Consumption Equivalence_Welfare analysis

Optimal Policy
23

I am not sure I am able to follow. It seems you are using unconditional welfare for the comparison, but IRFs are about conditioning what happens after a particular shock, starting from a particular point of the state space.

24

@jpfeifer Oh yes my bad. I am still puzzled with the estimation of welfare gain/loss from the estimation of CE conditional welfare for non logged Utilities. To change the above computation to a conditional welfare (and match the IRFs) will it possible to just replace Welfare by Utility in the below general expression ?
W_pos=strmatch(‘Welf’,M_.endo_names,‘exact’);
W_cond=oo_.dr.ys(W_pos)+0.5*oo_.dr.ghs2(oo_.dr.inv_order_var(W_pos))

25

Yes, that should be correct. See Unconditional versus conditional Welfare measure

27

Conditional welfare tells you about what will happen on average in the future, starting at some point. That involves averaging over positive and negative shocks. But your IRFs are for just a positive shock.

28

Thanks a lot

29

Hi all,
Thanks all for the interesting topic.

I wonder if I can use the same way to obtain welfare gain with the log utility with habit formation. My guess is yes but I am not sure. Thank you so much.

30

Usually, yes, because the comparison is relative to steady state, where the habits boil down to a constant factor.

31

Thank you so much,

But my concern is about consumption equivalence if the CE param only applies for C_t then we have a problem. Am I correct?

32

What exactly is the concept you are using?

33

Oh, I just realized I used the wrong concept. I am so sorry for that

I am interested in welfare gain in terms of consumption equivalent (CE). Thanks a lot.

34

Hi,
Can everyone know if I can treat a log utility with habit formation similar to log utility to pindown consumption equivalence welfare gain?
Thank a lot

35

You need to show us your equations.

36

Hi,

Many thanks.

I am talking about this kind of utility U_t= ln(C_t-hhC_{t-1}) . I guess I cannot extract CE in the traditional way.

37

You need to provide a lot more details on the problem you are facing. You cannot expect people to fill all the gaps.

38

I am sorry for that.

I want to do optimal policy using welfare with utility in the following form. W_t=ln(C_t−hhC_{t−1}). I want to extract the consumption equivalence in terms of welfare gain. For log utility, it is straightforward:

E[W^p_t(\zeta)] = \frac{1}{1-\beta}ln(1+\zeta) + E[W^p_t] \\ \frac{1}{1-\beta}ln(1+\zeta) + E[W^p_t]+E[W^{ap}_t(\zeta)]
I wonder if the same trick can be applied to the W_t=ln(C_t−hhC_{t−1}).

Thank you so much.

39

You did not explain your notation.

40

I am so sorry,
I made a mistake there. Let’s say I want to evaluate the consumption equivalence of welfare gain E[W^p_t(\zeta)]. So, from what I understand from this post, \zeta measures the maximum fraction of consumption that the agent would be willing to forgo in an economy p to join the economy in which ap is active E[W^{ap}_t].
For a log utility, it is straightforward:
E[W^p_t(\zeta)] =\sum^{\infty}_{i=0} \beta^i ln((1+\zeta)C_t) =E[W^{ap}_t]

E[W^p_t(\zeta)] = \frac{1}{1-\beta}ln(1+\zeta) + E[W^p_t]=E[W^{ap}_t]

\zeta= exp((1-\beta)(E[W^{ap}_t] -E[W^p_t]))-1

And we can easily get \zeta.

I wonder if the same trick can be applied to the W_t=ln(C_t−hhC_{t−1}) to get \zeta as I am not sure we can put \zeta also in front of C_{t-1}

I am sorry for not expressing it in a better way. Thanks a lot

41

No, that can generally not be done analytically, because you are asking for a fraction of stochastically changing consumption in each period, not a fraction of steady state consumption.

42

See section 5 of the paper:

El-Khalifi, A., Ouakil, H., & Torres, J. L. (2022). Efficiency and Welfare Effects of Fiscal Policy in Emerging Economies: The Case of Morocco.

You will find an application for measuring social welfare costs, and the method of calculation.