
Values of
a(P,X) ,b(P,X), and
g(P,
X),
defined in § 2, depend on the price vector
P
and the production vector
X. From definitions (4)(12), it follows that, for any arbitrary, nonzero constants h, w,

Let us now ask whether it is possible to find a vector of production
X, such that the values of
a,
b, and
g
or some linear combinations of them are independent of prices; and so find a price vector P, such that the values of
a,
b,
and g or linear combinations of them are independent of
X. 
From eq. (1)
X_{0} = (I 
A)^{1}
Y_{0,
}
and
H(m,n,r)X_{0} = l_{0}Y_{0}
.

Therefore, for any three nonnegative parameters m,n,r of which at least one is greater than zero, there exists a vector of total production X_{0}³ 0 such that eq. (19) holds independently of the price system chosen, provided that P' Y_{0} > 0. Note that if the matrix H(m,n,r)(IA)^{1 }is irreducible, then the vector X_{0 } is positive and uniquely determined.

For each group of three nonnegative parameters
m,n,r with at least one greater than zero, there exists a nonnegative price vector, such that eq. (21) holds independently of the production vector, provided that
P'_{0}Y>0.
_{}
m* = l_{0}^{1}m,
n* = l_{0}^{1}n,
r* = l_{0}^{1}r. (22) From eq. (20), then _{} P_{0} = (1 + n*)A'P_{0} + r*B'P_{0} + m*C'P_{0} (23)

Thus, prices P_{0} cover costs of materials and form income as a sum of three components: the first is proportional to cost of materials, the second to capital, and the third to wages. Prices defined by eq. (23) are therefore called threechannel prices.

where
l_{a} is the maximum characteristic root of matrix
A;

The fact that the two chosen parameters lie within the limits given by eq. (24) is not, however, a sufficient condition for the nonnegativity of the third parameter. We shall return to this problem later.
is unity. 



