Measures of welfare gains/losses

Yes indeed, my ricardian and non ricardian agents have a different level of base consumption. I get your point now. Should I normalize base consumption at my SS to 1 ? So then, I divide my consumption results for every period by the level of SS consumption ?

Thanks @jpfeifer

I am not saying they need to have the same base level. All I am saying is that you need to specify/explain the concrete metric you want to employ. That may be utils or absolute or relative consumption.

I’m not sure what to answer to that. I would go for relative consumption as I’m comparing between two dates the gains @jpfeifer

That sounds reasonable.

So now, how do I perform that ? Just by comparing the gain of relative consumption between two dates for my two agents ? @jpfeifer

It’s more complicated as we are talking about welfare, so the path of utility matters. Usually, you do something like

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Thank you @jpfeifer

I find it quite difficult to replicate your files as I do not compare two policies, but the gain of welfare between two dates after a shock.

That distinction does not matter and you should not use my files. Rather, it’s about the concept. You should know the level of welfare and of consumption in steady state. You also know how welfare changes between two dates. That should allow for backing out the consumption equivalent.

Thanks @jpfeifer . I think I got it.

SS End of period End of period
Cr 30772,1339
Cnr 27210,7542
Wr 1,988598784 1,988632544 30772,65628 consumpt eq
Wnr 1,987875615 1,987883907 27210,86772 consumpt eq

Here you have an example. You have consumption for the two types of households and welfare at SS. Then I have Welfare for the end of period computed from the utility function, and welfare in consumption units computed by a simply rull of three. Is it ok?

One more question: for the computation of welfare, should I include beta or just use the CRRA specification without the parameter beta and the sum ?

Thanls

The consumption equivalent would be the percentage of steady state consumption to achieve that utility level.

If 30772,65628 steady state consumption would generate a welfare of 1,988632544 as at the end of the period, then you would have a consumption equivalent of (30772.65628-30772.1339)/30772.1339*100= 1.6976e-03 percent.
If you do it in percent, then the beta and the sum do not matter.

Thank you @jpfeifer

  1. But is my computation of 30772,65628 is correct in the first place ? The welfare at the end of period, according to the CRRA utility, is 1,988632544. To get welfare in consumption equivalent I did the following calculation: (1,988632544*30772,1339)/1,988598784 = 30772,65628

  2. If the computation is okay, I have, as you computed, a welfare gain of 1,6976e-03 % in the end of period compared to the SS. I find this value rather low as the household consumed 30772,1339 at the SS and at the end of period it consumed 30955,1884. So there is a gain of more than 200 in terms of consumption

  3. How do I know which household gains the most in terms of welfare? To achieve the welfare at the end of period, the ricardian has a SS consumption equivalent of 0,0017% and the non ricardian 0,0004%. Does that mean that the ricardian gains more in terms of welfare ?

Best regards

Good morning @jpfeifer , would you have an answer for question 3? Thank you! Best regards

  1. Usually, this is wrong. You seem to be comparing utilities directly instead of the consumption levels giving rise to this utility. In W_{model}=W((1-\lambda)c_{ss}) you want to solve for lambda
  2. Welfare gains in most models are tiny. See e.g. Lucas (2003) presidential address to the AEA
  3. Welfare usually has arbitrary units as discussed above. If the metric you use is the share of steady state consumption, then the answer would be yes.

Thank you @jpfeifer

But with your formula, I end up with this:

Capture d’écran 2021-02-16 à 18.20.44

Can I compare two lambda, one for the ricardian and the other one for the non-ricardian? Because their SS consumption is not the same.

You are doing this for utility, but you need to compare welfare. In that case, the formula does not work due to an infinite sum.

Oh right !

So what’s wrong with my computation ? I thought I had to express the utility of the end of period in consumption SS equivalent.

The utility at SS corresponds to a value of 1,988598784, consumption at SS to a value of 30772,1339, and utility at the end of period to 1,988632544. In my point of view, expressing the end of period utility in terms of SS consumption give rise to this calculation: (1,988632544*30772,1339)/1,988598784 = 30772,65628

And then comparing the two consumption SS equivalent gives me the gain of welfare between two periods: (30772.65628-30772.1339)/30772.1339*100= 1.6976e-03 percent.

I still don’t get my mistake @jpfeifer

Welfare aka lifetime utility is a more complicated object as the time path of consumption obviously matters. Risk-averse agents prefer smooth consumption paths. For that reason, you cannot use the period utility for any comparison.

Oh okay. Well I think I’m just going to talk about the increase of consumption I have and not talk about welfare as the computation seems to be tricky.

Thank you for your patience @jpfeifer

How can Cr and Cnr be different in steady-state? It means the utility function specified for agent r and agent nr are different?

Cr 30772,1339
Cnr 27210,7542

These conditions

\frac{U_{N_{r, t}}(i)}{U_{C_{r, t}}(i)}=-\frac{W_{t}}{P_{t}}
\frac{U_{N_{nr, t}}(i)}{U_{C_{nr, t}}(i)}=-\frac{W_{t}}{P_{t}}

and the fact that both types of agents receive the same wage does not mean that Cr = Cnr in steady-state?

Some papers also impose Cr=Cnr with statements like, “The steady-state consumption and hours worked are assumed to be the same among both types of households so that, for a steady-state share of liquidity-constrained households of one-half, they are equal to their aggregate counterparts”. Good practice? Thanks!!

In the end, it is your model. Normalizations like that are common.

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