Samuelson emphasized that the Bergson-Samuelson SWF is about a
single profile of preferences. For a fixed profile $U=(U_1,\ldots,U_N)$,
\begin{equation*}
W_U:(c_1,\ldots,c_N)\mapsto\mathbb{R}
\end{equation*}
whereas an Arrow SWF maps all profiles and allocations to a
preference order,
\begin{equation*}
A: (c_1,\ldots,c_N,U_1,\ldots,U_N)\mapsto\mathbb{R}
\end{equation*}
But Kemp and Ng (1976) and Parks (1976) showed that the same problem
arose for a single utility profile if the preference profile were
rich enough.
Bergson function must make interpersonal comparisons or
be dictatorial, even if allowable utility functions are restricted
to be linear affine transformations of one another.
Parks (1976)
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