Welfare Comparison

Hi Jan78,

You can have a look at my answer at
Consumption Equivalence_Welfare analysis.

As to your second question, your measure of welfare is simply the present discounted value of flow utility

W_t=E_t\left\lbrace \sum_{i=0}^{\infty}\beta^{i}U\left(C_{t+i},N_{t+i}\right)\right\rbrace

or, to express it recursively

W_t=U\left(C_{t},N_{t}\right)+\beta E_t\left\lbrace W_{t+1} \right\rbrace.

For example, assuming a very simple utility function that includes labor supply, you have that

W_t=E_t\left\lbrace \sum_{i=0}^{\infty}\beta^{i}\left(\log C_{t+i}-\chi\log N_{t+i}\right)\right\rbrace.

Using my notation in the link above, you can write unconditional welfare in benchmark economy in a recursive manner as

E\left[W_{t}^{p}\left(\xi\right)\right]=E\left\lbrace \left(\log C_{t+i}\left(1+\xi\right)-\chi\log N_{t+i} +\beta E_t\left\lbrace W^{p}_{t+1} \left(\xi\right) \right\rbrace \right)\right\rbrace

which shows you that \xi retains a “consumption equivalent” interpretation even if you have other terms in the utility function :slight_smile:

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