4.1. The
Nincome price system
Let us examine the value of
a(P,X) defined by eq. (10). As shown in § 3, there exists a vector
X_{0}³ 0, such that the ratio of aggregate material costs to income, that is
a(P,X), is independent of the price vector; and analogously that there exists a vector
P_{0}³ 0, such that
a(P,X) is independent of the vector of total production
X.
In other words, the relation between total material costs and national income expressed in prices
P_{0} will not change with any changes in the vector of production
X. For proof, choose
n > 0,
m =
r = 0 in matrix H(m,n,r)(I  A)^{1} .
If we introduce values of the parameters into eq. (23), we obtain the following price relation:
(1 +
n*)A'P_{0} =
P_{0} (25)

The price system given by equation (25) is called an
Nincome price
system. (3) Nincome
prices are proportional to material
costs. In other words, the relation
between income and material costs is the same for all prices and is
equal to the parameter
n*. The parameter
n* is uniquely defined by eq. (25).
0 £
n* £(1/l_{a})  1,
where
l_{a} is the maximum characteristic root of matrix
A. The vector of
Nincome prices,
P_{0} is the characteristic vector of matrix
A'. Note that the uniform relation of income to material costs
n* depends only on matrix
A and, therefore, cannot be influenced by changes in production vector
X. From eqs.
(25) and (26) it also follows that, at
Nincome prices,
a(P_{0},X)
= l/n*.
Here, if matrix
A is irreducible,
P_{0} is positive and uniquely defined. 