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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 different
 outputs (including equipment) and X., Y,........Z the
factors of production used.
Assuming that R (A, B,......,C) is of first-order homogeneity,
 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