Can we normalize a production function in the same way we can normalize a utility function? For example, consider the CES function $$ F(x; A, a, \rho, \nu) = A \left( \sum_{i=1}^n a_i \, x_i^\rho \right)^{\nu/\rho} $$ where $x$ and $a$ are vectors of length $n$; and $A$, $\rho$, and $\nu$ are scalars. $x$ represents inputs and $A, a, \rho, \nu$ are parameters. If $F$ is a utility function, then we can make the normalization $\sum_{i=1}^n a_i = 1$, while keeping the same preferences. Can we do the same if $F$ is a production function or in that case the normalization matters?
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$\begingroup$ Yes, this can be done wlg if you reparameterize $A$. Your function is homogeneous in $(A,a_i)$. $\endgroup$– BertrandCommented Sep 30, 2021 at 5:30
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$\begingroup$ @Bertrand Please post answers as answers, and elaborate on this; it may not be clear to some why reparameterizing $A$ is necessary. $\endgroup$– GiskardCommented Sep 30, 2021 at 6:10
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1$\begingroup$ @Giskard: I was reluctant as it looks a bit like an exercise... $\endgroup$– BertrandCommented Sep 30, 2021 at 6:30
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Contrarily to utility functions, production functions are cardinal, and so they are not arbitrarily normalized. In the case of your CES production function, however, if we consider
$$ F(x; A, a, \rho, \nu) = A \left( \sum_{i=1}^n a_i \, x_i^\rho \right)^{\nu/\rho}, $$ with $\sum_{i=1}^n a_i \neq 1$ it is easy via a simple reparameterization of $(A,a_i)$ to rewrite it as $$ G(x; B, b, \rho, \nu) = B \left( \sum_{i=1}^n b_i \, x_i^\rho \right)^{\nu/\rho}, $$ with $\sum_{i=1}^n b_i = 1,$ and such that for any $x$ we have $$ F(x; A, a, \rho, \nu) = G(x; B, b, \rho, \nu). $$
