zero standard deviation for welfare with cost-push shock and efficient steady state

Optimal Policy
1

Dear Dynare Forum,

I have a question regarding a result I repeatedly encounter when studying optimal monetary policy under a specific scenario.

When I solve the Ramsey problem in a non-linear model with an efficient steady state and introduce a cost-push shock, I find that the standard deviation of the approximated theoretical moments of the utility function (and therefore of welfare) is always zero—regardless of the size or persistence of the shock. Interestingly, this seems to occur only in the case of cost-push shocks with an efficient steady state (i.e., under an optimal firm subsidy). When I instead consider other types of shocks (such as TFP or preference shocks) or a cost-push shock without the optimal subsidy, I do not obtain the same outcome.

Is this a standard result? If so, could you please provide some intuition as to why this occurs?

To illustrate, I attach below a standard OMP code (from Born & Pfeifer, 2018) where I introduce a cost-push shock in the form of a perturbation “Xi” to the marginal cost “MC”. However, I observe the same dynamic in other, simpler codes as well.

Thank you very much for your help.

Born_Pfeifer_2018_welfare.mod (24.7 KB)

2

It looks like this is a rather general result. See the discussion in https://olivierloisel.com/monetary_economics/Chapter%202.pdf

3

Dear Prof. Pfeifer,

Thank you very much for your answer.

If I understood correctly, this is a general result because:

  • We are considering an efficient steady state,

  • The introduced shock is “nominal” (a cost-push shock),

  • We deviate from the divine coincidence case, which introduces a trade-off between inflation and output-gap stabilization.

Therefore, the solution to the Ramsey problem can always achieve the same level of welfare in each period (i.e., the unconditional welfare). This is because the policy instrument (in this case, the nominal interest rate) can be adjusted without incurring in “real” disturbances, given the efficient steady state. As a result, the variance and standard deviation of welfare (in terms of theoretical second moments) are zero.

Is this interpretation correct, or am I overlooking something?

In the model I attached earlier, note that I do not work with the standard derived quadratic loss function in inflation and the output gap, but directly with the household utility function. What I observed is that household consumption and labor exhibit the same standard deviation when solving the OMP problem, which I assume explains why the standard deviation of the utility function—and therefore of welfare—is zero.

Thank you again for your time.