# Normalization of production function

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?

• Yes, this can be done wlg if you reparameterize $A$. Your function is homogeneous in $(A,a_i)$. Sep 30 at 5:30
• @Bertrand Please post answers as answers, and elaborate on this; it may not be clear to some why reparameterizing $A$ is necessary. Sep 30 at 6:10
• @Giskard: I was reluctant as it looks a bit like an exercise... Sep 30 at 6:30

## 1 Answer

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 $$F(x; A, a, \rho, \nu) = G(x; B, b, \rho, \nu).$$