Maximizing conditional welfare and correction term

Optimal Policy
1

Dear all
I would like to understand better the role of the correction term in a 2nd order stochastic simulation. Running the code with stoch_simul(order =2), the first three lines of the POLICY AND TRANSITION FUNCTIONS table look like this:

                              Welfare               c                     h

Constant -96.123756 0.850356 0.332889
(correction) -0.108595 -0.001571 -0.000444

If I understand correctly, the constant of the Welfare is what we look at if we want to maximize conditional welfare (where the condition set is the steady state) over a grid of policy parameters; this constant is given by the steady state plus 0.5*correction. However I do not understand the nature of the correction term; in particular I notice that:

a) Using the steady-state formula for Welfare:
Welfare=(c-kappaL/(1+phi)(h)^(1+phi))^(1-sig)/((1-beta)(1-sig)) and using the values of the constant for c and h, I do not get the value of the constant of Welfare.

b) If I change a policy parameter (which does not affect the deterministic steady-state) I get

                            Welfare            c                    h

Constant -96.073505 0.843540 0.332894
(correction) -0.058344 -0.008388 -0.000439

My utility function is, as usual, increasing in c (consumption) and decreasing in h (labor); Welfare is higher compared to the previous simulation; however in this second case households will consume less and work more, if look at the constant term.

I think that I do not fully get the role of the correction term and what exactly the constant is, since I cannot explain these results. What do you thing Professor Pfeifer?

Valerio

2

The point you are looking at is not the deterministic steady state, but rather what people call the “stochastic steady state”. It is the point that agents choose in the absence of shock when they take into account the possibility that shocks from the respective distribution can arise at every point in time. Due to that, it takes uncertainty (and the welfare losses/gains that might arise from this) into account. Your point a) is related to this uncertainty correction. The “constant” contains this correction, which does not directly map into c and h. If you read the Schmitt-Grohé/Uribe (2003) paper on second order approximations, the constant is

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