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?
and
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…