How do you show that the utility function
$$U = (X + A)^p (Y + B)^q$$
gives the same result than
$$ U= p \log (X+A) + q \log (Y+B) $$
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Sign up to join this communityHow do you show that the utility function
$$U = (X + A)^p (Y + B)^q$$
gives the same result than
$$ U= p \log (X+A) + q \log (Y+B) $$
The "proof" that you're looking for consists of a few steps.
The second utility function is a log transformation of the first;
Log transformations are order preserving (monotone);
This #2 means that the each bundle is ranked the same from most preferred to least preferred;
Since preferences are ordinal and utility functions are representations of ordinal preferences these two utility functions are the same because they represent the same set of preferences.
Hope that this helps.