oo_.planner_objective_value

[Ippei]

Hi,

Could you let me know what "oo_.planner_objective_value" reports in "ramsey_policy"?
It is not the unconditional (theoretical) mean of the welfare.
Is it steady state of welfare plus .5*Delta^2 in the solution of the second order approximation?
Is it steady state of welfare plus .5*Delta^2 plus higher order terms (namely parameter multiplied with variances) in the solution of the second order approximation?

Thanks in advance,
Ippei

[jpfeifer]

No, it is the conditional welfare.
The formula is (evaluate_objective_function.m)

Code: Select all

planner_objective_value = Wbar+Wy*yhat+W_u*u+W_yu*yhat*u ...
    + 0.5*(W_yy*yhat^2 + Wuu_u^2+W_ss);


That is, it is a full second order expansion. Note that the concept of welfare is

W_t=U_t+beta*E_t (W_t+1)

so that today's endogenous and exogenous states are contained in the information set at time t and affect U_t. You can specify these initial conditions with histval.

[Ippei]

Dear Johannes,

Many thanks for your swift reply.

I would like to compute the unconditional welfare using "ramsey_policy," which is trivial with the previous dynare code by Levin and Lopez-Salido. This is because W(t)=U(t)+beta*W(t+1) must be added in the system of equation. Adding W(t)=U(t)+beta*W(t+1) in "ramsey_policy" causes a problem since this becomes a redundant constraint.

Given the solution of the second order approximation:
Y(t)=.5*DELTA^2+AA*Y(t-1)+BB*u(t)+.5*CC*(Y(t-1)xY(t-1))+.5*DD*(u(t)xu(t))+EE*(Y(t-1)xu(t)),
where all variables are deviations from the Ramsey steady states,
is the only way to compute the unconditional welfare that
E[Y(t)]=.5*DELTA^2+.5*CC*var(Y)+.5*DD*var(u)+EE*covar(Y(t-1)Xu(t)))?

On the other hand, just to make is sure, what oo_.planeer_objective_value (the conditional welfare) with initial values being at the Ramsey steady state reports
E(t)[Y(t)]=.5*DELTA^2+BB*u(t)+.5*DD*(u(t)xu(t)).

Do I understand the procedures in dynare correctly? Also, if there is any easy way to compute the unconditional welfare, I would like to know this.

Sorry for posting a question again after your clear answer.

Cheers,
Ippei

[jpfeifer]

Sorry, but I will have to take a deeper look into this, which might take some time.

[MichelJuillard]

Hi Ippei,

in your previous post, there shouldn't be any u(t) because u(t) is observed at the beginning of the period and belongs by assumption to the information set of E_t()

If you want the unconditional welfare (a tricky concept), you could do the following
1) include utility = u(y_t), the period utility function, among the variable/equation of the model. This doesn't generate the same problem as adding the definition of welfare.
2) then you can compute E(W_t) = E(u(y(t)) + beta*E(W_{t+1}) as
E(W_t)= E(u(y_t))/(1-beta) because E() being unconditional expectation, E(W_t) = E(W_{t+1}) and you can read E(u(y_t)) as the second order approximation to the mean of u
3) In order to get the second order approximation of the model under Ramsey, you must indicate
ramsey_model(planner_discount_factor=....);
stoch_simul(order=2);

Hope it helps

Michel

[Ippei]

Dear Michel and Johannes,

Many thanks for your replies. I thought that I have not received answers. I really appreciate the way to compute unconditional welfare in the current Ramsey routine.

Cheers,
Ippei