On the idea of ex ante and ex post normalization of biproportional methods

Abstract.

Biproportional methods project a matrix A to give it the column and row sums of another matrix; the result is R A S, where R and S are diagonal matrices. As R and S are not identified, one must normalize them, even after computing, that is, ex post. This article starts from the idea developed in de Mesnard (2002) – any normalization amounts to put constraints on Lagrange multipliers, even when it is based on an economic reasoning, – to show that it is impossible to analytically derive the normalized solution at optimum. Convergence must be proved when normalization is applied at each step on the path to equilibrium. To summarize, normalization is impossible ex ante, what removes the possibility of having a certain control on it. It is also indicated that negativity is not a problem.