda * 2)*pi^2

It is the discounted sum of price dispersion equal to the discounted sum of the previous expression.

Thank you very much for the suggestion. Now it works.