A problem with welfare metrics

Dear Professor Pfeifer,

I want to measure the welfare variation using consumption equivalent. I know how to do this when there is only one type of household. While in the model with heterogenous household as patient and impatient, I am confused on how to deal with the consumption compensation parameter \lambda.

e.g. The two untility function of households with compensation parameters are:
{{E}_{t}}\sum\limits_{k=0}^{\infty }{\beta _{P}^{k}\left[ \log \left( 1+{{\lambda }^{P}} \right)c_{t+k}^{P}-\frac{1}{1+{{\phi }^{P}}}{{\left( n_{t+k}^{P} \right)}^{1+{{\phi }^{P}}}} \right]}
{{E}_{t}}\sum\limits_{k=0}^{\infty }{\beta _{I}^{k}\left[ \log \left( 1+{{\lambda }^{I}} \right)c_{t+k}^{I}-\frac{1}{1+{{\phi }^{I}}}{{\left( n_{t+k}^{I} \right)}^{1+{{\phi }^{I}}}} \right]}

There are two compensation parameters(\lambda^{P} and \lambda^{I}) now. I can calculate them respectively, while how should I get an aggregate compensation parameter? \lambda^{P}+\lambda^{I} seems not make any sense?

Thank your for your time.

In that case, you need to decide on the relevant counterfactual. There is no general answer. Do you want to make both agents indifferent? In that case, you would need two parameters. Alternatively, if there is a social welfare function, it may tell you the relative weights.

Dear Professor Pfeifer,

I think I may set the social welfare function as bellow like in many papers:
{{W}_{t}}=\left( 1-{{\beta }_{P}} \right)W_{t}^{P}+\left( 1-{{\beta }_{I}} \right)W_{t}^{I}

But I still have no idea how should I calculate the aggregate compensation parameter?

Looking forward to your reply and thanks again for your time.

Again, the answer will depend on the question you are asking. One single parameter will generally not be sufficient to reach two targets. Therefore, one consumption equivalent will not be sufficient to make both agents exactly indifferent. You can use one parameter to make overall welfare W_t the same, but that will imply trading off welfare between agents.