A question about welfare metrics?

Dear Professor Pfeifer,

I came across a problem when using DSGE model to do the welfare metrics. I use the compensation variation method. The utility form of household consumption is the one with consumer habit: log[C(t)-b*C(t-1)].

I know how to get lambda(which represents the variation parameter) when the utility form is as easy as logC(t), but what should I do when it contains consumer habit ? The variation is log[ (1+lambda)C(t)-bC(t-1) ] or log[(1+lambda)( C(t)-bC(t-1) ) ] ? It’s easy to get lambda if the latter one is right. But if the former one is right, how to get the analytical formula of lambda?

Any reply will be appreciated. Thank you very much!

For the former one, there is no analytical way. You would need to numerically solve it. But the second one seems more appropriate. You are asking for the permanent consumption compensating variation. That should also affect the lagged habit term, because consumption was also lower in the previous period.

Dear Professor Pfeifer,

Thank you so much for your reply ! I get it .