Quadratic welfare loss

Optimal Policy
1

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)

1 Like
2

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

3

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.

4

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.

5

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,

Valerio

nk2.mod (4.6 KB)

6

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.

7

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.