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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28 
The characteristic function can in fact be used to derive an 
expression for R,- As the reasoning is somewhat difficult it 
may be helpful to consider two relatively simple cases before 
going on to study the general case. 
Case 1. 
a) Assume in the first place that 
(117-3) R=R (A, B, ..., O)=R(X, Y, .... 2) 
where R is real national income, A, B,......,C represent diff- 
erent outputs (including equipment) and X., Y,........Z the 
factors of production used. 
Assuming that R (A, B,......,C) is of first-order homogen- 
eity, and that the function R (X, Y, ........ Z) is of E-order 
homogeneity, then 
(117-4) R (AA, AB, !.., XC)= AR (A, B, …, C) 
(117-5) R (uX, uY, … uZ)=uR (X. Y. … 7). 
Applying the Euler theorem, this gives 
(117-6) 
(117-7) 
> B R,=R 
YV R'v=F R _ 
[11] Allais - pag. 44