Can I compute the diference between welfare and welfare steady state?

[abcsk]

Dear professor,
I am computing the optimal monetary policy based on welfare maximization. I have two policy regimes. welfare_I and welfare_II denote the welfare of two models. And welfare_steady_I and welfare_steady_II denote the steady-state welfare values of two models.
I define
Dwelfare_I =welfare_I - welfare_steady_I;
Dwelfare_II =welfare_II - welfare_steady_II;

 Then I define

delta = (Dwelfare_II - Dwelfare_I) / Dwelfare_I;

My problem is, can ‘delta’ measure the welfare improvement effect of regime II over regime I?

[jpfeifer]

You can, but the number does not have a good interpretation. That is why people measure welfare improvements in consumption-equivalent units.

[abcsk]

Many Thanks!!! Why did you withdrawn the reply? Does it wrong?

[HelloDynare]

I would like to connect this topic with the one that I posted one year ago, and then ask some naive questions.

If “welfare_steady_I”\neq “welfare_steady_II”, how to compute the consumption equivalence?

@jpfeifer

Conditional welfare is something used when you ask whether a change in a policy regime is beneficial. In that case, you would start at or somewhere close to the steady state of the old regime and this is the starting point for the implementation of the new policy.

and

@jpfeifer

Unconditional welfare in contrast is not concerned with transition behavior. Policies happen in two different economies that have been running for a long time. Now you compare them.

Since steady states under two alternative regimes do not coincide, it seems that the unconditional welfare is more rational here?

Suppose we have a separable and additive CRRA utility form. Does it mean:
W^a_0-W^b_0 = \mathbb{E}_0 \displaystyle\sum_{t=0}^{\infty} \beta^t\left[ U\left(\left(1+\Delta^{unc}\right)C^b_{t}, N^a_{t}\right) - U\left(C^b_{t}, N^b_{t}\right) \right]

= \mathbb{E}_0 \displaystyle \sum_{t=0}^{\infty} \beta^t \left[ \frac{ \left(\left(1+\Delta^{unc}\right)C^{b}_{t}\right)^{1-\sigma} -{C^{b}_{t}}^{1-\sigma} }{1-\sigma} - \frac{{N^a_t}^{1-\sigma}-{N^b_t}^{1-\sigma}}{1-\sigma}\right].

\Rightarrow \Delta^{unc} = \left\{\left[ \left( W^a_0-W^b_0 \right)\left(1-\sigma\right)\left(1-\beta\right) +\left( {N^a}^{1-\sigma}-{N^b}^{1-\sigma} \right)\right] {C^b}^{\sigma-1}+1\right\}^{\frac{1}{1-\sigma}}-1

But, I never saw this sort of expression, containing labor, in the literature. It looks ugly…

[jpfeifer]

How did you eliminate the expected value if you are not in the steady state?